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(I accidentally deleted the original thread “Peikoff’s Dissertation” this morning. I’ll repost it now from my word-processing file of it. I’ll put the date of the original posts, each one, at the beginning of those posts reinstated now. I’ll leave the text of these re-posts as it was originally, without any updates of things on which I’ve since changed my mind.)

17 March 2017

The Status of the Law of Contradiction in Classical Logical Ontologism

Leonard Peikoff – Ph.D. Dissertation (NYU 1964)

There are no true contradictions, and there cannot be any. That is the law of contradiction, or principle of noncontradiction (PNC) as I shall call it. There is nothing and can be nothing that is both A and not-A at the same time and in the same respect. The last three decades, Graham Priest and others have argued specific exceptions to the law. These exceptions seem to be such that from them no possibility of observable, concrete true contradictions can be licensed. The debate over these circumscribed candidates for true contradictions continues. I shall in this study fence them off, without disposition, from our still very wide purview of PNC. There are reasons advanced in favor of these specific alleged exceptions to PNC, I should stress. It is not argued that we should just say true or false as we please of the contradiction reached in these cases. These are not situations for conventions such as the side of the road on which to regularly drive. (See Priest, Beall, and Armour-Garb 2004.)

Under the term classical in his title, Peikoff includes not only the ancient, but the medieval and early modern. By logical ontologism, he means the view that laws of logic and other necessary truths are expressive of facts, expressive of relationships existing in Being as such. Peikoff delineates the alternative ways in which that general view of PNC has been elaborated in various classical accounts of how one can come to know PNC as a necessary truth and what the various positions on that issue imply in an affirmation that PNC is a law issuing from reality. The alternative positions within the ontology-based logical tradition stand on alternative views on how we can come to know self-evident truths and on the relation of PNC to the empirical world, which latter implicates alternative views on the status of essences and universals.

Opposed to the classical logical ontologists are purportedly conventionalist approaches to logical truth in the first half of the twentieth century. Peikoff argues that infirmities in all the varieties of classical logical ontologism open the option of such conventionalism.

Firstly, Peikoff examines the views of Plato (427–347 B.C.E.) in their import for an explanation of our knowledge of PNC and its self-evident character and for the bases of PNC in reality. Peikoff then examines these imports in the views of Aristotle as well as in the views of the intellectual descendents of Plato and Aristotle to the time of Kant.

Peikoff cites a number of passages in which Plato invokes varieties of PNC as a general principle of the character of things that must always be acknowledged in reasoning. “The same thing will not be willing to do or undergo opposites in the same part of itself, in relation to the same thing, at the same time” (Republic 436b). “Do you suppose it possible for any existing thing not to be what it is? / Heavens no, not I” (Euthydemus 293b). To citations given by Peikoff, I add Republic 534d where Plato speaks of some persons “as irrational as incommensurable lines.” The incommensurability of the length of the diagonal of a square to the length of its side had been discovered by the time of Plato, and its proof is by showing that on assumption of commensurability of those lines there follows the contradiction that whatever number of integral units composing the diagonal, the number is both even and odd.

Peikoff rightly stresses that for Plato the perfect Forms are radically different from their empirical namesakes. Under the latter acquaintance, our knowing the Forms, so far as we do, is from memory of our full knowing of them in our existence before this life of perception, according to Plato:

“Consider, he said, whether this is the case: we say that there is something that is equal. I do not mean a stick equal to a stick or a stone to a stone, or anything of that kind, but something else beyond all these, the Equal itself. Shall we say that exists or not? / . . . Most definitely / . . . / Whence have we acquired the knowledge of it? . . . Do not equal stones and sticks sometimes, while remaining the same, appear to one to be equal and to another to be unequal – Certainly they do. / But what of the equals themselves? Have they ever appeared unequal to you, or Equality to be Inequality? / Never, Socrates / . . . / Whenever someone, on seeing something, realizes that that which he now sees wants to be like some other reality but falls short and cannot be like that other since it is inferior, do we agree that the one who thinks this must have prior knowledge of that to which he says it is like, but differently so? / Definitely. / . . . / We must then possess knowledge of the Equal before that time when we first saw the equal objects and realized that all these objects strive to be like the Equal but are deficient in this” (Phaedra 74).

Perceptibly equal things are deficient in that they can appear unequal in some occasions of perception. The Form Equal by contrast is always just that. Perceptibles “no more are than are not what we call them” (Rep. 479b).

Plato does not clearly isolate PNC, but he was getting onto an ontological basis for it, so far as he did grasp PNC, by his characterizing what I should call his faux contradictions of empirical objects—faux because he fails to give square reality to situational and temporal determinates of objects and to our contexts of thought and speech about objects—as both being and not being, which is to say, deficient in being. It is fair enough to say, as Peikoff concludes, that for Plato PNC has the same standing in ontology and in our knowledge as such Forms as Being, Same, Other, Equal, and Inequal. Additional support, I notice, for that standing of PNC in Plato would obtain had Plato called out Identity as a Form, where Identity means what was said above at Euthd. 393b: an existing thing must be what it is. As later thinkers would observe, Identity in that sense entails PNC.

Peikoff places Plato at the head of a sequence of philosophers who held PNC to be not learned from scratch by our experience in this world. They hold the principle to be in some sense innate and to be based on realities independent of the world we experience by the senses. In the innate-PNC sequence, Peikoff places later Stoicism (see Crivelli 2009, 393–94), Neoplatonism, early Christianity, Cambridge Platonism, and Continental Rationalism. Nearly all of these, I should note, are in a very different intellectual situation than Plato’s in that they have, directly or indirectly, Aristotle’s development of logic. The latter two certainly had as well his Posterior Analytics and Metaphysics. They had thereby Aristotle’s various formulations and accounts of PNC. They stand on the shoulders of both Plato (and Neoplatonism) and Aristotle, with innate-PNC being one of their leanings toward Plato along a line of difference with Aristotle. They had as well, unlike Plato or Aristotle, Euclid’s Elements, further mathematics beyond Euclid, and further developments in logic.

By the time of Republic, Plato had evidently abandoned his view that we recognize Forms in our present life because we knew them well in a previous life free of the perceptual and variation spoilers of being (Tait 2005, 179). The recollection from a previous life is no longer mentioned. It remains for Plato that the Forms, such as are engaged in geometry, are accessed only by intellect, and not to be found in sensory experience nor abstracted from sensory experience. Peikoff was aware that some scholars had begun to question whether Plato had held on to his early express view that the realm of Forms was a world in which we had lived in a previous life and from which we now have some recollection of our previous knowing. Peikoff took Plato’s view as uniform on the recollection doctrine we saw in Phaedra. I’m persuaded to the contrary view. Peikoff rightly points out that through much of the history of philosophy the recollection view and the other-world-of-Forms view had been taken for Plato’s view, and Plato’s influence, pro or con, was under that picture. I think, however, that the separateness of a purely intelligible realm of Forms, a realm not also a prior world of life, Forms separate from empirical classes participating in them, is enough for saying Plato heads a line in which knowledge of necessary truths such as in geometry or in the rules of right reasoning (importantly PNC), even if their elicitation is by sensory experience, must be innate. That much, given Peikoff’s analysis of the significant senses of innate, is enough for sharp contrast with Aristotle and his line, and the dominance of the Good over all other Forms suffices, in a foggy way, for their normativity in the empirical world (Rep. 504d–11e, 533b-d; Philebus 20b–22e, 55d–60c, 64c–67a; Denyer 2007, 306–8).

I mentioned the great difference, in Plato’s view, between the perfect Forms and their empirical namesakes. The bed one sleeps in is physically dependent on its materials and construction, but the bed constructed depends on the Idea or Form Bed, and the particular constructed bed is ontologically deficient in being when compared to the invariant full-being Bed, the Form on which the particular constructed bed’s being and name depends (Rep. 596–97).

It is the rational, best part of the soul that measures and calculates, helping to rectify illusions in perceptual experience and to bring us nearer truth of being (Rep. 602c–603a). In geometry we employ diagrams, but our arguments and concern are for the Forms of these figures, not the particular constructed, material figures (Rep. 510b–511a; on the “mathematical intermediates” controversy, see Denyer 2007, 304–5; Tait 2002, 183–85). Even higher than our rational capability for geometry is our rational capability for proceeding from Forms to Form-Form relations to the first principle of all Being—and the necessary ultimate spring and harmony of all knowing—which for Plato is a Form, the Good. This purportedly highest process of knowing is called dialectic, a notch above thought even in geometry (Rep. 510b–511e; further, Denyer 2007, 306–8).

Reviel Netz concludes “Greek mathematical form emerged in the period roughly corresponding to Plato’s lifetime” (1999, 311). He reports Hippocrates of Chios (not to be confused with the father of Greek medicine) as “first to leave writings on Euclidean subject matter,” say, around 440 B.C.E. (275). Hippocrates is credited with introducing the indirect method of proof into mathematics, which relies expressly on PNC. Netz concludes that “much of Greek mathematics was articulated in the Euclidean style” by around 360 B.C.E. (ibid.). Euclid’s Elements itself did not appear until about 300 B.C.E. Aristotle (384–322 B.C.E.) was attentive to this mature Greek mathematics, and he put it to some use in inference to and justification of the first principle that is PNC. Plato in his discussions of magnitudes and quantity (counts) stays rather distant from the systematization and rigor being given to mathematics in his day. Plato does make Form-hay from the circumstance that the idealized determinateness and exactitude supposed in geometry makes way for such knowledge as the relationships established in the Pythagorean Theorem (Meno 85–86), relationships that cannot be established so definitively by simply measuring sides of sensible triangles and squares, but require, rather, the operation of intellect on its own.

Peikoff’s Platonic line of logical ontologists hold PNC to be innate knowledge, not learned from scratch from experience of the sensible world. Peikoff conceives this line to also consist in holding that essences provide what regularity there is in sensible nature. In Phaedo Plato has Socrates say: “I am speaking of all things such as Size, Health, Strength and, in a word, the reality of all other things, that which each of them essentially is” (65d). In this dialogue, Plato invokes a notion of the contrary, within which can be read the contradictory, when he has Socrates invoke the principles (i) what one is explaining cannot have explanations giving the thing to be explained contrary qualities and (ii) an explanation must not itself consist in incompatible kinds of things (97a–b, 101a–b). Here Plato argues that the only adequate explanations are explanations by the regulative essences of things (e.g. the fineness of fine things), or we might also say, by the regulative Forms (e.g. the Fine) in which sensible and mathematical things participate, directly or indirectly (95e–102b; see Politus 2010.) I notice the implication in these parts of Phaedo that PNC, as within the prohibition of incompatibilities in explanations or in things explained, is a principle whose ultimate ground must lie in the realm of essence, or Form, not in the realm of the sensible world, lest explanation fall into the swamp of the sensible.

Peikoff observes that in Plato’s view the eternal, necessary essences, or Forms, do not require mind for their existence, but for the Neoplatonists and from Augustine to Cudworth and Leibniz, these essences and all necessary truths, such as PNC, do require mind for their existence (cf. Peikoff 2012, 24–25). In the line of logical ontologism extending from Plato, necessary truths exist in the eternal mind of God, they are prescriptive for the created empirical world, and they hold in the nature of that world. Their ultimate source and residence is the divine mind.

Peikoff draws out four arguments advanced in the Platonic line for why PNC cannot be learned from sensory experience. One of them is that PNC is a necessary truth. The principle states not only that there are no true contradictions, but that there cannot possibly be any true contradictions. In the Platonic line, let me add, such a necessity could no more be known merely from empirical induction than could be known in that way the necessary truth that any triangle in the Euclidean plane must have angles summing to exactly two right angles. These philosophers and theologians take such necessity to flow from the divine eternal mind, the permanent residence of such eternal, necessary truths. I observe, however, that their view that physical existence per se and in the whole of it is contingent because there are contingent things within this our world is an invalid inference. I say that ‘existence exists’ can be a necessity at least partly the ultimate base and reference of the truth and necessity of any necessary truths. On this corrective, Peikoff had things to say in his essay “The Analytic-Synthetic Dichotomy” in The Objectivist three years after completion of his dissertation (also Peikoff 2012, 12; further, Franklin 2014, 67–81).

I should add that for Plato, the necessity of necessary truths does not descend from a divine mind, lord of existence, mathematical and empirical, but from the Good, lord of all Forms and their traces in our reasoning on the mathematical and physical world. The Good is the Form dependent on no others. It is self-sufficient and is self-evident in a general way to human reason. It is the necessity that is source of all orderly necessity (Rep. 505c, 508d–509a, 511b–d; Philebus 20d, 60c, 64b–65a; further, Demos 1939, 35, 106, 307, 335). In my view, from Rand, all good is set in the highly contingent organization that is life. Then, I add, since the good does not have the ontological standing given it in Plato’s view, it cannot of itself (only a necessary-for) be the base of the sort of necessity had in necessary truths, truths such as the principle that, necessarily, there are no true contradictions.

To be continued.



Charles, D., editor, 2010. Definition in Greek Philosophy. Oxford.

Crivelli, P. 2010. The Stoics on Definition. In Charles 2010.

Demos, R. 1939. The Philosophy of Plato. Scribners.

Denyer, N. 2007. Sun and Line: The Role of the Good. In The Cambridge Companion to Plato’s Republic. G. R. F. Ferrari, editor. Cambridge.

Franklin, J. 2014. An Aristotelian Realist Philosophy of Mathematics. Palgrave Macmillan.

Netz, R. 1999. The Shaping of Deduction in Greek Mathematics. Cambridge.

Peikoff, L. 1967. The Analytic-Synthetic Dichotomy. In Ayn Rand: Introduction to Objectivist Epistemology. Expanded 2nd edition. 1990. Meridian.

——. 2012. The DIM Hypothesis. New American Library.

Plato [d. 347 B.C.E.] 1997. Plato – Complete Works. J. M. Cooper, editor. Hackett.

Politus, Y. 2010. Explanation and Essence in Plato’s Phaedo. In Charles 2010.

Priest, G., Beall, J. C., and B. Armour-Garb, editors, 2004. The Law of Non-Contradiction. Oxford.

Tait, W. 1986. Plato’s Second-Best Method. In Tait 2005.

——. 2002. Noēsis: Plato on Exact Science. In Tait 2005.

——. 2005. The Provenance of Reason. Oxford.

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14 May 2017

(I’m going to be talking more Rand in this segment, and I want the reader to keep in mind that Peikoff managed to hammer out his dissertation without any mention of Rand or her ideas, though her frame was also his in the years he was writing his dissertation.)

Aristotle I

Peikoff scrutinizes the broadly empiricist thinkers Aristotle, Aquinas, and Locke in their aspect of opposition to the Platonic views that necessary truths, such as the impossibility of contradictions in reality, are (i) innate in the human mind and (ii) features of essences accessible only by intellect and objectified beyond the particulars accessible by sensory perception. Aristotle recognized that the precise knowledge by demonstrations we make from true precise premises require principles constraining inference that are themselves true and precise and not themselves demonstrable.[1] Like all knowledge, in Aristotle’s view, these indemonstrable, necessarily true principles, such as PNC, must somehow derive from sensory experience. This somehow Aristotle sketched is a process begun in perception and capped by what has long been called intuitive induction.[2]

In the decades I lived in Chicago, there were university libraries that allowed the general public, if well behaved, to come in and read and xerox. The one with the most generous access was DePaul, which happened to be only an L-stop away from where I lived. One day I was there perusing bound volumes of The New Scholasticism, and I came across therein an article by Leonard Peikoff titled “Aristotle’s ‘Intuitive Induction’.”  I knew what went by the name intuitive induction, that it was also known as abstractive induction, that it was a genre among other genre of induction, and that it “exhibited the universal as implicit in the clearly known particular” (APo 71a8).[3] Peikoff’s published article was composed from portions of his dissertation.[4]

My quotation from Posterior Analytics (APo) just now was from the translation in the Oxford volumes edited by Ross (1910–52), which Peikoff had relied on in his dissertation and article.[5] In 1984 a Revised Oxford Translation of Aristotle was completed in which three of Aristotle’s books—Categories, de Interpretatione, and Posterior Analytics—unlike all the other books, were not mild emendations to the earlier translations into English, but were entirely new translations into English. The new translation of APo is by Jonathan Barnes. Some years after that translation, he thought he could do a little better, and he made a second translation. His translation of the familiar old-time Oxford “exhibiting the universal as implicit in the clearly known particular” becomes  (i) a class of inductive arguments “proving the universal through the particular’s being clear” and (ii) “proving something universal by way of the fact that the particular cases are plain.” Richard McKirahan paraphrases APo 71a8–9 on the sort of inductive arguments at issue as revealing the universal “through the fact that the particular is obvious” (1992, 237).

The obvious example, I say, would be from geometry, such as proving of any and every triangle that its interior angles sum to two right angles. Some proof of that result was known to Aristotle, and hopefully, any modern reader of philosophy knows the Euclidean proof.[6] I should mention that any three stars (not all in a single straight line) in a portion of the clear night sky reflected in still water determine a triangle in a Euclidean plane. The figure triangle exists in the world, whether by nature alone or by our constructions indicating that figure.[7] And for all such triangles, it is a fact that if they lie in a Euclidean plane their interior angles sum to two right angles. (Refer to this fact as 2R.) A triangle is a particular—one clear, plain, and obvious—and we can prove the fact 2R about triangles, a character of triangles holding necessarily for all of them.

Is the principle of noncontradiction a fact of the world in the way the sum of angles in a triangle is a fact in the world? Not exactly, I should say. That my right hand has five appendages is part of the character of the hand itself. That five fingers are not seventeen fingers is a fact, although one dependent not only on the character of five-fingered hands, but on auxiliary relations of five-fingered hands to something pretty far afield. Cases of noncontradiction run arbitrarily far afield: a five-fingered hand is not an opera, not an empty region of space, and so forth. A malformed human hand might lie on gradations between a typical hand and other natural or artificial instruments for grasping, but there is no such gradation between a typical hand and an opera. Full-scope noncontradiction depends for its existence in part on thought of negations arbitrarily far afield, negations untied from unities of real physical organization. PNC has some existential dependency on thought. 2R does not.

Objectivists could put it this way: Noncontradiction is a ramification of identity. The latter is not per se dependent on thought, I say, as they say. Fundamentally, identity is a fact like 2R, notwithstanding the circumstance that 2R is a demonstrable fact, whereas identity and noncontradiction are primitive principles (presumed, even if unstated) of demonstration (i.e., discursive demonstration, such as demonstration of 2R). An intuitive induction from sensory perception to the principles of identity and noncontradiction is not the same as intuitive induction cum demonstration from sensory experience with triangles to 2R. An intuitive induction from experience to the principle of noncontradiction cannot be a demonstrative proof, though it must be as precise and settled as demonstrations that rely on it. Intuitive induction to principles of identity and noncontradiction are more like proof-lacking inductions to “any three points not colinear determine a plane” and “nothing comes from nothing.” Although, those two facts grasped by intuitive induction do not depend at all on the cognitive power(s), the intuitive induction, under which they are cognized. In that they are like 2R or identity and unlike PNC. We should notice with Netz that, whether or not they are made explicit, certain intuitive propositions—intuitive in the sense of being obviously and necessarily true—are employed in the starting points and inferences of Greek mathematical proofs.[8]

Objectivists and some other moderns (e.g. Leibniz, Baumgarten, and Kant) have thought of noncontradiction as ontologically dependent on identity. Aristotle in Prior Analytics shows he knew that not all valid deductions exercise noncontradiction. Rather, the most perfect syllogistic forms of deduction exercise merely universal instantiation or transitivity of identity. Yet he says in Metaphysics:


Such a principle is the most certain of all, which principle this is, we proceed to say. The same attribute cannot at the same time belong and not belong to the same subject in the same respect . . . . If it is impossible that contrary attributes should belong at the same time to the same subject . . . and if an opinion which contradicts another is contrary to it, obviously it is impossible for the same man at the same time to believe the same thing to be and not to be; for if a man were mistaken in this point he would have contrary opinions at the same time. It is for this reason that all who are carrying out a demonstration refer it to this as an ultimate belief; for this is naturally the starting-point even for all other axioms. (1005b17–34)

One can, I say, think of a belief (or anything else) and its contradictory at the same time, where “same time” has a small, but nonzero duration. That would be on the duration-order of working memory. But only a mentally defective person could believe a thing and its contradictory within that scale of duration. Peikoff interpreted Aristotle in this passage to be arriving at the proposition, that one cannot believe a thing and its contradictory at the same time, by instantiation of the principle of noncontradiction in application to all existents, in this case the existent human mind.[9] That seems a shaky interpretation and a shaky conception of the human mind unless we have passed on from mere description to proper functioning of human mind. Peikoff’s 1964 position on this point, as straight description of mind, though it was in error, does not affect his characterization of logical ontologism or his contrast between its Platonic and Aristotelian wings.

Aristotle’s claim that “all who carry out a demonstration refer it to this [PNC] as an ultimate belief; for this is naturally the starting-point even for all other axioms” is close-but-no-cigar. The ultimate recognition for demonstration (which for Aristotle is a genre of syllogism), I say, is recognition of a principle of identity as rich as Rand’s or approximately that rich.

Rand wrote in 1957:


To exist is to be something, as distinguished from the nothing of non-existence, it is to be an entity of a specific nature made of specific attributes.[10] Centuries ago, the man who was—no matter his errors—the greatest of your philosophers, has stated the formula defining the concept of existence and the rule of all knowledge: A is A. A thing is itself. You have never grasped the meaning of his statement. I am here to complete it: Existence is Identity, Consciousness is Identification. (AS 1016)

The distinction of existence and identity is independent of consciousness, independent of identification. The distinction between existence and identity, as well as the inseparability of the former from the latter, are fundamental facts of the world.[11] Existence in its identity shows the elements of that identity to be without contradiction or self-contrariety.[12]

The Law of Identity in Rand’s usage of the title encompassed: A is A, a thing is itself, a thing is what it is, and existence is identity. By “greatest of your philosophers,” Rand meant Aristotle. Unlike moderns such as Leibniz, Baumgarten, Kant, or Rand, Aristotle did not connect a law of identity, in so many words, with his principle of noncontradiction.[13] Aristotle also did not connect the law of identity that speaks to the distinctive natures of things with a formula such as “A is A” or “A thing is itself.” Aristotle would say “A thing is itself” is nearly empty and useless, and he would not connect that proposition to “A thing is something specifically,” which he thought substantive and important.[14]

In Topics he holds that each and every thing is predicable of itself, predicable essentially and necessarily. Specifically, this predication is the thing’s definition. In this he means only that a thing and its definition refer to the same thing.[15] He does not convey the further thought that a thing is necessarily and nothing but the instanced definition together with all other instanced specific identity of the thing, along with any particularities of the thing, such as location. He does not convey that further thought from Rand I think right: that all those together compose the existence of the thing without remainder.

Aristotle was the founder of logic, and his great contribution thereto was his theory of correct inference, which is largely his theory of the syllogism. Though he did not realize it, the formula “A is A” in the form “Every A is A” can be used to consolidate the kingdom of the syllogism. By about 1240, Robert Kilwardly was using “Every A is A” to show conversions such as the inference “No A is B” from the premise “No B is A” can be licensed by syllogism.[16] Aristotle had taken these conversions, like he had taken the first-figure syllogistic inferences, to be obviously valid and not derivable.[17] Aristotle takes first-figure syllogisms to be obviously valid and the paragons of necessary consequence. The mere statement of these syllogisms makes evident their conclusion as following necessarily. Using conversions as additional premises, Aristotle shows that all syllogisms not first-figure can be reduced to first-figure ones. Their validity is thereby established, by the obvious validity of the first-figure ones and by the irreducible obvious validity of the conversions.[18] In this program, which is in Prior Analytics, Aristotle uses also the principle of noncontradiction; for some of his reductions of second- and third-figure syllogisms to first-figure employ indirect proof, specifically proof per impossibile. However, the per impossibile steps only establish a premise that can then be employed in a direct proof of reduction to first figure.[19] The principle of noncontradiction, like the first-figure inferences and the logical conversions, is self-evident. The principle of noncontradiction is not the entire or main base of valid logical inference, I observe. Rather, I maintain, identity is directly the main base, and indirectly identity is base when noncontradiction is base, for the former is base of the latter. Notice also: That the logical conversions were centuries later shown to be derivable from first-figure syllogisms by using A is A as a premise does not imply that the conversions are not also self-evident.[20]

There are places in which Aristotle connects “A thing is something specifically” or “A thing is what it is” with the principle of noncontradiction: “The same attribute cannot at the same time belong and not belong to the same subject in the same respect” (Metaph. 1005b19–20). Though not given the pride of place given it by Rand, there is some recognition that existence is identity in Aristotle: “If all contradictories are true of the same subject at the same time, evidently all things will be one . . . . And thus we get the doctrine of Anaxagoras, that all things are mixed together; so that nothing really exists” (1007b19–26).[21] Aristotle acknowledges on occasion that any existent not only is, but is a what.[22] He contradicts that principle, however, when he says: “That which is primarily and is simply (not is something) must be substance” (Metaph. 1028a30).

The art of noncontradictory identification is logic, in Rand’s conception of it. I take some issue with that definition, for avoidance of contradiction is not the main rule of deductive inference. That main rule is directly identity itself. Mathematical induction, also, does not rest on noncontradiction, but is a variety of identity. Then too, the rule of noncontradiction itself rests on the fact(s) of identity. This asymmetric dependence was evidently recognized in Rand 1957, wherein she had it that existence exists and is identity and that “existence exists” is the basis of logic. She took consciousness to be fundamentally identification and took logic to be the genre of consciousness-endeavor noncontradictory identification. That differentia noncontradictory is an inadequate span of the modes of inference in the discipline of logic. I suspect Rand was led astray by Aristotle’s “all who are carrying out a demonstration refer it to this [PNC] as an ultimate belief; for this is naturally the starting-point even for all the other axioms” which is only a few lines of Aristotle beyond the lines she quotes in the closing scene of 1957.

The inferences of first-figure syllogisms are, I maintain, licensed directly by identity alone, in Rand’s ample sense of identity, and without recourse to noncontradiction. Nathaniel Branden and Leonard Peikoff in their Objectivist writings erred in trying to support Rand’s definition of logic, with its differentia of the noncontradictory, by appeal to noncontradiction rather than directly to identity as basis of the inference in a certain first-figure syllogism.[23] That certain one is the inference-form of the familiar case: Socrates is a man, all men are mortal, and therefore, Socrates is mortal. Peikoff 1991 and Branden c.1968 rightly point out that denial of this inference would lead to contradiction,[24] but that is not to the point of first, most direct basis.[25] One already knows that these first-figure inferences are valid, that their conclusions necessarily follow, without invoking PNC, just as Aristotle had rightly observed in Prior Analytics and had messed up in Metaphysics. Another class of deductions not fitting Rand’s definition is the direct proof of mathematical identities, such as the trigonometric identities. All such proofs conclude 1=1, showing the initial proposed identity true. No appeal to noncontradiction is made; identity is invoked directly and is the entire basis of proofs of mathematical identities.

That identity in a broad Randian sense of the term is more fundamental than and is ground of PNC, though underground in Peikoff’s dissertation, does not undermine his characterization of Aristotle’s logical ontologism. Then too, characterization of PNC as being not only a fact of the world but a fact partly dependent on operation of thought in the world—my own added characterization—does not degrade Peikoff’s characterization of Aristotle’s logical ontologism, though my ontology of PNC may in the end suggest reformation in Peikoff’s divisions of schools of thought in the history of philosophy of logic.

In the next installment, I’ll continue with Aristotle and with Peikoff’s treatment of him, beginning with intuitive inductions to necessary truths including PNC. I want to close the present installment by noting the change in translation of APo. II 19 by Barnes concerning the traditional intuition in intuitive induction. The older translation relied upon by Peikoff 1964, 66, reads: “From these considerations it follows that there will be no scientific [i.e. deductive] knowledge of the primary premises, and since except intuition nothing can be truer than scientific knowledge, it will be intuition that apprehends the primary premises” (APo. 100b10–12). Barnes final translation reads: “Hence there will not be understanding of the principles; and since nothing apart from comprehension can be truer than understanding, there will be comprehension of the principles” (APo. 100b10–12). In the Barnes translation, scientific knowledge has become understanding; primary principles have become principles; and intuition has become comprehension. Each of these differences is significant, and Barnes argues for them. On the last of those three alterations in translation, Barnes argues against the traditional English of nous into intuition. He remarks in part:


The commentators translate nous by “intuition.” The word “intuition” is a term of art; when it has a determinate sense (and does not merely stand for knowing we know not how), it implies a sort of mental “vision”; intuition is mental sight; intuited truths are just “seen” to be true; intuiting that P is coming to know that P without any ratiocination and without using sense-perception—it is “seeing” that P. The term “intuition” has a hallowed connection with B 19; indeed, the classical distinction between “intuitive” and “demonstrative” knowledge, which is common property to rationalists and empiricists, derives ultimately from this chapter. (1992, 267)

Barnes argues that induction factors into Aristotle’s answers on whether we have innate knowledge of indemonstrable principles that are starting-points of demonstrations and, if not, how knowledge of such principles is acquired. He argues that nous is answer to a different question of Aristotle’s: what is our state that knows those principles? Under Barnes picture, Aristotle has us in the state Barnes calls understanding when we know theorems and has us in the state nous, which Barnes calls comprehension, in our knowledge of indemonstrable principles. “Understanding is not a means of acquiring knowledge. Nor, then, is nous. / . . . ‘Intuition’ will not do as a translation for nous; for intuition is precisely a faculty or means of gaining knowledge. Hence in my translation I abandon ‘intuition’ and use instead the colourless word ‘comprehension’ (268).

We can be sure that such issues of translation of Aristotle, and consequent divergent characterizations of Aristotle’s views, have been acute not only in translations into modern languages, but into Arabic and into Latin centuries ago.

To be continued.


[1] APo. 72b19–24, 99b20–21.

[2] APo. 99b35–100b5.

[3] Boydstun 1991, 36.

[4] Mainly pages 63–79 of his dissertation.

[5] The translations in Richard McKeon’s The Basic Works of Aristotle are from the Ross edition.

[6] APo. 71a19–29, 85b5–15, 91a3–4; Metaph. 1051a24–27; Euclid’s Elements I.32.

[7] On Memory 450a1–4; Metaph. 1089a25–26.

[8] Netz 1999, 182–85,189–98.

[9] Peikoff 1964, 156–57. See further, the translation and commentary of Kirwan 1993.

[10] Cf. Avicenna 1027: “It is evident that each thing has a reality proper to it—namely, its quiddity” (I.5.10). Think whatness for the traditional quiddity (quidditas, ); see e.g. Gilson 1939, 199.

[11] Cf. Heidegger’s ontological articulation and disclosedness in Haugeland 2013, 197–98, notes 6 and 7.

[12] AS 1016; ITOE App. 240, 286–88.

[13] Leibniz 1678; Baumgarten 1757 [1739], §11; Kant 1755, 1:389; 1764, 2:294. Rand, in the “About the Author” postscript to AS, and N. Branden, in Basic Principles of Objectivism, erroneously thought Aristotle held the tight bond of identity and noncontradiction that had actually come to be recognized only with Leibniz and his wake.

[14] Metaph. 1030a20–24, 1041a10–24.

[15] Top. 103a25–29, 135a9–12.

[16] First mood of the second figure; Kneale and Kneale 1962, 235–36; see also Kant 1800, §44n2. It was through Kneale and Kneale 1962 that I learned of Kilwardly’s recognition of the logical serviceability of “A is A” in the form “Every A is A.” In his 1964 dissertation, Peikoff did not make use of this book by the Kneales. Relying on older books on the history of logic, Peikoff noted in the Introduction to his dissertation that the law of identity specifically formulated as such was apparently not in play until end of the thirteenth century (works of Antonius Andreas). Placing first recognition of the law of identity a century or so earlier by more recent historical studies of logic, such as by the Kneales, still locates inception of the law’s recognition in the medieval era, as alleged in Peikoff’s older histories.

[17] Lear 1980, 3–5.

[18] Lear 1980, 1–14.

[19] Lear 1980, 34–53; Bonevac 2012, 68–72.

[20] On Aristotle’s alternative method ecthesis for reducing second- and third-figure syllogisms to first-figure, see Malink 2013, 86–97. This method rests directly on identity, not indirectly via noncontradiction.

[21] See also Metaph. 1006b26–27, 1007a26–27. Let EI designate Rand’s “Existence is Identity.” Aristotle, Avicenna, Henry of Ghent, John Duns Scotus, Francis Suárez, Spinoza, Leibniz, Baumgarten, Kant, and Bolzano also reached principles close to (EI), though not the Randian rank of (EI) or near-(EI) among other metaphysical principles. A Thomist text Rand read had included: “What exists is that which it is” (Gilson 1937, 253). That is a neighbor of Rand’s “Existence is identity.” Neighbor Baumgarten: “Whatever is entirely undetermined does not exist” (1757, §53).

[22] Metaph. 999a28; 1030a20–24, 1042b25–28; APo. 83a25–34.

[23] Branden c. 1968, 67–69; Peikoff 1991, 119, though Peikoff had not made this error in explicating this syllogism in his dissertation 1964, 134.  Leibniz errs in this way as well (1678, 187). But on another occasion, Leibniz writes, after listing some “Propositions true of themselves” (such as A is A), writes “Consequentia true of itself: A is B and B is C, therefore A is C” (quoted in Kneale and Kneale 1962, 338).

[24] See further, Buridan 1335, 119–20.

[25] See also Kneale and Kneale 1962, 357, and their conclusion that “the principle of noncontradiction is not a sufficient foundation for all [syllogistic] logic.”


Aristotle c. 348–322 B.C.E. The Complete Works of Aristotle. J. Barnes, ed. 1984. Princeton.

Avicenna 1027. The Metaphysics of The Healing. M. E. Marmura, trans. 2005.  Brigham Young.

Barnes, J., trans. and comm., 1992. Aristotle – Posterior Analytics. 2nd ed. Oxford.

Baumgarten, A. 1757 [1739]. Metaphysics. 4th ed. C. D. Fugate and J. Hymers, trans. 2013. Bloomsbury.

Bonevac, D. 2012. A History of Quantification. In Logic: A History of Its Central Concepts. D. M. Gabbay, F. J. Pelletier, and J. Woods, ed. Elsevier.

Boydstun, S. 1991. Induction on Identity. Pt. 1. Objectivity 1(2):33–46.

Branden, N. c. 1968. The Basic Principles of Objectivism Lectures. Transcribed in The Vision of Ayn Rand. 2009. Cobden.

Buridan, J. 1335. Treatise on Consequences. S. Read, trans. 2015. Fordam.

Euclid c. 300 B.C.E. The Elements. T. L. Heath, trans. and comm. 2nd ed. 1925. Dover.

Gilson, E. 1937. The Unity of Philosophical Experience. Ignatius.

——. 1939. Thomist Realism and the Critique of Knowledge. M. A. Wauk, trans. 1986. Ignatius.

Haugeland, J. 2013. Dasein Disclosed – John Haugeland’s Heidegger. J. Rouse, ed. Harvard.

Kant, I. 1755. A New Elucidation of the First Principles of Metaphysical Cognition. D. Walford and R. Meerbote, trans. In Immanuel Kant – Theoretical Philosophy, 1755–1770. 1992. Cambridge.

——. 1764. Inquiry Concerning the Distinctness of the Principles of Natural Theology and Morality. D. Walford and R. Meerbote, trans. In Immanuel Kant – Theoretical Philosophy, 1755–1770. 1992. Cambridge.

——. 1800. The Jäsche Logic. J. M. Young, trans. In Immanuel Kant – Lectures on Logic. 1992. Cambridge.

Kirwan, C., trans. and comm., 1993. Aristotle – Metaphysics, Books , , and . Oxford.

Kneale, W., and M. Kneale 1962. The Development of Logic. Oxford.

Lear, J. 1980. Aristotle and Logical Theory. Cambridge.

Leibniz, G. W. 1678. Letter to Herman Conring – March 19. In Gottfried Wilhelm Leibniz: Philosophical Papers and Letters. L. E. Loemker, trans. 2nd ed. 1969. Kluwer.

Malink, M. 2013. Aristotle’s Modal Syllogistic. Harvard.

McKirahan, R. D. 1992. Principles and Proofs – Aristotle’s Theory of Demonstrative Science. Princeton.

Netz, R. 1999. The Shaping of Deduction in Greek Mathematics. Cambridge.

Peikoff, L. 1964. The Status of the Law of Contradiction in Classical Logical Ontologism. Ph.D. dissertation.

——. 1985. Aristotle’s “Intuitive Induction.” The New Scholasticism 59(2):185–99.

——. 1991. Objectivism: The Philosophy of Ayn Rand. Dutton.

Rand, A. 1957. Atlas Shrugged. Random House.

——. 1990. Introduction to Objectivist Epistemology. Expanded 2nd ed. Meridian.

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2 November 2017

Aristotle II

In my own picture, if one is reading this and knows horses as horses and trees as trees, one knows that horses are not anything but horses, not anything such as trees. If one has concepts, such as the concept horse, then one knows identity, at least a thin identity, and knows classes, whether or not one yet realizes one is dealing in those general patterns identity, class inclusion, and class exclusion. Knowing horses and trees as such, one knows already that horses are necessarily horses and necessarily not anything other than horses, such as trees.[1] Then too, if one is reading this and has the concept horse, one knows validity of the inference from “All horses have blood vessels” and “Bucephalus was a horse” to “Bucephalus had blood vessels.” Also, being a horse, Bucephalus was necessarily not rooted in the soil like a tree.

If one has concepts—even ones held, in early development, according to weighted sums of typical features, not yet according to natures and definitions[2]—I say one has the knowledge grounding recognition of elementary logically valid inference.[3] Where there is knowing things under concepts, there is knowing some what-it-is and what-it-is-not. If equipped with concepts—even before learning to read and before express understanding of grammar—one knows at least dimly that contradiction is false of all things said of the world under concepts, necessarily false.

Rightness and necessity in our later grasp as concepts our concepts of things, things as they are, are inherited rightness and necessity from those earlier concepts of things as they are. Grasping logic requires grasping concepts as concepts. Then my view is that rightness and necessities of logic are heirs ultimately of rightness and necessities in our concepts not as concepts, but merely as of things as they are. My own picture then is a variety of logical ontologism.[4]

Peikoff sets Locke in that broad stream as well. Locke did not have the vantage of our contemporary scientific research into early cognitive development. On his somewhat inconstant model of human cognition and its ontogeny, one has no knowledge which one had not been self-consciously aware of in its acquisition.[5] Moreover, Locke’s tendency towards nominalism in universal concepts sets him to reject possession of universal concepts of things empirical as funding logical necessities such as noncontradiction (or syllogistic inference[6]). Locke would reject the conveyance of empirical necessities to logical necessities by attainment of empirical concepts. Rather, he would have PNC be a generalization of our early notice of particular empirical distinctions and necessities, which were made without knowledge of PNC.[7] Further, in his congeniality towards nominalism, Locke has PNC with its formality and necessity grounded rather more in keeping our reflections on the world straight than in reflecting the world.[8]

Locke rejected the realist theory of universals in both its extreme and moderate forms.[9] Locke understood Aristotle, or anyway the Aristotelianism of his own era, as moderate realism. He argued against our access to any such things as specific, substantial form or real essences.[10] We work with nominal essences, by the lights of Locke, and his own theory of universal concepts has been classified as conceptualism, wherein general words stand for general ideas.[11]

Marco Sgarbi 2013 shows that highly empiricist Aristotelian logic texts flourished in Britain in the 16th and 17th centuries. Frances Bacon criticized the strain therein subordinating the world to the mind and the mind to its concepts.[12] Insofar as Locke took concepts as tightly bound to the mind-independent world, he is located, as Peikoff locates him, in the tradition of logical ontologism, specifically in its Aristotelian wing. Conceptualism is an alternative to realism in theory of universals and in theory of science. Insofar as a variety of conceptualism creates any breach or looseness between what Newton called phenomena (such as orbits of planets) and the mind, it leans away from logical ontologism, I should say.[13]

Peikoff 1964, like Locke 1690, takes Aristotle in the usual way, which is as a moderate realist[14] in which universal concepts, such as horse or tree, derive from particulars in perceptual experience, particulars containing real essences, which are real forms. Aristotelian forms are the definite delimitations joined with fundamental indefinite matter in any actual particular.[15] Unlike Platonic Forms, Aristotle’s forms, even the forms essential to a thing being the kind of thing it is, are not residents of a realm separate from this world of particulars around us. Those essences reside in the particulars around us, and they are accessible to us through appropriation by intellect attending to things sensed. Essential form is of itself never separate from its matter; it is contemplated as separate by the intellect. Contra Plato, essences according to Aristotle do not themselves reside in a realm separately and independently from the realm of sensible particulars, and essences are not accessed by intellect alone.

Where concepts and propositions, including the propositions that are logical principles, were thought to be bonded to and guided by mind-independent reality through their incorporation of Aristotelian form from particulars, supposed conformity to the world by concepts and propositions and logical principles was at stake should the existence of Aristotelian forms be rejected.[16] They were indeed rejected by moderns not Scholastic, excepting Leibniz. They were rejected, as I have mentioned, by Rand and Peikoff. Rightly so. Locke, as we have seen, rejected Aristotelian accounts of abstraction from perceptual particulars via mental absorption of Aristotelian forms, Aristotelian essence.

Peikoff observes that Aristotle attempted to account for PNC as deriving from our assimilation of essential forms in particulars, yet account for such assimilation being reliant on our knowing PNC and for PNC being foundation of all knowing.[17] (A parallel tension appears in Rand’s proposition Existence exists as most fundamental axiom, yet as conceptually derived from perception.[18]) Locke rejected any need for any of the accounting after the yet. Before the yet, he rejected Aristotle’s forms and essences as existing and as sourcing the necessity of PNC we find in thought or in the world. Locke proposed necessity of PNC is perceived, by sense, in distinct particulars. I agree with Locke that there are directly perceived physical necessities. Peikoff, Kant, and virtually the entire bench of philosophers are right, however, to dismiss Locke’s idea that logical necessity is among the types of necessity perceived directly in sensory perception.

There is another view on the ontogeny of PNC, my own view (supplementing what I wrote in the first three paragraphs of this installment), which is only a stone’s throw from Locke’s. That is the view that impossibility of performative contradictions have metaphysical and epistemological priority with respect to formal PNC. One was getting a grip on incompatibility of one’s alternative physical acts before reaching the one-word stage of language development.[19] Further, one has action- and image-schemata (and working memory) preceding and continually supporting one’s concepts. Furthermore, all defenses of PNC eventually invoke impossibilities in actual performances.

Back to older philosophy. Logical necessity for Aristotle resides in character of concepts, predications, definitions, and inferences. These require abstractions from examination of groups of singulars. In Aristotle’s view, perception of a single object not yet conceptualized will not yet open conceptual necessities, such as PNC logical necessity, even though the basis of PNC stands in the intelligible forms and essences shared by and residing in each perceptible singular.[20]

Aristotle had located the source of logical necessities in active operations of intellect and not in the perceptual, memorial, and imaginative supports of active intellect. This stance of Aristotle, which Locke rejected, would please Leibniz, whom Peikoff places in the Platonic line of logical ontologism.[21] Recall that the Platonic line takes some ideas to be innate. In their view, PNC is an innate guiding principle we posses, not a principle derived from or delivered by sensory perceptions. And they take logical necessities to reflect necessity in relations between essences or essential forms obtaining apart from any participation of concrete existents in those eternally true patterns. I want to indicate Leibniz’ shredding of Locke’s outlook on PNC, along with my display of Peikoff’s diagnosis of the failures of both Platonic and Aristotelian logical ontologism.

Leibniz argued persuasively against Locke’s thesis that we know nothing that was not explicitly known by us at least in its initial appropriation.[22] Locke had rejected the Platonic picture in which there is knowledge of geometry or logic possessed by us implicitly prior to it becoming explicit to us. By way of blocking possibility of innate knowledge, Locke had declined possibility of implicit knowledge. It is implausible, I say (as would most moderns), that we have no implicit knowledge hanging about things known explicitly in their initial acquisition. Yet, contra Leibniz, this is no license for thinking any ideas (as distinct from faculties) to be innate.

Demise of doctrines of the innateness of ideas, including necessary truths, cuts down the Platonic line in their defense of the view that logical truths are grounded in something fixed and independent of our knowing those truths. Likewise in ruins became the Platonic support of PNC ontologism by reification of universals and essences, whether residing in an other-worldly place and whether constituting or inhabiting God’s this-world-independent understanding.[23]

Leibniz challenged Locke’s position that PNC is simply an empirical generalization from particular oppositions in experience such as that bitter is not sweet or that wormwood is not sugarplum or that the nurse is not the cat.[24] Leibniz objects that such oppositions of sense have not the absolute certainty of freedom from illusion or other defect as has PNC. I should say against Leibniz that that is no airtight showing that PNC is not derived by empirical generalization. A triangulated result can be more sure than its individual elements of evidence towards that result; we fare well with Whewell’s consilience of inductions.[25] Be that as it may, Leibniz was correct, I say, to hold that PNC is not derived merely by empirical generalization because ideas of being, possible, and identity intertwined in PNC are at hand in any of our general concepts. Leibniz errs, to be sure, in his rush from that picture to innateness of such ideas and PNC.

Leibniz argues well against Locke’s tendency towards nominalism in universals, essences, definitions, conceptual taxonomies, and logical principles. Leibniz submits their bases to be in the similarity of singular things in reality and in possibilities that are independent of our thinking.[26] But Leibniz’ own realist account, with its Platonic and Aristotelian elements, is upset with the upset of those elements and, as well, of his particular amalgam of them.[27]

Aristotle had written in Physics:


What is plain and clear at first is rather confused masses, the elements and principles of which become known to us later by analysis. Thus we must advance from universals to particulars; for it is a whole that is more knowable to sense perception, and a universal is a kind of whole, comprehending many things within it, like parts. (184a22–25)

Following out this line of thought, Aquinas thought of being and its opposite nonbeing as contained in some way in any knowledge we might have, however elementary the knowledge.[28]


As Avicenna says, that which the intellect first conceives as, in a way, the most evident, and to which it reduces all its concepts is being. Consequently, all the other conceptions of the intellect are had by additions to being. But nothing can be added to being as though it were something not included in being—in the way that a difference is added to a genus or an accident to a subject—for every reality is essentially a being. (Aquinas 1256, I.1)

Aquinas maintained the Aristotelian view that PNC is learned by an integrated employment of sense experience and reason. However, on Aquinas' view,


the function of the experiential component is not to provide the learner with instances of the Law; rather, it provides the learner with at least one instance of an existent. From this point on, the intellect, by abstraction and intuition, generates the knowledge of the Law of Contradiction. (Peikoff 1964, 85–86)

The Aquinas over-writing of Aristotle is a right strand, I say, in an adequate theory of acquisition of PNC and logical ontologism. Aquinas is able to support the theses on both wings of the Aristotelian tension Peikoff highlights across the yet I mentioned in connection with Locke. One does not ascend to grasp of PNC by inductive steps, according to Aquinas. Rather, the more comprehensive precedes the less so, in both sense and intellect.

Our first possible empirical knowledge of any singular is that it is. And, to this knowledge, we must have (in some form) the awareness of being; and once this latter is attained, the Law of Contradiction is, for all practical epistemological purposes, thereby known. . . . Sensory experience is unquestionably necessary to the cognition of the law of contradiction . . . and yet experience teaches us no truths prior to our cognition of the Law since, in the act of grasping the first (singular) truth that it teaches, we have, simultaneously, grasped the Law. (Peikoff 1964, 86–87)

Aquinas’ utilization of Aristotle in metaphysics and epistemology of logical ontologism land the ship in wreckage for modern thought since the time of Newton and Locke. Aquinas had nous, or reason, as our essentially human capability in cognition, but nous itself as derived from realities transcending nature. Peikoff shows Aristotle’s sayings in that same voice.[29] For Aquinas formal structure among particulars are latched to Aristotle’s hylomorphism, his fundamental metaphysics of prime matter and form, essential and incidental form, which all would be coming to wreckage.[30]

Peikoff argues the unraveling of Platonic logical ontologism into logical conventionalism (as of mid-twentieth century) to have been mediated significantly by Kant. Unraveling of the Aristotelian logical ontologism is mediated significantly by Locke.[31] The Aristotelian moderate realism of universals collapses. Aristotelian Matter, Form, and their join collapse. Concrete particulars, individuated by “matter” nigh well itself “unknowable” apart from form, is a soggy base for any universals, including those entrained in PNC and funding its objectivity. An assembly of super-strong axioms resting ultimately on things unknowable is problematic.[32]

In the next installment, I want to convey and assess Peikoff’s account of Kant’s contribution to the transition to conventionality in philosophy of PNC. I hope to touch on not only conventionalist theories to the time of Peikoff’s dissertation, but on those flourishing today and their historical setting. I plan to add a coda that is an inventory of the elements and works in Peikoff’s dissertation that plainly contributed to things addressed in the early ’60’s in the Rand/Branden journals, points in Rand’s epistemology (1966–67), and points, with morphisms, in Peikoff’s own writings from his “Analytic-Synthetic Dichotomy” (1967) to The DIM Hypothesis (2012).


[1] Cf. Sullivan 1939, 52–53, 62.

[2] Boydstun 1990, 34–36.

[3] Cf. Salmieri 2010, 160n12.

[4] See also Rasmussen 2014, 337–41.

[5] Locke 1690, I.1.5, IV.1.9; Peikoff 1964, 87–102.

[6] Locke 1690, IV.7.8.

[7] Locke 1690, I.1.15, 25, 3.3, IV.1.4, 2.6, 7.9–10; Peikoff 1964, 103–13.

[8] Locke 1690, IV.7.4, 10–14, 8.2; Peikoff 1964, 216–26.

[9] Locke 1690, III.3.11.

[10] Locke 1690, III.6.23–26, 10.14–15.

[11] Aaron 1952, a fine work I learned of through Peikoff 1964. Similarly, Salmieri 2008 renders Locke as conceptualist, rather than as realist (51n118). Hobbes is today also argued to be a conceptualist, rather than a nominalist (Sgarbi 2013, 190–92). The conceptualism and empiricism of Locke is presaged in the century before him by the British logicians John Case and Giulio Pace, who were in the lineage of the Paduan Aristotelian Jacopo Zabarello (Sgarbi 2013, 91–97, 101–6). / Binswanger 2014 takes moderate realism most generally as holding there are non-specific properties of things in the world independently of our cognizance of such properties. He argues Locke falls into that general bin in spite of himself (102–4). Binswanger divides theory of universals (Rand’s Objectivist theory aside) into the jointly exhaustive bins of realism and nominalism, as had Armstrong 1978. A conceptualist theory could then belong in either of these bins, depending on the particulars of the conceptualist theory. Salmieri 2008 argues the division of concept theories into realist and nominalist (anti-realist) is not a distinction adequate for classing theories of concepts in accordance with their comparative degrees of similarity and their comparative degrees of difference (52–54).

[12] Sgarbi 2013, 169.

[13] Conceptualism in which general ideas are thought of as unbound to the world is what Rand meant by Conceptualism in her 1966 and what David Armstrong meant by Concept Nominalism in his 1978.

[14] The usual view that Aristotle held to moderate realism is disputed by Greg Salmieri (see Gotthelf 2012, 302n19). Salmieri argues in his dissertation that Aristotle thought our concepts of natural kinds, such as horse or tree, stand to their instances not under identity of shared essential, nonaccidental form, identity of a shared kind-form, or identity of a shared real essence resident in each particular instance. Rather, as a relation of determinables to determinates (2008, 44–51, 56–122). (See citation of Johnson, Prior, and Searle in Boydstun 2004n5. Salmieri aligns Aristotle with the Johnson version of the determinable-determinate relation.) Salmieri’s cast of Aristotle’s relation of horse to an instance such as Bucephalus as a complex of determinable-determinate relations incidentally locates Aristotle closer to Rand than traditional moderate realism is close to Rand. Determinable-determinate relations, I should mention, have a realist underlining, for determinable-determinate relations are along dimensions, such as length or material hardness, plainly real and accessible. / Against the traditional interpretation of Aristotle as a moderate realist holding to shared essences of kind by instances of the kind is also Lennox 2001, Chapter 7. For Peikoff 1964, prevailing interpretations of Aristotle across the long arc of the history of philosophy are the pertinent interpretations to his tracing of logical turns in that history.

[15] Aristotle, Ph. 193a30–94b15, 199a30–33, 209b22–23; Metaph. 1033b24–1034a8, 1036a27–31.

[16] Peikoff 1964, 212–16.

[17] Aristotle, APo. 71b10–72b4, 73b17–74a4, 75b21–36, 77a5–35, 81a37–b9, 85b16–23, 87b28–88a17, 99b15–100b17; De An. 429a10–30a26; Metaph. 1005b9–08b27, 1015a20–b15, 1018b30–34, 1035b32–36a11, 1040b25–27, 1061b34–62b11; NE 1143a32–b6; Peikoff 1964, 60-79, 119–20, 124–35, 213–14.

[18] Lennox 2005.

[19] Boydstun 1991a, 39; 1991b, 34.

[20] APo. 71b35–72a6, 87b28–88a11; Metaph. 982a23–25, 1029b1–12; Peikoff 1964, 63–75; Barnes1993, 95–97; McKirahan 1992, 30–33. 127, 219–22; Ferejohn 2013, 76–80; Salmeiri 2010, 158–60.

[21] On the Platonism of Leibniz, see Mercer 2001, 173–205, 243–52.

[22] Leibniz 1704, 76–78, 359–61, 411–12.

[23] Peikoff 1964, 187–200, 205–9.

[24] Leibniz 1704, 86–87, 101–2, 412.

[25] See also Zabarella, in Sgarbi 2013, on the perfection of first principles in intellect from received empirical inductions (65–70). On Whewell’s consilience, see Snyder 2017. https://plato.stanford.edu/entries/whewell/

[26] Leibniz 1704, 286–96.

[27] Leibniz 1704, 317–19, 343–48.

[28] Cf. Spinoza: “The first thing that constitutes the actual being of a human mind is nothing but the idea of a singular thing which actually exists” (1677, 2p11).

[29] Peikoff 1964, 119–23.

[30] On the separation capability of Form from Matter, greater for Christian Aristotelians such as Aquinas than for Aristotle, see Peikoff 1964, 148–56.

[31] Peikoff 1964, 212–35.

[32] Peikoff 1964, 67–68, 126, 214–15; Lewis 2013, 177n4, 185, 251; Reeve 2016, 400–402n691, 428n796. A similar problem is argued in Ferejohn 2013 for Aristotle’s conception of substances as most-primary and as simple beings resisting definition, yet they are to be explanatory grounds of other, complex beings (172–73).


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Aquinas, T. c.1256–59. De Veritate. J. V. McGlynn, translator. 1994. Indianapolis: Hackett.

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Armstrong, D. 1978. Nominalism and Realism. Volume 1 of Universals and Scientific Realism. Cambridge: University Press.

Barnes, J., translator and commentator, 1993. Aristotle – Posterior Analytics. 2nd ed. Oxford: Clarendon.

Binswanger, H. 2014. How We Know –Epistemology on an Objectivist Foundation. New York: TOF Publications.

Boydstun, S. C. 1990. Capturing Concepts. Objectivity 1(1):13–41.

——. 1991a. Induction on Identity – Part 1. Objectivity 1(2):33–46.

——. 1991b. Induction on Identity – Part 2. Objectivity 1(3):1–56.

——. 2004. Universals and Measurement. The Journal of Ayn Rand Studies 5(2):271–305.

Ferejohn, M. T. 2013. Formal Causes – Definition, Explanation, and Primacy in Socratic and Aristotelian Thought. New York: Oxford University Press.

Gotthelf, A. 2012. Teleology, First Principles, and Scientific Method in Aristotle’s Biology. New York: Oxford University Press.

Leibniz, G.W. 1704. New Essays on Human Understanding. P. Remnant and J. Bennett, translators. 1996. Cambridge: University Press.

Lennox, J. G. 2001. Kinds, Forms of Kinds, and the More and the Less. In Aristotle’s Philosophy of Biology. Cambridge: University Press.

——. 2005. Axioms and Their Validation. Paper at APA session of The Ayn Rand Society.

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Locke, J. 1690. An Essay Concerning Human Understanding. New York: Dover.

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Mercer, C. 2001. Leibniz’s Metaphysics – Its Origins and Development. Cambridge: University Press.

Peikoff, L. 1964. The Status of the Law of Contradiction in Classic Logic Ontologism. Ph.D. dissertation.

Rand, A. 1966. Introduction to Objectivist Epistemology. H. Binswanger and L. Peikoff, editors. 1990. New York: Meridian.

Rasmussen, D. B. 2014. Grounding Necessary Truth in the Nature of Things. In Shifting the Paradigm: Alternative Perspectives on Induction. P. C. Biondi and L. F. Groarke, editors. Berlin: De Gruyter.

Reeve, C. D. C., translator, 2016. Aristotle – Metaphysics. Indianapolis: Hackett.

Salmieri, G. 2008. Aristotle and the Problem of Concepts. Ph.D. dissertation.

——. 2010. Aisthêsis, Empeiria, and the Advent of Universals in Posterior Analytics II 19. In From Inquiry to Demonstrative Knowledge – New Essays on Aristotle’s Posterior Analytics. J. H. Lesher, editor. Kelowna: Academic Printing and Publishing.

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PNC Ground Shifts to the Side of the Subject – Kant I

We have seen the weaknesses of the classical accounts of how PNC is grounded in the nature of objects apart from their subjects. Platonic and Aristotelian accounts of form and essence fell off the center stage of philosophy in the modern era. With them fell the accounts of the necessity and normativity of the principle of noncontradiction (PNC) utilizing them. Moreover, those accounts, and the more nominalistic account of Locke too, were inadequate to the task anyway. Peikoff 1964 maintains that Kant’s views on logic were a main highway to the subsequent modern view that logic, including PNC, takes its correctness and necessity most basically from the side of the human subject, not from objects existing apart from the subject.

A right-hand glove will not fit my left hand unless I turn the glove inside out. That is a fact about physical objects, including my natural and artificial instruments. My learning, retaining, and stating the fact entails facility in tacitly using set-membership relations. The fact is not dependent on those set-membership relations or on the abstraction process. With much more abstraction from the physical, one can learn that the glove-hand fact is a manifestation of spaces we call oriented spaces. Again, I cannot simultaneously be turning a right glove into a left and not doing that. Beyond the facility with sets and abstraction in stating that fact is comprehending that the fact and its statement instantiates PNC. Any account of the ontology and coming-to-knowledge of PNC that slights either the side of the object (facts) or the side of the conscious subject is bound to be inadequate, I should say.

Kant definitely slighted the side of the object. But consider the following statement attributed to Kant:

“Only artificial or scientific logic [not natural or popular logic] deserves this name [logic], then, as a science of the necessary and universal rules of thought, which can and must be cognized a priori, independently of the natural use of the understanding and of reason in concreto, although these rules can first be found only through observation of that natural use.”

This statement is in the Introduction of what we know as the Jäsche Logic. It was issued, in German, in 1800. Kant died in 1804. It was not written by Kant nor reviewed and approved by him. He had approved, however, this project of creating a manual for lecturers in logic based on his notes used for his own lectures, aiming presumably for what was being used from the notes by Kant in his lectures late in his career. That means lectures for logic consonant with Kant’s mature, Critical philosophy, which had been inaugurated in the Critique of Pure Reason in 1781. Kant’s approval of the creation of a manual from his own lecture notes had been awarded to one of Kant’s students Gottlob Benjamin Jäsche. The entire manual has been translated into English and included in the volume Kant’s Lectures on Logic (in the Cambridge series translating all of Kant’s works) in 1992 by J. Michael Young.

The Introduction of this manual had been translated into English in 1885 by Thomas Kingsmill Abbott. This is the source, the only source, Peikoff 1964 quotes as Kant’s own words, in translation, on the subject of logic. Peikoff gives the impression that, and I expect he thought that, this is Kant’s own writing. The parts he and we are concerned with likely are close to what was stated by Kant in his lectures. At least I find no contradiction with the rather detailed student notes known to us as the Vienna Logic, which are thought to be from the early 1780’s.

Today we have the advantage of all the superb translations of Kant’s works and of students’ Kant lecture notes into English through the Cambridge project (and translation of Critique of Pure Reason and Critique of Judgment by Werner Pluhar as well). Until recent decades, student lecture notes had no role in the understanding of Kant and no part in the influence of Kant, whether in German or English, since those notes were simply not generally available. The Jäsche Logic was put about in German under the title (here translated) Immanuel Kant’s Logic – A Manual for Lectures. https://archive.org/details/immanuelkantslo00jsgoog Across the nineteenth century and to the present, that has been available to German readers. It includes the passage I quoted from it. Abbott’s translation omits that passage. The Kant view available to German readers “. . . although these rules can first be found only through observation of that natural use” is concealed by Abbott to the English reader, such as Peikoff at mid-twentieth century.

The important sentence I quoted that is missing from Abbott should have appeared at the bottom of his page 7. (It appears at the bottom of page 12 in the German original.) Prior to that point, Abbott was giving a meticulous rendition of the introductory part of the work known in German as Immanuel Kant’s Logic – A Manual for Lectures. Where Abbott has the term knowledge or its variants, Young has cognition and its variants. Where Abbott has ideas, Young has representations. Where Abbott has semblance, Young has illusion (in characterizing the target of the dialectical logic, which is complementary to our concern here which is known in Kant and others as analytic logic). Those three differences are minor for our pursuit of what is Jäsche’s representation of Kant’s views on logic. The concurrence on substance in the two translations is considerable.

Peikoff quotes this much from page 2 of Abbott’s translation concerning Kant’s views on how we discover laws of logic:

“[We] set aside all knowledge that we can only borrow from objects, and reflect simply on the exercise of the understanding in general, [and] then we discover those rules which are absolutely necessary, and independently of any particular objects of thought, because without them we cannot think at all. These rules, accordingly, can be discerned a priori, that is, independently of all experience, because they contain merely the conditions of the use of the understanding in general, whether pure or empirical, without distinction of its objects. . . . The science, therefore, which contains these universal and necessary laws is simply a science of the form of thought.” (Cf. KrV A52–55 B76–79)

There is a sentence at Peikoff’s elision points, and there is one more sentence in this paragraph after the final sentence he quotes here. Starting at the elision, we read as follows:

“Hence, also, it follows that the universal and necessary laws of thought can only be concerned with its form, not in anywise with its matter. The science, therefore, which contains these universal and necessary laws is simply a science of the form of thought. And we can form a conception of the possibility of such a science, just as a universal grammar which contains nothing beyond the mere form of language, without words, which belong to the matter of language.”

That last sentence gives us some idea of what Kant means by saying that reflection on the exercise of the understanding enables us to discern absolutely necessary rules of our thought such as the constraint against contradictions. This reflection, then, is Kant’s replacement for Aristotle’s ‘intuitive induction’. Before school age, we follow elementary grammar in speaking our native language. We conform to that language’s grammar a good deal, and it has become habitual. We learn expressly what grammatical forms we are following and should be following from grammar school (after we have learned to write). Some earlier humans had to have reflected on the language, such as Latin or German, to have discovered its grammar. Kant’s analogy on the use, express statement, and normativity of grammar with the use, express statement, and normativity of logic that Jäsche and Abbott here publicize is corroborated as standard in Kant’s lectures on logic by student notes, the Bloomberg (early 1770’s), the Dohna-Wundlacken (1792), and the Vienna. The D-W notes indicate that because logic must contain a priori principles, “logic is a science and grammar is not, because its rules are contingent” (page 432 in Young 1992). I should mention that in Kant’s various remarks on logic, talk of the necessary v. the contingent is shorthand for (what is earlier stated as) the absolutely necessary v. the contingently necessary.

Kant penned an incomplete monograph (published after his death in 1804) What Real Progress Has Metaphysics Made in Germany Since the Time of Leibniz and Wolff? Therein he writes: “As grammar is the resolution of a speech-form into it’s elementary rules, and logic a resolution of the form of thought, so ontology is a resolution of knowledge into the concepts that lie a priori in the understanding, and have their use in experience . . . .” (page 354 of Henry Allison translation in Cambridge’s Kant Theoretical Philosophy after 1781). In Prolegomena to Any Future Metaphysics that Will Be Able to Come Forward as Science (1783), Kant writes of how he discerned those fundamental a priori concepts of the understanding that are used in human intelligibility in experience:

“To pick from ordinary cognition the concepts that are not based on any particular experience and yet are present in all cognition from experience (for which they constitute as it were the mere form of connection) required no greater reflection or more insight than to cull from a language rules for the actual use of words in general, and so to compile the elements for a grammar (and in fact both investigations are very closely related to one another) without, for all that, being able to give a reason why any given language should have precisely this and no other formal constitution, and still less why precisely so many, neither more nor fewer, of such formal determinations of the language can be found at all.” (ibid. 115, translator Gary Hatfield)

To be continued.

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PNC Ground Shifts to the Side of the Subject – Kant II

I mentioned that Kant’s own logic lecture notes compiled by Jäsche were always available to German readers from 1800. We have seen that Kant therein, in his Introduction to the discipline of logic, made an analogy between logic and grammar. (I see now that Capozzi and Roncaglia have also drawn attention to this analogy in the third chapter, p. 143, of The Development of Modern Logic [2009, L. Haaparanta, editor].) Logic is the form of thought, with contents of thought its matter; as grammar is the form of language, with particular words its matter. A book of Kant’s in 1798 includes his view on the relation between thought and language. That book is Anthropology from a Pragmatic Point of View, which was always available in German, but did not come into English translations (two) until the decade after Peikoff’s dissertation. From the Anthropology in a third translation, the Cambridge translation (2007) by Robert Louden:

“All language is a signification of thought and, on the other hand, the best way of signifying thought is through language, the greatest instrument for understanding ourselves and others. Thinking is speaking with oneself . . . consequently it is also listening to oneself inwardly (by means of the reproductive power of the imagination). . . . Those who can speak and hear do not always understand themselves or others, and it is due to the lack of the faculty of signification, or its faculty use (when signs are taken for things, and vice versa), that, especially in matters of reason, human beings who are united in language are as distant as heaven from earth in concepts.” (300)

Peikoff in crafting his dissertation did not have, in English, this Kant passage on the close relationship of language to thought. We’ve seen he also did not have available that paragraph missing (typesetting?) from the Abbott translation of the Jäsche Logic. That is the paragraph in which Kant maintained that the universal and a priori rules of thought, that is, the rules of logic, could only be found in observation of their natural use in particular cases of reasoning. Peikoff had available in English Kant’s analogy between how logic is discovered and how grammar is discovered. This analogy is mentioned, as we have seen, in the Abbott translation of the Jäsche Logic Peikoff used. As we have seen, the parallel of grammar-logic discovery is set in further parallel, in Kant’s Prolegomena, to how fundamental categories of the understanding (necessary factors in making percepts [“appearances”] in experience into that experience) are discovered. Peikoff elected not to address these passages indicating Kant’s notion of the reflective act by which one could (mainly Aristotle, who did) originally discover the rules of logic together with their character of absolute necessity and normativity.

Peikoff rightly observes that Kant cannot draw forth logical, universally necessary principles from the mind as flat empirical generalizations of the mind’s operations. Locke’s idea we’ve put off the table, the idea that among our sensory perceptions of physical necessities there are straight perceptions of instances of PNC in the world. Also off would be any indirect discernment of PNC (i) in the constitution of the world or (ii) in the constitution of the mind by the method of empirical generalization. We must conform to rules of elementary logic in all right thinking, including in right empirical generalization of mental operations. Kant quite agreed, and Peikoff addresses (180–81) Kant’s conviction on this point. (Not that Kant denigrates the senses in the way of Plato or the Rationalists, but in each area of his philosophy, it is plain since I first began to study him fifty years ago that Kant sings the imperial purple of the a priori, whether synthetic or analytic, in comparison to empirical generalization.) The corresponding point for the logical ontologist is stated by Aristotle (in the course of arguing a different issue): “It is a wrong assumption to suppose universally that we have an adequate first principle in virtue of the fact that something always is so or always happens so” (Phy. 252a2–3). Aristotle’s account of coming to know PNC by an intuitive induction, not by empirical generalization, was quite opaque, not very illuminating. Kant is facing the same problem in resting PNC simply in the constitution of the mind and then trying to explain how we come to know the principle is an absolutely necessary one. And a normative one.

Peikoff notices subsequent Kantians’ return nevertheless to empirical psychology for grounding PNC in the constitution of mind. Peikoff exhibits such a move in Henry Mansel’s Prolegomena Logica https://archive.org/details/prolegomenalogic00mansrich (1860). Concerning this work, I’ll mention that Prof. Mansel should report to the Bureau of Transcendental Licensing and turn in his card. C. S. Peirce 1864, which I mentioned at the end of the thread “Peikoff Dissertation Prep,” was mistaken in its assessment that in the Prolegomena Logica “the Kantian conception of logic is developed in the most consistent and beautiful manner.” Mansel’s philosophy surrounding logic is in a manner mildly more realist than Kant and by that it is more pleasing to Peirce. It is indebted to Kant, but it leaves behind Kant’s concept of the noumenal self, Kant’s notion of form (distinct from Aristotle’s), and Kant’s formal and transcendental idealism. Mansel’s idealism, which he represents as under the sway of Kant’s, is as much or more under the sway of Berkeley’s. Mansel is more Humean than Kantian concerning the character of physical laws, such as Kant had exhibited in Metaphysical Foundations of Natural Science (1785). Mansel read Kant in German, including the entirety of Jäsche‘s Logic, and in German he read also successors of Kant concerning the character of logic, such as Wilhelm Krug and Jakob Fries. In footnote after footnote, Mansel specifies his deviations from Kant on the nature of logic. Mansel 1860 misses or declines taking up Kant’s lead to the absolute necessity-but-normativity of logical rules by parallel with the contingent necessity-but-normativity of grammar (65–67, 79–81, 92–97, 135–45, 151–63, 172–80, 192–96, 201–4, 208–9, 225–26, 246–48, 263–69, 278–80, 286–94, 356–59).

A logical ontologist at least has no great problem explaining how one can fail to conform to PNC. The absolute necessity of this rule for thinking comes from the total absence of contradictions in reality together with the mind’s ability to fail in its effort to always keep out contradictions within and among all its pictures of reality. To be entirely true so far as one has gotten a comprehension of reality, when contradictions are found in one’s comprehension, the comprehension must be revised. Kant has trouble explaining how the rules of logic take their absolute necessity from law of the mind’s operation, yet the rules are guides for right thinking, rules that the mind can violate. Peikoff 1964 points out (183–86) that Kant notes this difficulty in his lectures as shown in Jäsche’s Logic. How does Kant try to solve this problem?

To be continued.

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PNC Ground Shifts to the Side of the Subject – Kant III

I’d like to pause, before answering that question, to mention that a reasonably complete theory of the rules of elementary formal logic, how we come to know them and their character of absolute necessity, and how it is possible to violate them would need to cover not only PNC. It should include in its scope also the fallacies of affirming the consequent (AC) and the fallacy of denying the antecedent (DA). (“If someone recently sat in this chair, then it will be warm; and it is warm, so someone recently sat in this chair.” “If it’s raining, then the sky is not entirely clear; but it’s not raining, so the sky is entirely clear.”) All adult interlocutors, however meager their formal education, know in practice that they should not violate PNC in their reasoning. But the unschooled seem oblivious to those other rules for their right reasoning and keeping to reality. This is especially so when the reasoning is not about such concrete matters as in my examples, but about more abstract matters as come up in disputations in politics and religion in which they mainly want the conclusion and are not keenly interested in whether a particular reasoning to it is valid. They generally take care to avoid PNC even in heated argument. Its invalidating character is ever close with them, and they know it’s ever close to all the participants or observers.

I suggest that PNC is more obviously mandated (than AC and DA are mandated) by the metaphysical principle of identity as to the which and the what in reality and that PNC is more obviously required for keeping hold of those identities and for communication concerning them. Although avoiding the fallacies of AC and DA also rests on those aims and on those aspects of identity in reality, they are less primitive and less fundamental for discursive cognition than the logical principle of noncontradiction. Nevertheless, the principles of barring AC and barring DA have the same absolute, perfectly general necessity and normativity as PNC.

Having acknowledged the tension between having logical principles such as PNC be at once absolutely necessary laws of human mind, yet crossable by that mind, Kant in the Jäsche Logic addresses how such error is possible. The faculty of understanding would make no errors were its judgments never under illusions it forms in its commerce with the faculties of sense (also KrV A293–94 B350–51). The sensory inputs themselves are not erroneous, for only judgments can be true or false. Kant is in keeping with Descartes’ view that errors all arise from allowing our will to outrun our understanding. We alone are responsible for all our errors.

That analysis of error is fine for a wide class of errors, but not, I say, for the class into which contradictions fall. Formal contradictions are judgment against judgment, and the rather obvious sources of contradictions in one’s judgments are limitations of memory and not drawing out all the implications of one’s various judgments. The latter source can range from evasion to plain economics of mental reflections in the course of a human day or life. Like most any philosopher before him, Kant can dig into our motives for the willful portions of such errors. He cannot explain and seems reluctant to admit the existence of one’s contradictions not willful.

Might Kant’s analogy help here, his analogy between logic and grammar, each discovered and become explicit by reflection on their natural employment, consisting of rules descriptively necessary yet normative? No. The problem is that when Kant speaks of the necessity of the rules of grammar being contingent rather than absolutely necessary, he does not mean that rules of grammar are probabilistic rules. He means they might have been otherwise, and that makes the analogy converge on congruence in the crucial respect. The grammar is as necessary within a language or range of languages as PNC is necessary in any possible setting. He cannot explain (or even acknowledge?) an error of grammar not willful any more than he can explain a contradiction not willful.

Peikoff 1964 does not attempt to delve into these various doctrines of Kant concerning the character and sources of error. He takes it, like some other contemporary philosophers, that one cannot succeed in holding onto the absolutism of logical rules while saying also that we can violate them and that they are due only to the constitution of the mind. So far, my mining of Kant on error confirms that estimation of Kant’s effort on his conundrum.

What about the kind of error Kant mentioned in the Anthropology in the preceding post? That was the error of mistaking linguistic signs for things they signify and vice versa. Such signs, Kant calls artificial, in contrast to natural indicators such as smoke for fire. Kant observed that people having common language can yet signify in their vocabulary concepts quite different one person to the next. He implies that this variance is due to infirmities in the faculty of signification, which rather suggests that if we were all working correctly in our linguistic significations, we should have no variance among persons in concepts signified by a word. I seriously doubt that, given the variance in individual backgrounds of experience and education and given the creativity in thought, especially in more abstract thought. Were Kant’s rigid connection between vocabulary and right concept correct, infirmity of word-concept powers would yet not explain how errors of logic or grammar are possible. The same goes under my denial of the word-concept complete rigidity of right signification, for then there is utter incommensurability between the would-be explanation and the thing to be explained, since the rules of logic and grammar are fixed, in Kant’s view, in all the heads talking and thinking to themselves and with others. Error of signification and its source (source pretty vague in Kant) does not help to explain error in logic or grammar.

The sort of error to which Kant draws attention most famously is the one that is mood lighting for his Critical philosophy. That is the error of letting reason run off into speculations about things as they are in themselves, things as they are beyond the bounds of possible experience. Kant’s advertisement for his critique of reason by reason is that all fundamental contradictory positions on metaphysical questions before his 1781 are resolvable once we realize that opposing answers are addressing the question in different senses. One side is addressing a question about a thing as it is in itself; the other side, as that thing is an object of possible experience (A395, Bxxvii). This error is an extrapolation from the kind of error Descartes and others had cautioned against: making judgments on things for which we are not in a position to judge. Rather, we should withhold judgment and not let our will outrun our understanding. Kant’s casting as error reason overstepping the so-called phenomenal district, reason stepping into the so-called noumenal district, relies on correctness of PNC. This overstepping error, Kant’s sweetheart, provides no help to resolving his problem of how absolute necessity of PNC is on account of the way the mind operates yet that mind is able to commit contradictions.

So I concur with the conclusion of Peikoff and others he cites that once Kant had the constitution of the subject the sole source of the purely formal and purely a priori, he was not able to stably maintain an absolute necessity of PNC and other principles of logic together with their normativity, which latter entails our ability to not adhere to such principles. I add that this same irresolvable mess arises for every other sort of cognition purely formal and purely a priori, whether analytic or synthetic, once Kant has squarely located their source purely in the constitution of mind, in its fundamental dynamics, not at all in the constitution of the world.

(In the next installment, I’ll cover Peikoff’s story of the shift of PNC ground to the side of the subject beyond Kant and the role of Kant in that further development to 1964. I’ll assess his account of Kant’s role and carry the story of the ground shift away from logical ontologism to the present.)

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26 February 2019

PNC Ground Shifts to Side of the Subject – On to Conventionalism

Peikoff laid out the varieties of logical ontologism, these being the various ways in which it had been thought that principles of logic, such as PNC, are general ways the world is. The principle of noncontradiction (PNC) is a guide for us to adhere to in thinking, and under ontology-based theories of logic—Plato to Leibniz—PNC is right to follow in one’s thought because it is a fact of the world independently of human mind, an everywhere fact of the world. If our aim in thought is grasp of the world, PNC is a fact of the world we must hold onto for success in that aim.

Let us notice that in claiming (not-A and B ) both the not and the and can be in the world. An object not having support and that object’s falling to the floor is a fact of the world. If our aim is keeping objects from falling, we must see to it that they are adequately supported. For that A and B, in usual household life, do not (not-A and B). In the formula (not-A and A), we move to what Peikoff called a formal aspect of the world. Logical ontologism would have it: do not (not-A and A) in thought if our pursuit is getting and keeping a grip on the world, because [not (not-A and A)] is an everywhere formal fact of the world, a necessity given, regardless of our aims.

I had written in “Aristotle I” that for my own part I thought that, notwithstanding its objectivity, PNC has some dependency on thought which 2R (the fact that the angles of a triangle in the Euclidean plane sum to two right angles) does not have. That was because cases of noncontradiction run arbitrarily far afield, as far as our free imaginations: a five-fingered hand is not an opera, and so forth for anything at all not a five-fingered hand. That composition was nearly two years ago, and I’ve changed my mind. PNC is not partly dependent at root on operations of mind, notwithstanding PNC’s unlimited scenes of pertinence.

Peikoff’s 1964 dissertation talk of formal aspects to the empirical world, without embrace of Aristotelian or Kantian schemes of form, is talk and conception that has proven valuable to me in development of my own metaphysics in my book in progress. Independently of Peikoff 1964, it is also talk and conception now taking hold and developing in philosophy of logic and mathematics by Gila Sher, as in Epistemic Friction (2016). As of mid-20th Century, however, as Peikoff 1964 observed, principles of logic in modern philosophy had become sourced no longer in the world, but in the subject.

To be continued.

Edited by Boydstun

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20 March 2019

PNC Ground Shifts to the Side of the Subject – Conventionalism I

Peikoff addressed logical conventionalism in a sense broad enough to include the various approaches to logical truth within what he took to be the most influential movements of Anglo-American philosophy in the twentieth century to the time of his dissertation (1964, 165n). Those would be pragmatism, logical empiricism, and the analytic movement. For exemplification of philosophies upholding conventionalism in fundamental character of logical truths, Peikoff delves into Dewey’s Logic: The Theory of Inquiry (1938); C. I. Lewis’ Mind and the World Order (1929) and An Analysis of Knowledge and Evaluation (1946); A. J. Ayer’s Language, Truth, and Logic (1946 [1936]); and E. Nagel’s Logic without Metaphysics (1956).

There had been an analogue of conventionalism in logical and mathematical principles within a minority of earlier thinkers who, wanting to guard doctrine of the omnipotence of God and taking the truth and necessity of formal principles to emanate from divine selection of them, endowed formal principles with an ultimate arbitrariness. Those principles would be perfectly unchanging, however, as far as human thought is concerned. Anyway, such an ultimate situation was from a brew of theology with an extra-heavy dose of vacuum imagination.

The conventionalisms Peikoff addressed took shape and took hold in part and in some sense due to inadequacies of the various forms of logical ontologism that had been on offer (186–87, 210–11, 235–36, 239). The rejection of those logical ontologisms was reasonable, as Peikoff illustrated, even if we consider them only from within the discipline of classical philosophy from Plato to Kant.

The conventionalist replacements for logical ontologism were presaged by (i) nominalist strands in epistemology,* and (ii) substantial additions to logic itself and to geometry in the latter part of the nineteenth century. To those two in the vista of Peikoff 1964, I should add (iii) the spectacular empirical success—from Maxwell to Einstein (GR) and Schrodinger (wave QM) / Heisenberg (matrix QM)—won by casting physical relations in terms of portions of modern analytic mathematics. In my own assessment, it is (ii) and (iii), against the background of (i) and inadequacy of old-time logical ontologism that are the main and sufficient preparations for the crop of conventionalist characterizations of logical truth by logical empiricists to mid-twentieth century.

“My concern has not, of course, been to maintain a primarily causal thesis; it has not been my intention to argue, for instance, that Cudworth’s difficulties with God or Locke’s problems with Aristotle’s forms were the causal factors centrally responsible for the dominance in our century of the conventionalist approach to logical truth. Such a thesis would hardly be tenable; the creation of non-Euclidean geometries, to cite just one example, was undoubtedly more influential in this connection than the sort of difficulties I have discussed. . . . My concern has been rather, by considering a few aspects of the question, to suggest that, as a matter of fact, the seeds of conventionalism were implicitly present in the formulations of the classical logical ontologists, and that there was a logic to this presence.” (Peikoff 1964, 240)

Peikoff took Kant to be “the philosopher most responsible for the demise of logical ontologism in the history of philosophy” (165). In a roundabout way, I concur. The demise of ontological essences, Platonic forms, and Aristotelian forms and formal causes had transpired before Kant, as far as the modern stream of philosophy was concerned. In that fall was also the fall of logical ontologism. Kant’s weight on the demise was through his own imposing, positive system of theoretical philosophy replacing Aristotle’s (and replacing modern systems such as the system of Leibniz). Kant weight on that demise and Kant shadows on the future saliently include: His success in bringing to much attention a philosophical division of sense and understanding and of the synthetic and the analytic (not what we mean by synthetic/analytic in geometry); his subject-rooted theory of how geometry is possible; his replacement of Aristotle’s categories as in the world with categories as belonging to human understanding in its approach to sensory experience of the world; lastly and most profoundly, in my estimation, his particular replacement of Aristotelian ontological form with subject-side form.**

“Although many—but not all—classic philosophers subscribed to the necessary-contingent or rational-empirical dichotomies in their classification of propositions, this was not for them the equivalent of the logical-factual dichotomy; to this latter the vast majority did not subscribe, nor could they have, since they regarded the laws of logic as themselves matters of fact (i.e., ontological in character, not ‘mere’ matters of fact).” (13)

The philosophers Peikoff examines in their conventionality of logical principles do not regard these salutary principles as arbitrarily selected, although, as with Kant, their basis is not some fact(s) holding independently of human existence and consciousness. Peikoff quotes logical empiricist Ayer denying that analytic propositions “provide any information about any matter of fact. In other words, they are entirely devoid of factual content” (Ayer 79; Peikoff 170). Ayer does not follow Kant’s proposal that analytic patterns are from invariant organization of the human mind. Rather, granted various linguistic conventions, “the laws of logic which flow from them are necessary and incontestable truths” (Peikoff 174).***

From what we have seen of Kant in previous segments of this essay, he would take conditioning truth and necessity of logical principles such as PNC on any conventional structure of language as inadequate to deliver the necessity logical truths possess. Any indebtedness of logic to structure of language cannot be indebtedness to anything conventional in language, because conventionality is contingency, not absolute necessity, not the necessity Kant attached to the a priori.

Ayer and Kant agree that logical truths are a priori and analytic. An example of an analytic statement from Ayer is: “Either some ants are parasitic or none are” (79). “Either some are or none are” is so no matter what facts of the world it is being applied to. And no observation in experience could refute this logical truth. On that much Kant and Ayer could agree.

Kant famously did not think that analytic truths are the only sort of a priori truths. The other sort of a priori truths, he called synthetic. He took the analytic and the synthetic to exhaust the sorts of a priori truth. The exemplar of synthetic a priori truths was geometry of Euclidean space, the only structure of physical space known until Einstein’s general relativity (1916). That space (spatial slices of the four-dimensional [semi-] Riemannian spacetime manifold of variable curvature) is not intuitive in the ready-to-hand way that Euclidean space is intuitive (in carpentry or, differently, in Euclid), thus not congenial to Kant’s conception of space as pure form of outer, sensory intuition (in Kant’s technical sense of intuition). Logical empiricism arose in an intellectual scene in which Kant’s exemplar of synthetic a priori truth lay in shambles. Moreover, by that time, David Hilbert had staked pure geometry as purely abstract and independent of any physical application or sensory experience.

Hans Reichenbach in 1920 correctly observed that Kant had held a priori truth to be not only necessary and unrevisable, but constitutive of the concept of the object as object of knowledge. That last character of the a priori was not toppled by Einstein’s revolution. At least it was not toppled in the obvious way that universality and unrevisability were toppled with respect to physical Euclidean geometry. At first Reichenbach thought such constitutive principles were at hand in modern application of mathematics to physics, but he soon became persuaded that those principles of application were neither true nor false. They were simplicity- and tractability-based conventions. By the 1920’s, the last toeholds of Kant’s intuitive, synthetic a priori in geometry and in geometry’s physical exemplification had been dissolved by Reichenbach, Schlick, and Carnap.****

To be continued.


* In the installment “Aristotle II”, I conveyed Peikoff 1964 on inadequacies of nominalism in provisioning a theory of logic. See also Paul Forster’s Peirce and the Threat of Nominalism (2011, 24–28, 38).

** On transformation of Aristotle’s matter-form distinction from Leibniz to Kant, see Marco Sgarbi’s Kant and Aristotle – Epistemology, Logic, and Method (2016, 79–94).

*** Cf. Herder c.1767 and 1772 in Michael Forster’s Johann Gottfried von Herder – Philosophical Writings (2002, 48, 100).

**** Michael Friedman’s “The Evolution of the A Priori in Logical Empiricism” in The Cambridge Companion to Logical Empiricism, Richardson and Uebel, editors (2007, 95–108).

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(I don't know the original date of this post.)


I’ll hold off remarking on pragmatism until we get to Dewey and Lewis.

Concerning the classical ontologists, “they regarded the laws of logic as themselves matters of fact (i.e. ontological in character, not ‘mere’ matters of fact)” (Peikoff 1964, 13).

The classical philosophers basing logic in ontology (such as Plato, Aristotle, Aquinas, and Leibniz) would want to have PNC both as an ontological fact of the world and as a norm, a consciously followed constraint, for ascertaining any fact, whether itself or other facts, whether facts empirical or mathematical. With the variations in ontology between various theories basing logic in ontology are variations in what is ontological form. I think it is always what philosophers say about the ontology of form that is key to their ontology of PNC and their account of how PNC is also a norm.

Below is Peikoff’s representation of Aristotle’s ontology at work in a syllogistic inference. I should like to mention that this text is my personal favorite in Peikoff’s dissertation. Also, I’d like to mention that, as Jonathan Lear showed from Prior Analytics, the certitude of the validity of the syllogism below, and the other first-figure ones, is the base certitude of validity by which Aristotle, using some self-evident logical conversions, certified validity of the syllogisms of the other figures. Lastly, in their lectures and writings concerted with Rand; Branden and Peikoff point to contradictions that occur if one denies the conclusion of this syllogism below while affirming its premises. It is a good assignment for the future to work out the moments of Aristotelian form in rendering those contradictions.



All A is B. [major premise]

This is an A. [minor premise]

Therefore, this A is B.

The major premise here represents a necessary connection learned on the level of apprehension of a separated formal structure; the minor premise represents the recognition of that structure in (or, as the structure of) a given particular stuff or material. (And the conclusion, derived from combining the two premises, represents our recognition that any property necessarily attaching to the “separated” structure will thus necessarily attach to the particular thing in question which possesses that structure.) Each premise then—and this is the key point for our present purposes—represents an apprehension of particulars, but in quite different ways. The difference is difficult to state simply; here are some possible formulations of it: the major represents an apprehension of the structure of particulars, the minor, of the structure in particulars; the major, of the particular qua possessed of a certain formal character, the minor, of a the particular qua particular; the major, of an aspect, feature or element of the particular, the minor of the particular itself; the major, of form, the minor, of form in matter.*

* This apprehension of form in matter, which follows the apprehension of the separated structure, is to be distinguished from a quite different apprehension of the form-matter amalgam which precedes the apprehension of separated structure, viz., the initial sensory perceptions of particulars qua particulars which occur as the prerequisite of the performance of the abstracting process. There are actually three stages in the process for Aristotle: a) Sense-apprehension of the particular qua particular prior to any abstracting process. This is an undiscriminating apprehension of the form-matter amalgam as a whole, and thus “accidentally” of the formal element of the particular . . . . b) At some point, after repeated experiences and memorial retention, we come to discriminate the form from the matter and “separate it out” in thought; we are then able, by contemplating the separated structure, to apprehend the necessary connections among its features. This is the level of rational cognition. c) Finally, we return to the particular and reintegrate the form with the matter, once again, as in stage a, perceiving the form-matter amalgam as a whole. Only we are now able to apprehend the form in the matter; i.e., to apprehend it as a distinctly discerned structure in this particular stuff. The perception of form in matter at this stage is thus authentic, not accidental; and we are thus able to apply to the particular the knowledge of necessary connection gained in stage b.

(Peikoff 1964, 134–35)


Under Aristotle’s account, we learn the truth of PNC by observing instances of it and performing an intuitive induction to it (also called an abstractive induction). PNC has to be a law prior to the operation of thought in order to be discovered by such observation and abstraction. The normativity of PNC in Aristotle’s account is from the purpose of thought, which is the comprehension of existence. To serve as guide to that purpose in the way PNC serves, PNC must, in Aristotle’s view, be a first principle in existence. We must not think a thing has and has not a certain character at the same time because, as Joseph puts it, “we see that a thing cannot have and not have at once the same character; and the so-called necessity of thought is really the apprehension of a necessity in the being of things’” (Peikoff 1964, 162).

I’ll be looking at Dewey’s expansive notion of logic in turn when we come to it in this series. Looking also at Lewis and at Peikoff’s extractions from both of them. I don’t expect to take up Wittgenstein, and Peikoff also did not. But I thought I’d mention just now a book from Penelope Maddy The Logical Must – Wittgenstein on Logic (2014).

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14 April 2019

PNC Ground Shifts to the Side of the Subject – Conventionalism II

Logical empiricists rejected Kant’s synthetic a priori as a class of propositions, and they rejected as well Kant’s role of intuition in arithmetic and geometry. All a priori propositions were analytic for these twentieth century philosophers. Having taken that position, they took some modern philosophers before Kant, such as Leibniz or Hume, to have been on the right track; they saw Kant as a derailment. I should note that none of these twentieth century philosophers giving a significant nod to convention in logical principles, such as PNC, were epistemological skeptics. They found attractive Hume’s wall between abstract reasoning and matters of fact, which was similar to epistemological dipoles of their own. They could applaud his clipping wings of metaphysics. They treasured mathematics and modern empirical science, and they did not give an inch to religion.

L. E. J. Brouwer threw a Kantian intuitionist spanner into the logical empiricist program in the late 1920’s. He formulated an intuitionist, constructivist, and finitary conception of mathematics which implied the invalidity of a significant portion of classical mathematics that had been developed by that time. Carnap tried to legitimate in broad perspective both the Brouwer system and classical mathematics by characterizing them as devoid of factual content and by leaving them to win the day according to which could best serve the formal deductive needs of empirical science.[1]

The selection between Brouwer’s intuitionist mathematics and classical mathematics for the boosting of empirical science is not arbitrary. What is better suited or best suited to some end, such as boosting empirical science, is a matter of objective fact. The following, I notice, share nothing with the arbitrariness that enters convention (just as the need for a left-or-right-side driving rule shares nothing with the arbitrariness that enters the choice of which side): what is more rather than less convenient, what is more rather than less simple, and what is more rather than less practical. One slip into a subject-siding error in this neighborhood would be to say that because a model or a theory is more convenient or simple or practical, it is more likely to be true (or, stepping with Protagoras, it is what truth is).

Within pure, unapplied geometry, it is false to say there is no correctness or incorrectness concerning hyperbolic geometry, elliptical geometry, and Euclidean geometry; all three are true within the discipline of pure mathematics. The impulse to consider these geometries as somehow not only distinct from, but as opposed to each other within pure geometry is wrong thinking. There is some truth to the conclusion of Poincaré and, later, the logical empiricists that question of which, between hyperbolic, elliptical, or Euclidean, within pure geometry is true (concerning their differences, not their commonalities) is a meaningless question. The context, I say, that makes such a question meaningless is a context in which there are facts, albeit facts not empirical. So Carnap was wrong to say additionally that purely formal disciplines and their systems are themselves devoid of factual content. It is misleading to confine usage of fact to empirical fact, just as it would be misleading to confine circumstance or form to empirical circumstance or empirical form.

(My own view is that there are formal lays of the physical and ways of ours with the physical that are not empirical lays of the physical and not our empirical ways with the physical. Specifics are reserved for my book in progress.)

Henri Poincaré died in 1912. He lived long enough to assimilate modern geometry and special relativity, and the Minkowskian geometric character of SR spacetime into his epistemological views on physics and on mathematics. Poincaré did not live to see the advent of general relativity (1915), with its condensations of the principle of inertia into spacetime geometry and gravitational force into inertial force, its spacetime structure affecting motions of mass-energy, and its distribution of mass-energy dictating spacetime structure, all at play in one super-fertile physical equation, Einstein’s field equation. Poincaré had over-extended the role of convention in both physical theory (kinematics and dynamics, including SR) and modern geometry. Those over-extensions have been soundly refuted, even without setting that much physics and physical geometry within general relativity. Those repudiations aside, general relativity was utterly devastating to the roles Poincaré had purported for conventions in physics and physical geometry.[2] The logical empiricists sometimes situated conventions in logical truths in ways self-consciously similar to ways Poincaré had (mistakenly) thought he found sturdy niches for convention in physics and geometry.

That does not mean that every such mimic of Poincaré mistaken. I should say only beware, for separating what is conventional and what is not is not always easy, within one’s present context of knowledge. But the more important point I want to make for our present examination of possible connection of PNC ontological and epistemological character from Kant to conventionalism is that Poincaré held a generalized version of Kant’s synthetic a priori status for arithmetic, geometry, and fundamental mechanics. This generalized version with its niches for convention possessed those niches only due to advances in mathematics and science since the time of Kant. No shifting of ontology to the side of the subject nor deflation of ontology by Kant, in his specific ways, seems to be required for Poincaré to have made his conventionalist moves. And the logical empiricists made their conventionalist moves on logical truths, including PNC, without any reliance on, indeed in flat denial of the Kantian class of the a priori that is also synthetic. Kant’s critical philosophy further sealed the tomb of logical ontologism, but in my assessment thus far, Kant prepared no ground and planted no seeds for the spring of twentieth-century conventionalisms in the character of logic or its applications.

But what about Kant via tributaries from neo-Kantians (viz., Marburg ones) into logical empiricism?

To be continued.

[1] Friedman 2010, 669–76.

[2] Ben-Menahem 2006, 40–68; Friedman 2010, 642–64; Gray 2013, 525–33.


Ben-Menahem, Y. 2006. Conventionalism. Cambridge.

Friedman, M. 2010. Synthetic History Reconsidered. In Discourse on a New Method. M. Domski and M. Dickson, editors. Open Court.

Gray, J. 2013. Henri Poincaré – A Scientific Biography. Princeton.

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23 May 2019

PNC Ground Shifts to the Side of the Subject – Conventionalism III

To set myself the task “weed this patch of periwinkles” I may need to use language.  The two popular weeds there at this season are dog violets and a native vine I don’t know the name of. Getting to the nub of that weed-vine among the thicket of periwinkle vines and pulling out the former without pulling out the latter is a challenge. Names and language do not seem to be enlisted in executing the task; they enable only my report of this work. The weed-vine and the periwinkle are of different leaf shape and color. Tug gently on the end of the weed-vine reaching for the sun. You won’t be able to see the weed-vine you’re tugging but a few inches before it disappears (leafless in this portion of it) among the thicket of periwinkle vines hugging the earth and putting down their roots continually along their way. But as you tug on the weed-vine, you’ll be able to find with your other hand that single vine being tugged. It is tightly tensed and in synchrony with any rhythm of tugs you apply with the other hand. Repeat from there, and eventually you arrive at the nub of the weed-vine and pull out that vine by the root.

Pause at a step in which you have the single obscured weed-vine in each hand. Pull with the one hand, feel the pull in the other. That is a perceived connection between two distinct events. At this point, philosophers from Plato and Aristotle to Hume and Kant stick up their noses. Not Locke.

That applied force can be conveyed along a vine is a physical necessity. That different things in general (as example, weed-vines and periwinkle vines) are not same things is another type of necessity, logical necessity, however neatly it coincides with physical necessity. Logical necessity holds unconditionally and in all contexts. What I’ve called physical necessity is traditionally taken to be necessity under some sort of limiting conditions, and this necessity has been called a contingent connection, reserving necessary connection for logical (and other formal) necessity. The real distinction, I think contrariwise, should be in what aspects of things we are accessing and the different ways these two aspects are accessed.

Peikoff 1964 points out that Locke avoided the contingent/necessary terminology. Locke instead applied probable/certain to the division. We have seen in my section Aristotle II that Locke maintained we have by sensory perception instances of the general fact that different things are not same things and that a thing is never both A and not A at the same time and in the same respect. Philosophers, including Peikoff in 1964, are correct to fault Locke’s blurring under probable/certain a clear understanding that ampliative inductive generalizations over perceived instances do not suffice to land the absolute necessity in general principles of logic or pure mathematics. Peikoff notes on page 218 the parallel criticism in Hume’s famous dictum that we do not find in sense perception any necessary connection between distinct events (distinct impressions, in Hume’s own parlance and perspective). Countering Hume’s quandary, Kant attempted a radical subject-sided formulation of necessities such as the necessity in a principle of causality, a reformulation in which Kant would have objective temporal order of distinct events get the necessity of that order from a necessity of causal structure demanded by human mind. (Cf. Peikoff 2012, 32–33.)

Locke had fogged up by his softening of the distinction between (i) the physical necessities one can sense and manipulate with the weed-vine in one’s hands and (ii) formal and metaphysical necessities. Nevertheless, I maintain Locke right in taking (i) to be the driver of (ii) and not the other way around, as philosophers from Plato to Kant and beyond would have it. British empiricism has its good sense even if it was never good enough.

Locke was not really of one mind in this. Peikoff lays out an opposite strand also in An Essay Concerning Human Understanding: IV 3.31, 4.6, 4.8, 9.1, 11.13–14. “What is Locke doing in such passages as these? He is now contrasting eternal truths and existential truths. The former are to be discovered only by ‘the examining of our own ideas’, and ‘concern not existence’ . . .” (222). Peikoff points out that the likes of platonist Cudworth or Leibniz had also maintained such a division, but for them consideration of our own ideas accesses the eternal truths as immutable relations in the divine understanding. Eternal truths such as the laws of identity and noncontradiction, as well as the essences of existing things, are givens to the human mind, independently of our self-examinations accessing them. But for an empiricist such as Locke, rejecting that rationalism, and joining considerable nominalism (the conceptualist wing of nominalism) concerning universal ideas to the empiricism, the divide between matters of fact and the eternal, formal truths can make conventionalism concerning the ground of logic “almost inevitable” (223).

The leading German spokesman for conventionalism in science, geometry, and logic in the early years of the twentieth century was Hugo Dingler: “The application of the law of contradiction rests on my free will . . . and this is just what is called a stipulation [Festsetzung]” (1919, 14-15; quoted in Carus 2007, 120n14). “There is no other way to guarantee the general validity of a law other than its stipulation by the will” (1919, 13; Carus 119). Peikoff would not likely have known much about this history in 1964, much beyond, that is, what Popper wrote against it in his 1934 The Logic of Scientific Discovery. I want to point it out because Dingler rejected as unfounded Kant’s basis of the necessity in geometry as arising from synthetic a priori judgments and Kant’s picture of how certain laws are a priori conditions of the possibility of any experience (Wolters 1988). Dingler is nonetheless a redo of Kant, of the first Critique, with conscious choice (of alleged conventions) replacing Kant’s mandatory structure in any sensory intuition and in any conceptualization of things external to mind. Though crucial, fundamental organization of mind on Dingler’s view is voluntary, and although Kant would shake his head over such free play as that, it remains that the organization is an a priori condition for the possibility of any experience or knowledge.

Carnap will resist such radical conventionalism in the 20’s and 30’s. I’ll return in the next installment to the course of Logical Empiricism and the role of (still overextended) conventionalism in their characterization of logic and in the characterization by Dewey and by C. I. Lewis. I expect to yet dig into the fate of conventionalism concerning logic to the present day. Jumping out of chronological order, just now I want to be sure to mention—to show that conventionalism in logic remains a current and a concern in philosophy today—the section 6.5 “Logical Conventionalism” in Theodore Sider’s Writing the Book of the World (2011 Oxford).


Carus, A. W. 2007. Carnap and Twentieth-Century Thought – Explication as Enlightenment. Cambridge.

Dingler, H. 1919. The Foundations of Physics: Synthetic Principles of Mathematical Natural Philosophy. Union for Scientific Publishing, Berlin and Leipzig. (In German.)

Peikoff, L. 2012. The DIM Hypothesis. New American Library.

Wolters, G. 1988. Hugo Dingler. Science in Context 2(2):359–67.

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30 October 2019

PNC Ground Shifts to the Side of the Subject – Conventionalism IV

Leonard Peikoff’s dissertation sets out a developmental conceptual story across the centuries of philosophy. His pages present the story of the theories put forth for the place, character, and origin of PNC and the points on which those theories become untenable, suggesting replacement theories.

That contradictions cannot obtain in reality is a necessary truth. The existence of necessary truths poses a problem for philosophers.

“One way of putting the problem is as follows: How can man, who has access to only a very limited portion of a universe whose character is independent of what man thinks about it, nevertheless know with certainty that certain truths will obtain throughout all the regions of space and across the whole course of future time? . . . The conventionalists, in principle, find the solution in denying that necessary truths provide information about any world, real or phenomenal, and in construing them instead as expressive of relations between meanings which men themselves have created; i.e. the conventionalists protect necessary truths from the vagaries of an uncertain world, not by claiming a particular insight into the structure of that world nor legislative power over it, but rather by cutting them off from the world and making them independent of what occurs in it.” (Peikoff 1964, 230)

It is only conventionalism that cuts off necessary truths from the world that was or is provocative. The conventionalists Peikoff mentions were as guilty of equivocation in describing their position as “conventionalist” as were their critics. When they pass off the practicality, convenience, or effectiveness of using PNC and other necessary truths in thinking or communication as showing the “conventionalism” of their position, they are being imprecise (possibly for showiness). Ayer is an example of that, and I’ll look at him in a moment and in the next installment. Firstly, I want to point out that when one takes necessary truths to be tools, having (in our supposed choice of them among supposed alternatives) highest practicality, convenience, or effectiveness, one is certainly not cutting them off from the world and making them independent of what occurs in it, even were one to imagine that is what one is doing. Aristotle pointed out that if one wants an artifact to be effective as a saw, making its teeth of iron is the smart choice. Why? Because of the nature of the world!

It is only certain parts of what conventionalists put forth as conventionalism that actually goes to their position of making necessary truths independent of the structure of the world. Any points at which they are speaking of the practicality, convenience, or effectiveness of necessary truths go against, not to, the provocative part in their pronouncements that conventionality makes necessary truths what they are.

At the end of his dissertation, Peikoff sets out some points on which a new sort of ontologism of PNC would need to diverge from previous ones to block the alternative of conventionalism. This is a highly informed set of constraints—shown to be highly informed by the dissertation—to apply for a viable new ontologism, and Peikoff silently knows that Rand’s epistemology to be issued in a couple of years after his dissertation, along with his own addendum (1967) to that epistemology, will be satisfying those constraints.

Peikoff 1964 does not undertake an exposure of the weaknesses in the conventionalists positions he notes as having displaced ontologism to mid-twentieth century. I shall critique those conventionalist approaches. Conventionalisms too, not only previous ontologisms, had their inadequacies, which by now in philosophy have been exposed, opening the area for reformed ontologisms.

For logical empiricist Ayer, the path to an account of logical and mathematical truths and their necessities is the exclusive and exhaustive division of all truths into either empirical ones or analytic ones, where analyticity is conceived in a very thin way. Like logical empiricists Reichenbach, Schlick, and Carnap before him, Ayer rejected Kant’s synthetic class of necessary, a priori truths. The only necessary, a priori truths Ayer acknowledged were analytic ones. Sebastian Rödl observes that this rejection of the existence of synthetic a priori knowledge is of a piece with rejection of the idea of logic as including general forms of right connection of thought to the world (e.g. Kant’s transcendental logic or Rand’s theory of proper concepts and definitions), leaving only right deductive inference (formal calculii) as logic (Rödl 2012, 3, 22–27, 33–39, 43–45).

In the view of Ayer, analytic propositions “are entirely devoid of factual content. And it is for this reason that no experience can confute them.

“When we say that analytic propositions are devoid of factual content, and consequently that they say nothing, we are not suggesting that they are senseless in the way that metaphysical utterances are senseless. For, although they give us no information about any empirical situation they do enlighten us by illustrating the way in which we use certain symbols. Thus if I say, ‘Nothing can be colored in different ways at the same time with respect to the same part of itself’, I am not saying anything about the properties of any actual thing; but I am not talking nonsense. I am expressing an analytic proposition, which records our determination to call a color expanse which differs in quality from a neighboring color expanse a different part of a given thing. In other words, I am simply calling attention to the implications of a certain linguistic usage. Similarly, in saying  that if all Bretons are Frenchmen, and all Frenchmen Europeans, the further statement that all Bretons are Europeans is implicitly contained. And I am thereby indicating the convention which governs our usage of the words ‘if’ and ‘all’.

“We see, then, that there is a sense in which analytic propositions do give us new knowledge. They call attention to linguistic usages, of which we might otherwise not be conscious, and they reveal unsuspected implications in our assertions and beliefs.” (79–80)

What is an analytic proposition (or analytic truth) according to Ayer? What was his definition of analyticity?

Kant had held that all necessary judgments had to be known a priori. What made an analytic judgment a necessary judgment was that its predicate was conceptually contained in its subject. The concept of a body contains the concept that bodies are extended in space. “Bodies are extended” is necessarily so. Kant’s concept of containment is not entirely determinate. He takes “all bodies are extended” as analytic. What about “all smart women are smart”? Is that a variety of analytic containment? If one denies that all smart women are smart, one plainly has contradicted oneself. Kant required analytic statements to be contradictions upon denial, but it is unclear whether that requirement is a fundamental criterion for analyticity or only a variety of his containment criterion for analyticity (Juhl and Loomis 2010, 4–8).

Bolzano tried to formulate a clearer concept of analyticity, one more centered on logic. If a proposition’s truth value remains constant under any substitution of its terms not belonging to logic itself, it is analytic. Also, the the rules of logic are analytic. Frege took up that concept of analyticity, imported it into his logic wider than subject-predicate logic, and argued that, contra Kant, propositions of arithmetic are analytic, not synthetic a priori. Russell identified an important flaw in that Frege program and made innovations to keep the program afloat (ibid., 11–18; further, Burgess 2005, 34–46; Potter 2000).

Logical empiricists such as Ayer were heirs of this logic-centered concept of analyticity and its proposed undergirding of arithmetic. Then too, they took to heart Wittgenstein’s analysis of logical truths (such as PNC) in Tractatus, a work informed by logical ideas of Frege and Russell, but a work crafting “a single, unified relation between language and the world” (Juhl and Loomis 2010, 19). “Unlike Frege, Wittgenstein did not treat logical truths as statements or propositions at all. Rather, he saw such truths as ‘tautologies’ which, while they might show the ‘logical scaffolding of the world’, do not themselves say anything” (ibid.)

“The fact that language, and the world it pictured, possesses certain ‘formal’ features was thought by Wittgenstein to be shown (although not said) in the fact that certain expressions are tautologies. But the Vienna Circle [1926–1938] was dissatisfied with this conception. Wittgenstein’s talk of ‘showing formal properties of the world’ smacked of the metaphysics they, as empiricists, were concerned to avoid. Rather, Circle members (Moritz Schlick and Rudolf Carnap in particular) proposed treating the truths of logic as expressions of the conventions governing a given language. Their role was thus not one of saying anything about the way things are—on this point they agreed with Wittgenstein—but rather that of spelling out the relations of implication among statements. And to the extent that mathematics could be reduced to logic following Frege and Russell, a similar account could be given of mathematical truths as well—they too express relations between statements. /There thus emerges a new conception of analytic truths as expressions of the conventions governing language.” (Juhl and Loomis 2010, 20–21)

Ayer’s philosophy tutor at Oxford, beginning in 1929, was Gilbert Ryle. Ryle introduced Ayer to the works of Russell and to Wittgenstein’s Tractatus Logico-Philosophicus. Ayer was captivated by Tractatus. Circumstances converged such that Ayer was given two terms leave of absence from Oxford, and at Ryle’s urging, Ayer headed for Vienna. Carnap was absent from the Circle for that interval, but it was in Vienna that Ayer became impressed with the work of Carnap to that point. In 1933 Ayer lectured at Oxford on Wittgenstein and Carnap. Other influences on Ayer were Popper and C. I. Lewis. Ayer’s Language, Truth and Logic was published in 1936. It was to become one of the most famous English-language philosophy books of the twentieth century.

The edition of it used by Peikoff in his dissertation and in “The Analytic-Synthetic Dichotomy” was the same as ours today: the second edition, 1946.

(Continuation with Ayer in the next installment.)


Ayer, Alfred Jules [1936] 1946. Language, Truth and Logic. New York: Dover.

Burgess, John P. 2005. Fixing Frege. Princeton: Princeton University Press.

Juhl, Cory and Eric Loomis 2010. Analyticity. New York: Routledge.

Peikoff, Leonard 1964. The Status of the Law of Contradiction in Classical Ontologism. Ph.D. dissertation, New York University.

Potter, Michael 2000. Reason’s Nearest Kin – Philosophies of Arithmetic from Kant to Carnap. New York: Oxford University Press.

Rödl, Sebastian 2012. Categories of the Temporal – An Inquiry into the Forms of the Finite Intellect. Translated by Sibylle Salewski. Cambridge, Massachusetts: Harvard University Press.

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5 November 2019

PNC Ground Shifts to the Side of the Subject – Conventionalism V

For the second edition of his book (1946), Ayer did not revise the text, but he wrote an Introduction in which he clarified some points and stated his revised positions on other points. Ayer had taken the usual view contra Mill that necessary propositions or necessary truths had to be a priori, not derived from experience (1946, 74–85). That view and Ayer’s view that only analytic propositions were such a priori ones had not changed. He clarified his view that a priori propositions do not describe any facts, saying he means facts that verify those propositions, as empirical propositions can be verified.

“I now think that it is a mistake to say that they [a priori propositions] are themselves linguistic rules. For apart from the fact that they can properly be said to be true, which linguistic rules cannot, they are distinguished also by being necessary, whereas linguistic rules are arbitrary. At the same time, if they are necessary it is only because the relevant linguistic rules are presupposed. Thus it is a contingent, empirical fact that the word ‘earlier’ is used in English to mean earlier, and it is an arbitrary, though convenient, rule of language that words that stand for temporal relations are to be used transitively; but, given this rule, the proposition that, if A is earlier than B and B is earlier than C, A is earlier than C becomes a necessary truth.” (1946, 17)

How in the world could he avoid seeing the feebleness of claiming necessities of transitivity for these propositions comes from acceptance of an arbitrary rule of language? Is it not straightforward to discern that the necessity in the transitivity of these propositions reflects merely a necessity in temporal relations which are topic of these propositions? Ayer had great intellectual sympathy with Hume, and one might wonder if Ayer was at this stage holding on to a view of temporality as contingent, close to Hume’s in the Treatise (which view Kant thoroughly demolished; see Rödl 2012, 111–68; see also Honderich 1987). Be that as it may, it is clear Ayer recognized the conventions taken as source of logical necessity must be arbitrary if they are to serve in maintaining a chasm between logic and empirical facts.

Even if Kant’s reason for thinking arithmetic to be not analytic is off the mark, Ayer did not make a good case that arithmetic is analytic in his sense (expression of convention governing language), thence necessarily true regardless of the constitution of the physical world empirically present. Among Wittgenstein’s scribbles circa 1937–1938, he observed that various sorts of items, such as apples or beans, are such that by counting portions of a collection of those items and counting totals of that collection, one can demonstrate the correctness of summation in arithmetic.

“This is how children learn sums; for one makes them put down three beans and then another three beans and then count what is there. If the result at one time were 5, another 7 (say because, as we should now say, one sometimes got added, and one sometimes vanished of itself), then the first thing we said would be that beans were no good for teaching sums. But if the same thing happened with sticks, fingers, lines and most other things, that would be the end of all sums. / But shouldn’t we then still have 2+2 = 4? —This sentence would have become unusable.” (Wittgenstein 1978, 51–52)

Late in life, Ayer wrote “I am more baffled than enlightened by his [Wittgenstein’s] Remarks on the Foundations of Mathematics. Nevertheless he does make one point of the utmost importance: he calls attention to the dependence of arithmetic upon contingent matters of fact.” Ayer then rehearsed Wittgenstein’s fancy of a world without exemplification of our arithmetic sums, and Ayer asked: “Would the proposition 7+5 = 12 be false in such a world? No. It would still be valid within the framework of standard arithmetic. It would have become not false but useless. . . . it would, says Wittgenstein, be the end of all sums. I am not convinced that this would be so. . . . I do not find it inconceivable that someone would devise an arithmetic which was adapted to such natural facts” (1989, 484–85). Talk of such deliberate adaptedness seems to forget about the supposed chasm between the analytic and the world.

Ayer held out to the end the hope that arithmetic, being not from empirical generalization, is purely analytic (there being no other sort of necessary truths in the view of Ayer and other logical empiricists). It was not a tenable hope. As a contrast to sums among countable things like beans or like words in a sentence, we need not turn to a radical fancy such as Wittgenstein’s. We can turn to contrasting the sorts of sums in the actual world. A positive road grade of 153 yards per mile added to a positive road grade of 153 yards per mile does not sum to a grade of 306 yards per mile, but to 310. 3 yards per mile. Road grades are just as physically real as beans. Different sorts of summations are right for things different in certain of their magnitude-characters. There is no place, I maintain, indifferent to physical reality for pure mathematical summation to abide in “pure analyticity” nor to be based most fundamentally on arbitrary convention.

It was not only of the a priori, necessary principles constituting mathematics, but of such principles constituting logic, Ayer had maintained: “If they are necessary it is only because the relevant linguistic rules are presupposed” (1946, 17). Linguistic rules concerning whether nouns shall have genders with which varieties of definite article must agree (as in German) surely would not be the relevant sort of presupposition—relevant sort of acceptance by one as rule for one’s language—to source the necessity in logic we all put to work in our discursive thought. A linguistic source plausible enough to get off the ground would have to be some deeper aspect of  grammar, an aspect common to all languages.

Recall the picture by Kant that I recorded in Kant I (this picture omitted from the English translation available for Peikoff 1964 and Ayer 1946): “The science which contains these universal and necessary laws [of thought] is simply a science of the form of thought. And we can form a conception of the possibility of such a science, just as a universal grammar which contains nothing beyond the mere form of language, without words, which belong to the matter of language.” That remark engages an analogy in which grammatical form is to grammar-slots filled with particular nouns, verbs, and so forth, as logical form is to its occasions of use in particular topics. Beyond analogy, Sebastian Rödl plumbs deep grammar of thought, along lines of the first Critique (Rödl 2012, 4, 22–25). Deeper plumbs, whether of grammar of language or grammar of thought, pull logical empiricists such as Ayer willy nilly hopelessly far from arbitrary convention as base for logic and its necessities. Wittgenstein evidently suffered that same pull (Ben-Menahem 2006, 265–69). “To collapse correctness into propriety is to obliterate the essential character of thought” (Haugeland 1998, 317; further, 325–43; see also Rasmussen 1982; 2014, 337–41; Rand 1966–67, 47–48; Peikoff 1967, 104; 1991, 143–44).

(To be continued, with Ernest Nagel and Arthur Pap next.)


Ayer, Alfred Jules. [1936] 1948. Language, Truth and Logic. New York: Dover.

——. 1989. Reply to F. Miró Quesada. In Hahn 1992, 478–88.

Ben-Menahem, Yemina. 2006. Conventionalism. Cambridge: Cambridge University Press.

Biondi, Paolo C. and Louis F. Groake, eds. 2014. Shifting the Paradigm – Alternative Perspectives on Induction. Berlin: De Gruyter.

Hahn, Lewis Edwin. 1992. The Philosophy of A.J. Ayer. La Salle, Illinois: Open Court.

Haugeland, John. 1998. Truth and Rule-Following. In Having Thought – Essays in the Metaphysics of Mind. Cambridge, Massachusetts: Harvard University Press.

Honderich, Ted. 1989. Causation: One Thing Just Happens after Another. In Hahn 1992, 243–70.

Peikoff, Leonard 1964. The Status of the Law of Contradiction in Classical Ontologism. Ph.D. dissertation, New York University.

——. 1967. The Analytic-Synthetic Dichotomy. In Rand 1966–67, 88–121.

——. 1991. Objectivism: The Philosophy of Ayn Rand. New York: Dutton.

Rand, Ayn. 1966–67. Introduction to Objectivist Epistemology. Expanded 2nd edition. New York: Meridian.

Rasmussen, Douglas B. 1982. Necessary Truth, the Game Analogy, and the Meaning-Is-Use Thesis. The Thomist 46(3):423–40.

——. 2014. Grounding Necessary Truth in the Nature of Things. In Biondi and Groarke 2014, 323–58.

Rödl, Sebastian. 2012. Categories of the Temporal – An Inquiry into the Forms of the Finite Intellect. Translated by Sibylle Salewski. Cambridge, Massachusetts: Harvard University Press.

Wittgenstein, Ludwig. 1978. Remarks on the Foundations of Mathematics. Edited by G.H. von Wright, R. Rhees, and G.E.M. Anscombe. Translated by G.E.M. Anscombe. Cambridge, Massachusetts: MIT Press.

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2 December 2019

PNC Ground Shifts to the Side of the Subject – Conventionalism VI

(Nagel and Pap will not be reached in this installment after all.)

In the Preface to the first edition of his Language, Truth, and Logic (1936), Ayer writes: “Like Hume, I divide all genuine propositions into two classes: those which, in his terminology, concern ‘relations of ideas’, and those which concern ‘matters of fact’” (31). The propositions of logic and mathematics belong to the former, and they are necessarily true, as Hume would have it. That is clear from his texts. But Hume could not concur with Ayer’s further characterization of such propositions as a priori, where an a priori proposition is one known not only independently of this or that perceptual experience, but independently of each and every one of them. Ayer erred, as so many before him and after him, in identifying such a radical sense of the a priori with Hume’s notion of the a priori for propositions of pure mathematics and logic (including PNC) understood as concerning only relations of ideas.

Hume, like Berkeley and Locke before him, was a full-blown empiricist. Ayer’s route to holding forth the logical empiricist view as an empiricist view was to shrink fact to only facts ascertained by the methods of the empirical sciences, to characterize all necessary propositions as radically a priori, which class coincides with a logic-centered analytic, whose truth and necessity are not from relations to the world, but from arbitrary convention hand-waved (in Ayer’s case) from grammar. Then, Ayer proclaims his view as empirical: all knowledge derives from experience because his genre of analytic propositions do not state facts; they are not knowledge. (See also Friedman 1999, 5–6, 9, 18–19, 32–33, 116–18.)

Graciela De Pierris 2015 concludes of Hume:

“The addition in the Enquiry of a seemingly logical criterion for identifying ‘relations of ideas’ does not mean a departure from the sensible phenomenological model of apprehension and ultimate evidence. Hume does not uphold the logical law of non-contradiction in its own right, prior to and independently of phenomenological sensible factors. On the contrary, the status of non-contradiction as a criterion for the acceptance of propositions based on ‘relations of ideas’ depends solely on the phenomenological inspection of intrinsic characteristics of particular items ostensively present before the mind. Necessary a priori methods, in both the Treatise and Enquiry, are ultimately grounded on nothing but the sensible phenomenological model.” (102)

In consistency with their rejection of Kant’s containment-of-predicate-in-subject criterion for seeing which concepts stand in a logically necessary relation each to the other (see Convention IV), the logical empiricists should have spurned Hume’s model of pure relations of ideas (or thoughts). That is to say, they should have spurned his explication of the a priori character of arithmetic, geometry, and logic. Had Ayer understood Hume on this point in Enquiry as harmonious with Hume’s treatment in Treatise, perhaps Ayer would not have set up Hume for worship on analyticity.

In Hume’s picture, discernment of necessary union of items forming a sum or the union of triangularity with the 2R sum of those three angles stands us in high certainty, indeed our highest certainty. We stand on PNC with that highest certainty, and for that reason, PNC can have normative force. “Hume uses his version of the presentational-phenomenological model of apprehension, just as the tradition before him had done, to give a verdict about ultimate evidence” (De Pierris 2015,  228; see also 220–22).

Locke had taken reason as “the discovery of the certainty or probability of such propositions or truths, which the mind arrives at by deduction made from such ideas, which it has got by the use of its natural faculties; viz. by sensation or reflection” (EU IV.xviii.2). Reason investigates ideas and their degrees of certainty or probability of being true, insofar as said ideas are grounded in sensory experience of outer things or in reflection, where reflection means our notice of our own inner, mental operations (EU II.i.3–4, II.v).

I mentioned in Aristotle II that Peikoff argued the unravelling of Platonic logical ontologism into logical conventionalism (as of mid-twentieth century) to have been mediated significantly by Kant. He concluded the unraveling of Aristotelian logical ontologism had been significantly mediated by Locke (Peikoff 1964, 212–35).

I had written in Conventionalism III:

“Logical necessity holds unconditionally and in all contexts. What I’ve called physical necessity is traditionally taken to be necessity under some sort of limiting conditions, and this necessity has been called a contingent connection, reserving necessary connection for logical (and other formal) necessity. . . .

“Peikoff 1964 points out that Locke avoided the contingent/necessary terminology. Locke instead applied probable/certain to the division. We have seen in my section Aristotle II that Locke maintained we have by sensory perception instances of the general fact that different things are not same things and that a thing is never both A and not A at the same time and in the same respect. Philosophers, including Peikoff in 1964, are correct to fault Locke’s blurring under probable/certain a clear understanding that ampliative inductive generalizations over perceived instances do not suffice to land the absolute necessity in general principles of logic or pure mathematics. . . .

“Locke was not really of one mind in this. Peikoff lays out an opposite strand also in An Essay Concerning Human Understanding: IV 3.31, 4.6, 4.8, 9.1, 11.13–14. ‘What is Locke doing in such passages as these? He is now contrasting eternal truths and existential truths. The former are to be discovered only by “the examining of our own ideas,” and “concern not existence” . . .’ (222). . . . For an empiricist such as Locke, . . . [one] joining considerable nominalism (the conceptualist wing of nominalism) concerning universal ideas to the empiricism, the divide between matters of fact and the eternal, formal truths can make conventionalism concerning the ground of logic ‘almost inevitable’ (223).”

(That is not to say, at least in my view, that it makes almost inevitable a root conventionalism that is arbitrary. It need not make almost inevitable a conventionalism making logical principles entirely independent of constraints from the world and from our physical operations in the world.)

I now think, having studied De Pierris 2015, that Locke’s two sorts of distinctions are harmonious. Whether some knowledge is gotten from “examining our own ideas” or from empirical generalization (or from some combination of the two), the knowledge has placement with us as to its certainty or probability. If, as with Descartes before him and Hume after him, degree of certainty is the fundamental index to truth, index to reality won, then Locke was innocent of fundamental severance of formal truths from empirical truths. We of last century and this have been guilty of writing too much of the divides of the logical empiricists back into the distinctions made by Locke and those by Hume.

Locke can be wrong or vague about how we derive PNC from sensory experience, yet right in his view that sensory experience is found to always conform to PNC. That PNC and mathematics are found as well and with highest possible certainty in ‘reflection’ (even reflection “concerning not existence”) need not somehow compete with or belie one’s finding PNC to hold in sensory experience, just as Euclidean geometry is conformed to in carpentry or in Newton’s system of the world (see also Franklin 2014, 95–100). Insofar as Locke is presented with high, highest certainty that PNC is true of the real in every nook and cranny, then PNC can serve as a norm to conform to for correct thinking. Even were the incorrect nominalist leanings of Locke and the incorrect voiding of natural necessitation of Hume not yet diagnosed and dispelled, PNC (and other formal relations) with its mark of certainty and generality can be given to, not cast by, the human faculties envisioned by Locke or by Hume. Then it is not those empiricists who are rightly regarded as intellectual tributaries into logical empiricism, notwithstanding the distant lineage claims declared by the latter. (Further on Locke: chapter VI “Ideas and Psychology” in Yolton 1984.)

What about kinship of the logical empiricists to the classical British empiricist George Berkeley? Kenneth Pearce 2017 argues that for Berkeley words get to be meaningful by being used in pubic discourse, following conventional rules, but for practical ends. The reason a system of language can be effective practically is because with language we capture the grammar of nature as created by God. There is under this conception, I notice, no severance of our minds, including our mathematical minds, from the world due to conventions in language (Pearce 2017, 45–50, 80–83, 112, 151–52, 158–62, 195, 204). Berkeley’s view is contrary the logical empiricist thesis of arbitrariness of linguistic convention sufficiently deep in any formal truths to decouple them from the world.


De Pierris, G. 2015. Ideas, Evidence, & Method – Hume’s Skepticism & Naturalism Concerning Knowledge & Causation. New York: Oxford University Press.

Franklin, J. 2014. An Aristotelian Realist Philosophy of Mathematics – Mathematics as the Science of Quantity and Structure. New York: Palgrave Macmillan.

Friedman, M. 1999. Reconsidering Logical Positivism. New York: Cambridge University Press.

Pearce, K. L. 2017. Language and the Structure of Berkeley’s World. New York: Oxford University Press.

Yolton, J. W. 1984. Perceptual Acquaintance from Descartes to Reid. Minneapolis: University of Minnesota Press.

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25 December 2019

PNC Ground Shifts to the Side of the Subject – Conventionalism VII

From Gillian Russell’s entry LOGICAL PLURALISM (2019) in the Stanford Encyclopedia of Philosophy:

“There are several issues that someone who wanted to defend Carnap’s [1937] position today would need to address. A first concern about the view is that while we are working within the various languages we invent, we could be missing the ‘correct’ rules—the ones that were out there, in effect, before we invented anything. In the words of Paul Boghossian,

“‘Are we really to suppose that, prior to our stipulating a meaning for the sentence ‘Either snow is white or it isn’t,’ it wasn’t the case that either snow was white or it wasn’t? Isn’t it overwhelmingly obvious that this claim was true before such an act of meaning, and that it would have been true even if no one had thought about it, or chosen it to be expressed by one of our sentences?’ (Boghossian 1996)

“Carnap would perhaps not have taken this objection seriously, since, like the Wittgenstein of the Tractatus (e.g., §4.26, 4.641–4.465), he does not believe that logical truths and rules are ‘out there’, waiting to be discovered:

“‘The so-called “real” sentences, constitute the core of the science; the mathematico-logical sentences are analytic, with no real content, and are merely formal auxiliaries’. (Carnap 1937, xiv)

“Nonetheless, such a ‘conventionalist’ view of logical truth (and along with it, analytic truth) has been argued against by, for example, Quine, Sober, Yablo and Boghossian, and it no longer enjoys the popularity that it had in Carnap’s time (Quine 1936; Yablo 1992; Boghossian 1996; Sober 2000).”

One of the most noted essays of the twentieth century is Quine’s 1951 “Two Dogmas of Empiricism,” which argues the distinction between analytic and synthetic truths, so dear in logical empiricism, is untenable. Necessary truths we have in logic and mathematics cannot receive their necessity of being true merely in virtue of meaning, which is to say, by being analytic truths. Furthermore, for analytic there is no noncircular and enduring rule establishing its extension. A logical truth such as A is identically A, in Quine’s view, need not get its truth only by our say-so meaning of is identically, but could as well get its truth by its capture of the way the world is (Quine 1954, 113).

In a 1963 lecture “Necessary Truth” which Quine delivered on Voice of America (the lecture was published by that organization the following year as a pamphlet), he concluded: “In principle . . . I see no higher or more austere necessity than natural necessity; and in natural necessity, or our attributions of it, I see only Hume’s regularities, culminating here and there in what passes for an explanatory trait or the promise of it” (76). With that last clause, Quine rather takes back the ribbon he had just given Hume. Moreover, earlier in the lecture, Quine had augmented bare regularity with generality-within-a-domain (71, 74).

When I say “Necessarily, were Little Bo Peep’s sheep to come home of their own accord, they will be wagging their tails behind them” (here I abjure Standley’s wild thought they might come home walking in reverse), I speak in the essential form of natural necessity, according to Quine 1963. My necessity-sentence is a subjunctive resting on the nature of sheep. Quine’s view that logical and mathematical necessity are only wide-domain natural necessities acknowledged in subjunctive statements seem right to me. “Necessarily, were not both p and q true, and p were true, then q would be false.” The Randian 1957 necessities concerning leaf/stone, freeze/burn, and red/green can be put into that p/q form with ease. Where q is simply not-p (very wide domain), we have the bare PNC.

In 1967, four years after Quine’s “Necessary Truth,” Leonard Peikoff’s “The Analytic-Synthetic Dichotomy” appeared in The Objectivist. Roderick Long once remarked, relying on Quine 1951: “Rather than defending the existence of necessary factual truth, Quine had in effect denied that any truths were necessary, even the laws of logic” (2005, 226n3). No, not really. To say “necessarily, not both p and not-p” under a blank condition that its wide domain of application might in the future be shown to be not the widest domain, is not to in effect deny there are necessary truths, only to deny knowing that the presently known widest domain is not but part of a wider domain in which those necessary truths do not everywhere apply. I disagree with Quine—we do know the application of PNC is to the widest possible domain—but that does not alter the value of Quine’s insight that formal necessities (however broad, broad their purview) are natural necessities.

Peikoff 1967 attacks contemporary conventionalism concerning necessary truths, though not along the Quinean lines of attack. Rather, Peikoff attacks by attacking the old distinction, from Plato-Aristotle to the early moderns, that there is a distinction, a knowable distinction, between contingent facts and necessary truths. He takes contemporary conventionalism to merely replace metaphysical bases for necessary truths with “subjective choices.” “Their ‘contribution’ is merely to interpret [the traditional position] in an avowedly subjectivist manner” (1967, 108). (Again I stress that a conventionalism concerning logical truth that is not wholly arbitrary in its conventions and their combinations is not wholly effective in freeing logical truth from facts of the world and of our logical facility with those facts.)

Peikoff 1967 rightly attacks the traditional distinction, with its supernatural prop. He then takes on the embrace of the concept contingent facts by (many among) contemporary analytic philosophers. He would have it, rather, that all natural facts are necessary, and only man-made facts are contingent (106–11). Various divisions of all that is have arisen across the centuries under the same words necessary/contingent. Quine does not conceive of natural necessity, thence logical necessity as applying to everything not man-made. Quine’s necessity does not apply, save elliptically, to particular events or states; it applies properly only to whole conditional connections—“Necessarily, if p then q.”

“We must not suppose that a man is entitled to apply ‘necessarily’ to an assertion so long merely as he thinks there is some general truth that subsumes it” (Quine 1963, 70).

Suppose the raccoon Rocky is climbing the tree to reach the bird feeder. Then we can say of every raccoon x without exception that if x is Rocky, then x is climbing the tree to reach the bird feeder. Quine restricts necessity in his intended sense to disallow that because we got Rocky’s deed into that form, in a true report of what is afoot, we may straightly say in Quine’s sense “Necessarily, Rocky being indeed Rocky, he’s climbing the tree to reach the bird feeder.” “This line would allow us to attribute necessity to anything, however casual, that we are prepared to affirm at all” (1963, 71). The relation of Quine’s sense of necessity and the one in play in Peikoff 1967 remains work not yet done by anyone, I gather.

In his Introduction to the collection The Age of Alternative Logics (2009), editor Johan van Benthem writes:

“Modern logic shows a wide variety of perspectives, application areas, and formal systems which often go under the heading ‘alternative logics’. . . . Actually, terms like ‘alternative’ or ‘non-classical’ logic can easily be misunderstood . . . . To us, the diversity of logical systems today . . . signals a natural and respectable process of growth of the discipline, not of replacement or competition.” (1)

It is a mistake to suppose that because there are alternative logics, such as intuitionist logic or relevance logic, that the choice of their domain of application is unconstrained, in fact arbitrary.

We have noticed that Peikoff 1964 used “conventionalism” concerning fundamental logical principles somewhat broadly. The label was meant to encompass the stances of pragmatist and logical empiricist philosophers in the first half of the twentieth century. Peikoff rightly did not try to sweep Kant’s view of logic into the conventionalist bin.

The term “conventionalism” in Sidelle 1989 is far too inclusive. He used conventionalism to mean any view of necessary truth that does not take that necessity to reside in the mind-independent world. A major innovation in analytic philosophy had occurred by the time of Sidelle book: Kripke 1972. Without resort to Kant’s proposition-type or truth-type synthetic a priori, Kripke had upheld, from logical analysis of language, credible candidates for necessarily true statements not a priori and not analytic. Sidelle attempts to counter the common understanding that Kripke succeeded in putting some necessary truth into the mind-independent world.

To be continued.


Boghossian, P. A. 1996. Analyticity Reconsidered. Noûs 30(3):360–91.

Browne, G. M. 2001. Necessary Factual Truth. Lanham: University Press of America.

Carnap, R. 1937. The Logical Syntax of Language. London: Kegan Paul.

Kripke, S. A. 1972. Naming and Necessity. Cambridge: Harvard University Press.

Long, R. T. Reference and Necessity: A Rand-Kripke Synthesis? – Review of Brown 2001. The Journal of Ayn Rand Studies 7(1):209–28.

Peikoff, Leonard 1964. The Status of the Law of Contradiction in Classical Ontologism. Ph.D. dissertation, New York University.

——. 1967. The Analytic-Synthetic Dichotomy. In Rand 1966–67, 88–121.

Quine, W. V. O. 1936. Truth by Convention. In Quine 1976.

——. 1951. Two Dogmas of Empiricism. In Quine 1980.

——. 1954. Carnap and Logical Truth. In Quine 1976.

——. 1963. Necessary Truth. In Quine 1976.

——. 1976. The Ways of Paradox and Other Essays. Cambridge: Harvard University Press.

——. 1980. From a Logical Point of View. Cambridge: Harvard University Press.

Rand, A. 1957. Atlas Shrugged. New York: Random House.

——. 1966–67. Introduction to Objectivist Epistemology. Expanded 2nd edition. 1990. New York: Meridian.

Sidelle, A. 1989. Necessity, Essence, and Individuation – A Defense of Conventionalism. Ithaca: Cornell University Press.

Sober, E. 2008. Quine. Proceedings of the Aristotelian Society LXXIV:237–80.

Standley, J. 1952. https://www.youtube.com/watch?v=7aExUGQOnI0

Van Benthem, J., Heinzmann, G., Rebuschi, M., and H. Visser, editors, 2009. The Age of Alternative Logics – Assessing Philosophy of Logic and Mathematics Today. Dordrecht: Springer.

Wittgenstein, L. 1922 [1918]. Tractatus Logico-Philosophicus. C. K. Ogden, translator. New York: Routledge.

Yablo, S. 1992. Review of Necessity, Essence, and Individuation: A Defense of Conventionalism by Alan Sidelle. The Philosophical Review 101(4):878–81.

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5 January 2020

PNC Ground Shifts to the Side of the Subject – Conventionalism VIII

“I have never asserted that it is inconceivable that water isn’t H2O, but only that it is impossible that it isn’t H2O.” —H. Putnam

In the nineteenth century, Weber & Kohlbrausch, Maxwell, and Hertz established that light is electromagnetic radiation (within a certain range of frequencies). This was established by measurements and by mathematical representation of relationships between various physical, electromagnetic properties in media and in the vacuum. They established as well that radiant heat was electromagnetic radiation. Heat and light from the sun, for example, are electromagnetic radiation. Electromagnetic radiation (of some range of frequencies) is the light allowing the wheat to grow. For short, light = EM rad, where ‘=‘ is here ‘identically the same as’.

At least one, probably more, of the characteristics of the single thing that is light/EM rad is an essential characteristic of it. But light and EM radiation being the same thing is nothing essential about that thing. Rather, this being-the-same is total identity of the object investigated in optics and an object investigated in EM science. A is not a characteristic of A, though A is A.

In her 2017 paper on Kripkean necessity of identity, García-Encinas argues the identity discovered (my example) by Weber & Kohlbrausch, Maxwell, and Hertz subtly exhibits “how identity belongs to the inner and to the most profound structure of cognition and language” (52). That much having been argued, she suggests: “Despite the general tendency to the contrary, it could be the case that even logical truths, or truths that are usually believed to be necessary but devoid of metaphysical content, like x = x are finally grounded in metaphysics (and aesthetics). x = x could be a logical form of our intuition of Identity” (68). The indefinite article of “a logical form” would be the right choice of article in Rand’s lights, since “x is identically x” as the identity added to first-order predicate calculus with quantification (see Quine 1982, chapter 43) is only a proper part of the full A is A of logic employed in the discovery of the light/EM rad identity.

The single self-same thing light/EM rad exhibits wave-particle duality; has a wavelength inversely proportional to its momentum; has a definite role not only in electrodynamics, but in general kinematics of modern mechanics; has no rest-mass of its particle, the photon; has polarization character; and shows universal characteristic atomic and molecular spectra from matter throughout the observed universe. Whether any one or more of these characteristics is essential to light/EM rad being what it is, Objectivist metaphysics and epistemology takes them all as being necessary to what it is. They have a necessity transcending the necessity of an essential characteristic. This transcending necessity is a necessity merely denying cognitive validity of imagination-criteria for whether something is “contingent.” That I can imagine, without manifest contradiction, that the atoms and molecules naturally present here in the solar system are nonexistent elsewhere in the galaxy or that light and electromagnetic radiation are not (in any range of frequencies) the same thing is no insight into metaphysical or physical character. Such imaginings, since Descartes and until this day, so beloved by many philosophers (e.g. Sidelle 2002), are valuable only for entertainment, and they exhibit only what playfulness I can have over past ignorance and inquiry (cf. Peikoff 1967, 114–16).

Necessities in a posteriori truths as well as all logical necessities whatever are not the offspring of linguistic convention in stipulative definitions. That much, many contemporary philosophers have concluded. Those necessities are the offspring of existence in its identities, an assertion that needs to be fleshed out (see Long 2005, 213, on that need).

I should like to add (my own original connection) that the insusceptibility to trisection of angles in complete generality by straight-edge and compass constructions alone is not shown to be an absolutely necessary insusceptibility by stipulative definitions joined with their contradiction upon supposition of such trisection. The absolute necessity of that insusceptibility was shown by creative discovery of whole new areas of mathematics and their connection to Euclidean geometry (within which workers had failed repeatedly to produce general angle trisection using only straight-edge construction together with compass construction, and within which could not prove its absolute impossibility).


In his 2016, Greg Salmieri notes that it is curious that Peikoff 1967 does not mention Quine’s “Two Dogmas.” Salmeiri points out some ways the Rand-Peikoff diagnoses of and remedies for the errors in analytic-versus-synthetic doctrines differ from Quine’s. Salmieri understands the later challenge of AvS from Kripke and Putnam to have more in common with the Objectivist challenge, though Putnam differs importantly from Rand on definitions and essences, which looms large in the Objectivist challenge (2016, 304n34, 311n87). Salmieri points to the book-review article, in JARS in 2005, by Roderick Long for thoughts on some relations between Randian theory of meaning and those of Kripke and Putnam.

Long’s 2005 review of Greg Browne’s book Necessary Factual Truth was followed a year later by a substantial reply from Browne and rejoinder by Long (JARS V7N1). From May to September of 2007, Prof. Browne engaged in a very generous exchange (his own words coming to about 19,000) in a thread at Objectivist Living defending the rejection by Peikoff of AvS and defending his own kindred rejection of AvS. Browne had in his arsenal the Kripke-Putnam developments that had been savaging AvS in the years since Peikoff 1967. Browne vigorously countered, in that thread, devotees of Logical Empiricism (and of Popper) who criticized (and poorly understood the revolution afoot, such as in) Peikoff 1967.

Late in that thread, Robert Campbell entered it only to ask Browne if he had any thoughts on why Peikoff had not addressed the famous Quine paper in his Peikoff’s dissertation, which Campbell had lately acquired. Browne had not seen the dissertation and had not much to conjecture on that peculiarity. (Remember, Peikoff 1964 is not written as a champion of Ayn Rand’s philosophic views, but, in an even-handed way, by an author acknowledging his background preference for some rehabilitated sort of logical ontologism and pointing near the end of the dissertation to some of that rehabilitation, such as fresh thinking on the nature of definitions and essence; distance between Quine’s views on logic and on AvS and Randian Peikoff views would not be the reason for no Quine in Peikoff 1964.) I should suggest that Quine, Carnap, Russell, and Wittgenstein raise such a briar patch of technicalities that it was better (and enough for deserving a Ph.D.) to stick with the more accessible and manageable Ayer, Nagel, Dewey, and Lewis to get the dissertation (already more than an armful in history assimilated) finally completed.


Near the close of “Two Dogmas of Empiricism,” after saying once more that he rejects the distinction between the analytic and the synthetic, Quine enters a footnote directing the reader to a paper by Morton White “for an effective expression of further misgivings over this distinction” (1953, 46). That paper is “The Analytic and the Synthetic: An Untenable Dualism” (UD).[1] It appeared originally in Hook 1950, then again, in Linsky 1952. My page references are from the latter.

Morton White noted two kinds of statements that had lately been regarded as analytic. The first are purely formal logical truths such as “A is A” and “A or not-A.” The second are cases of “what is traditionally known as essential predication” (UD 318). He ponders especially the example “All men are rational animals.” That statement is logically the same as “Any man is a rational animal” or “A man is a rational animal.” This last expression of the proposition is one of Leonard Peikoff’s examples of a purportedly analytic statement in “The Analytic-Synthetic Dichotomy” ([A-S] 1967, 90).

White did not pursue in this paper whether it is correct to characterize logical truths as analytic (UD 318–19). It will be recalled that Peikoff held forth Rand’s conception of logical truth against that of A. J. Ayer, who had maintained: “The principles of logic and mathematics are true universally simply because we never allow them to be anything else. . . . In other words, the truths of logic and mathematics are analytic propositions or tautologies” (1946, 77; Branden 1963, 7; A-S 94, 101, 111–18).

As with Quine’s “Two Dogmas,” White undermined the distinction between the analytic and the synthetic by finding fault with various explications of what analyticity amount to. They concluded there is no durable articulate way of classifying propositions and truths as analytic in sharp contrast to synthetic.

One way of conceiving an analytic statement is as expressing a proposition deducible from a logical truth by substitution of a synonym of one of its terms. (i) Every A is A. Therefore, (ii) Every man is a man. With rational animal as synonym for man, we obtain (iii) Every man is a rational animal (UD 319).

Thence analyticity is explicated in terms of logical truth and synonymy. White rejects the view that whether man and rational animal are synonymous is a matter of arbitrarily selected convention. Similarly, that man and featherless biped are not synonymous is not a matter of arbitrarily selected convention. Natural language is not like an artificial logical language in which meanings of terms are set entirely by stipulation (UD 321–24).

White allows we certainly have some sort of working distinction between propositions such as, on the one hand, “Man is an animal” and “Man is a rational animal” and, on the other hand, “Man is a featherless biped” and “Man has two eyes” (Peikoff’s example, A-S 90). White concludes that distinction between those two classes of statement is not that statements in the first class are analytic, the latter not, where the analytic is defined as consequence of a logical truth under substitution of a synonym and we are given no objective criterion for synonymy (UD 318–24).

Could analytic statements be defined instead as those whose denials are self-contradictory? White queries how it is that “Man is not a rational animal” leads to “Man is not man,” yet “Man is a quadruped” does not lead to “Man is not man.” He again notes that appealing to synonymies in the language is not illuminating in the absence of objective criteria for synonymy (UD 324). If it is said that one’s sense of wrongness in “Man is not a rational animal” differs from one’s sense of wrongness in “Man is a quadruped,” White replies that that is surely only a matter of degree, not a sharp difference in kind. Between one’s response to contradiction of “Man is a rational animal” and contradiction of “Man is a biped,” there is not a sharp difference in kind. If self-contradiction upon denial of a proposition is the criterion for analyticity of the proposition, then there is no sharp divide between the analytic and the synthetic (UD 325–26).

Suppose we adopt the following criterion for analyticity. Were we to come across an animal we determine to be not a rational animal, we would dismiss it instantly as being a man. By contrast, were we to come across an animal we see is not a featherless biped (it is, say, a quadruped), but whose rationality is not yet confirmed or disconfirmed, we hesitate over whether this animal is a man. We know that we might give up the proposition “All men are featherless bipeds” if we learn this animal is rational (UD 326–28). White responds: “Now I suspect that this criterion will be workable but it will not allow us to distinguish what we think in advance are the analytic equivalences. It will result in our finding that many firmly believed ‘synthetic’ equivalences are analytic on this criterion” (UD 328).

White gives no example, but I think his point is illustrated by an analytic-synthetic pair of judgments, favorites with Kant: “All bodies are extended” (analytic) and “All bodies have weight” (synthetic). By the latter, given his knowledge of Newtonian physics, I think Kant rightly understands “All bodies not in gravitational orbit have weight.” Be that as it may, Kant and his contemporaneous intellectuals would dismiss as body just as quickly an entity lacking weight (in the appropriate setting) as they would dismiss as body an entity lacking extension.[2] The criterion of speed of dismissal upon counterfactual encounter fails to always sort what is taken for analytic from what is taken for synthetic.

White observes that the obscurity of proposed criteria for distinguishing analytic from synthetic statements, propositions, and judgments, is not fixed by incorporating the sound Millian point that what is synonymous with man, for example, varies with discursive context. In a biological discourse, “mammiferous animal having two hands” (Mill’s example) might be synonym for man. It remains that analyticity is not illuminated by proposing logical truth and synonymy as its base, not illuminated so as to yield a sharp divide, rather than a gradual divide, between the analytic and the synthetic. The arguments run against such an explication of the analyticity of “Man is a rational animal” will rerun for “Man is a mammiferous animal with two hands” (UD 329–30).

White saw the myth of a sharp divide between the analytic and the synthetic as affiliate of an older mythically sharp division: the Aristotelian division between essential and accidental predication (UD 319, 325, 330). This kinship was also recognized in Peikoff 1967 (A-S 95).

To be continued.


1. Nelson Goodman writes in a 1953 footnote: “Perhaps I should explain for the sake of some unusually sheltered reader that the notion of a necessary connection of ideas, or of an absolutely analytic statement, is no longer sacrosanct. Some, like Quine and White, have forthrightly attacked the notion; others, like myself, have simply discarded it; and still others have begun to feel acutely uncomfortable about it” (60).

2. Notice also that in modern physics of elementary particles, we take electrons and the other leptons to be bodies (matter) because they have weight (because of nonzero rest mass), yet they have no extension. The feature Kant took for analytic, we eventually took as dispensable, whereas the feature he took for synthetic, we have retained.


Ayer, A. J. 1946. Language, Truth and Logic. Dover.

Branden, N. 1963. Review of Brand Blanshard’s Reason and Analysis. The Objectivist Newsletter 2(2):7–8.

Browne, G. M. 2001. Necessary Factual Truth. Lanham: University Press of America.

García-Encinas, M. J. 2017. The Discovery that Phosphorus is Hesperus: A Follow-Up to Kripke on the Necessity of Identity. Analysis and Metaphysics 16:52–69.

Gendler, T. S., and J. Hawthorne, editors, 2002. Conceivability and Possibility. Oxford.

Goodman, N. 1953. The New Riddle of Induction. In Fact, Fiction, and Forecast. 4th edition. 1983. Harvard.

Gotthelf, A. and G. Salmieri, editors, 2016. A Companion to Ayn Rand. Wiley Blackwell.

Hook, S., editor, 1950. John Dewey: Philosopher of Science and Freedom. Dial.

Linsky, L., editor, 1952. Semantics and the Philosophy of Language. Illinois.

Peikoff, L. 1967. The Analytic-Synthetic Dichotomy. In Rand 1990.

Long, R. T. 2005. Reference and Necessity: A Rand-Kripke Synthesis? —Review of Brown 2001. The Journal of Ayn Rand Studies 7(1):209–28.

Quine, W. V. O. 1951. Two Dogmas of Empiricism. In From a Logical Point of View. 1953. Harvard.

——. 1982. Methods of Logic. 4th ed. Harvard.

Rand, A. 1990 [1966–67]. Introduction to Objectivist Epistemology. Expanded 2nd edition. Meridian.

Salmieri, G. 2016. The Objectivist Epistemology. In Gotthelf and Salmieri 2016.

Sidelle, A. 2002. On the Metaphysical Contingency of Laws of Nature. In Gendler and Hawthorne 2002.

White, M. G. 1952 [1950]. The Analytic and the Synthetic: An Untenable Dualism. In  Linsky 1952.

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19 February 2020

PNC Ground Shifts to the Side of the Subject – Kant IV-a

I made a mistake concerning Kant’s doctrines on logic. Relying on a smaller pool of Kant material in English, Peikoff 1964 had also made this mistake.

I asked of Kant in the Prep thread (2009 paper) as well as in this Dissertation thread: How can logic be rules necessary for correct operation of the understanding and reason if they simply are rules given by the understanding and reason, and they are rules descriptive and necessarily operative in all understanding and reason? “I concur with the conclusion of Peikoff and others he cites that once Kant had the constitution of the subject the sole source of the purely formal and purely a priori, he was not able to stably maintain an absolute necessity of PNC and other principles of logic together with their normativity, which latter entails our ability to not adhere to such principles. I add that this same irresolvable mess arises for every other sort of cognition purely formal and purely a priori, whether analytic or synthetic, once Kant has squarely located their source purely in the constitution of mind, in its fundamental dynamics, not at all in the constitution of the world.” (Kant III)

The world according to Kant—the empirical and the mathematical—cannot be found with structure contrary to the formal structure of logic provided by human mind. The source of form had moved indoors. Conformance of the world to logic is assured in Kant’s scheme; there is absolute rightness of PNC (and its cohort, identity) for the world we take up, even though PNC is ultimately sourced in pure mind. Objects of any sort can be given to us as objects only upon actuation of our conceptual, logical faculties. The portion of mind trying to discern the world beyond the world’s bare, mind-given logical form is able to commit contradictions, within Kant’s scheme. The pure understanding gives some laws, but the impure understanding may fail to always conform to them in its pursuit of specific empirical and mathematical knowledge.

We can still fault Kant in his view that we possess any “pure a priori” knowledge of any thing at all, in Kant’s radical senses of the a priori and the pure; we can deny that PNC or any other formal principle is such knowledge as that (on purity, see Chance 2018). But it is not the case that Kant’s scheme falters over the fact that humans make logical errors. (Notice Kant’s stress on human logical fallibility in lectures already from the early 1770’s; Lu-Adler 2018, 108.)

Peikoff quotes from the Abbott translation (1885) of the Introduction of Kant’s Logic (the Jäsche Logic😞How error is possible, since in the formal sense of the word, that is, how a form of thought inconsistent with the understanding is possible; this is hard to comprehend; as indeed in general we cannot comprehend how any faculty can deviate from its own essential laws” (44). But Peikoff and I failed to notice or anyway failed to address Jäsche’s words, reflecting Kant’s notes for his logic lectures, on how this problem is to be resolved (here staying with the Abbott translation): “Besides the understanding there is in us another indispensable source of knowledge. This is the sensibility, which supplies the material for thought, and besides works according to different laws from the understanding. From the senses, however, considered in and by itself, error cannot arise, since the senses do not judge. / Hence the origin of all error must be sought solely in the unobserved influence of the sensibility on the understanding, or, to speak more exactly, on the judgment. It is owing to this influence that in our judgements we mistake merely subjective reasons for objective, and consequently confound the mere semblance of truth with truth itself.” (ibid.)

Thinkers following on the heels of Kant then need not “return to an ontological interpretation of logic, thereby preserving absolutism in logic by founding it on facts of reality independent of the variable workings of human minds” else plunk for conventional choice of logical rules such as PNC, the fork proposed in Peikoff 1964. For instance Robert Hanna’s 2006 neo-Kantian, subject-sided, anti-conventionalist account of the nature of formal logic crafts a third way.


Peikoff’s reliance on Henry Mansel’s infirm understanding and adulterated representation of Kant was unfortunate. I refer specifically to Mansel’s Prolegomena Logica: An Inquiry into the Psychological Character of Logical Processes (1860). Mansel insists an empirical psychology must be put to work to “vindicate” logic. Mansel and others Mansel cites who took this post-Kantian turn were trying to keep a foot in the school of Kant (and touting themselves as significantly Kantian) while kicking that foot with the other, empiricist, Kant-undone foot. “To Psychlogy we must look for the explanation and justification of the peculiar features of Logic” (1860, 6). Not by the lights of Kant! For Kant all successful reasoning and understanding requires formal logic as a necessary presumed norm; psychology cannot attain and justify psychology’s special knowledge without thoroughly conforming itself to the general norms of formal logic, which are already presumed (further, Hatfield 1990).

When Kant was four years old, Christian Wolff was writing: “In order to demonstrate the rules of logic, principles must be taken from ontology. Furthermore, . . . we must learn from psychology what the cognitive faculty is and what its operations are. Hence it is also clear that, in order to demonstrate the rules of logic, principles must be taken from psychology. . . . Therefore, ontology and psychology should precede logic if everything in logic is to be rigorously demonstrated and if its rules are to be genuinely known.” (quoted in Lu-Adler 2018, 92)

In Wolff’s view, we have a natural aptitude to follow the prescriptions that are the principles of logic, but to cultivate that aptitude into a habit of their correct use requires learning explicitly what those rules are. The correct logical rules having become explicit are to be followed for knowledge in general, and they are discernible as the correct rules by their enablement of the complete demonstrations in mathematics, particularly in geometry (Lu-Adler 2018, 90–97).

Kant took under consideration the ancient and medieval views on logic as well as moderns such as F. Bacon, Locke, Leibniz, Wolff, and of course Baumgarten and Meier (Wolffian variants). Meier followed Wolff in taking logic-as-a-science (what Kant would later call pure general logic) to be based in part on principles of psychology. Kant was rejecting this already in the pre-Critical period of the Bloomberg logic lecture notes from the early 1770’s (Lu-Adler 2018, 110–11).

Kant regarded pure general logic as a science in his strong sense of science, in which the object of the science is treated “wholly according to a priori principles” (1786, 4:468). A science is termed proper science by Kant if it not treated only according to laws of experience and has more than mere empirical certainty. Proper science has apodictic certainty. Science more generally is “a system, that is, a whole of cognition ordered according to principles,” and “such principles may be either principles of empirical or of rational connection of cognitions into a whole” (4:467). In proper science, the connecting principles are not empirical, but the rational connection of ground to consequence. The empirical connections in chemistry do not possess the necessity that yields apodictic certainty, according to Kant. Proper science of nature has only as much apodictic certainty as there is in it a priori knowledge and application of mathematics. Empirical psychology is even farther from being a proper natural science than chemistry (4:469–71).

“A rational doctrine of nature . . . deserves the name of a natural science, only in case the fundamental natural laws therein are cognized a priori, and are not mere laws of experience. One calls a cognition of nature of the first kind pure, but that of the second kind is called applied rational cognition” (4:468, my underscore). That distinction under the terms pure and applied resembles the distinction under those same terms in Kant’s treatment of logic, although the two sorts of laws of nature, unlike principles of logic, are not norms.

To be continued.


Abbott, Thomas Kingsmill, trans. 1885. Kant’s Introduction to Logic. London: Longmans Green.

Allison, Henry and Peter Heath, eds. 2002. Immanuel Kant – Theoretical Philosophy after 1781. Cambridge: Cambridge University Press.

Chance, Brian A. 2018. Wolff’s Empirical Psychology and the Structure of the Transcendental Logic. In Dyck and Wunderlich 2018.

Dyck, Corey W. and Falk Wunderlich, 2018. Kant and His German Contemporaries. Vol. 1. Cambridge: Cambridge University Press.

Hanna, Robert. 2006. Rationality and Logic. Cambridge, Massachusetts: MIT Press.

Hatfield, Gary. 1990. The Natural and the Normative – Theories of Spatial Perception from Kant to Helmholtz. Cambridge, Massachusetts: MIT Press.

Kant, Immanuel. 1786. Metaphysical Foundations of Natural Science. Michael Friedman, translator. In Allison and Heath 2002.

Lu-Adler, Huaping. 2018. Kant and the Science of Logic. New York: Oxford University Press.

Mansel, Henry Longuevill. 1860. Prolegomena Logica – An Inquiry into the Psychological Character of Logical Processes. London: Oxford University Press.

Peikoff, Leonard. 1964. The Status of the Law of Contradiction in Classical Logical Ontologism. Ph.D. dissertation. New York University.

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14 March 2020

PNC Ground Shifts to the Side of the Subject – Kant IV-b

Bernard Bolzano’s masterwork Theory of Science issued in 1837. In this work, we find him objecting to Kant’s definition of logic as conveyed by Jäsche: “The science of the necessary laws of understanding and reason in general, or of the mere form of thinking, is logic.” Bolzano named seven writers of logic texts since Kant who had followed Kant in that definition of logic as a science. Taking thinking in its usual wide sense, Bolzano objected to that definition. It is not a plausible characterization of logic to say logic is merely the law-governed use of reason and understanding. One could then be regarded as engaging logic when thinking fallaciously or with an aim to evasion or when thinking in a whimsical entertainment. Quite better, in Bolzano’s assessment, were several writers who had required that such laws be restricted to those serving the chosen purpose of recognizing truth (the purpose of identification, in Rand’s vocabulary and explication) to warrant the title logic.

We have seen Kant’s view that “general logic . . . abstracts from all content of cognition, i.e., from all reference of cognition to its object” (KrV A55 B79). Bolzano’s own conception of logic was as the set of rules for dividing truths into perspicacious domains of specific sciences, for their cultivation, and for their perspicacious presentation. Logic so understood is itself a science as well. We use some of these rules before turning to isolate what they are. “Once these rules are known, every science, including the theory of science itself can be further elaborated and presented in writing. This amounts to no more than arranging certain known truths in an order and connection that they themselves prescribe” (§2).

Bolzano thought the ultimate goal of logic the discovery of truth (§7). In that ambition, one is reminded of the expansion of logic envisioned in Francis Bacon’s New Organon (although Bolzano, contra Bacon, did not regard Aristotle’s syllogistic as useless). It seemed to Bolzano that “one of Kant’s literary sins was that he attempted to deprive us of a wholesome faith in the perfectibility of logic through an assertion very welcome to human indolence, namely, that logic is a science which has been complete and closed since the time of Aristotle” (§9).  Bolzano 1837 looked forward to future developments of logic that would be a boon to all the sciences. It turned out that, after Bolzano, there were advances in deductive logic. However, these did nothing to advance or clarify knowledge in empirical science. They did illuminate mathematics and its connections to logic, and they illuminated and extended the Aristotelian (and Stoic) logic of old.

Bolzano criticized the Germans such as Kant, Jakob (1791), Hoffbauer (1794), and Maimon (1794) for their slippage from the topic-neutrality of a syllogism form presented as “all A are B, all B are C, therefore all A are C” to taking the objects A, B, and C for indeterminate as to all their characteristics. That is, they erred in taking A, B, and C as empty of absolutely all content. “If we think of an object as altogether indeterminate, then we cannot claim anything about it” (§7). The signs A, B, and C need the determination that B can be rightly predicated of A, and so forth. Logical form in Bolzano’s view was not fundamentally about thoughts, but relations of truths. Bolzano correctly objected as well to contemporaries trying to make logic into some kind of empirical science of the mind.

Today when we say science, we usually mean empirical science. The term science had been used more broadly until recent times, such that science encompassed also the organized disciplines of mathematics, ethics, and logic, especially when considered in their allegedly pure, necessary, nonempirical, and applications-suspended mode.

I said in “Kant IV-a” that Kant proposed to get out of a bind—the bind of holding logical rules to be absolutely necessary rules of human mind given a priori by human mind, yet rules capable of being transgressed in operations of human mind—by a distinction of pure general logic from its application, a distinction between pure and applied logic, and  by a dissection of the latter in terms of posited cognitive powers. I want to press on the soundness of Kant’s distinction of pure and applied logic (and whether problems for Kant in this area also bear against Hanna). I also want to press on Kant’s conception that necessity in empirical science is a function of application of mathematics and of basic (Kantian) metaphysics in the empirical science.

Aristotle had noted the import of necessity by import of geometry into his account of the gross form of the rainbow.* Although, that sort of geometry application was a tidbit compared to the use of geometry by Descartes in theory of the rainbow, let alone the use of geometry by Newton in remaking the world. It was amid these modern roles for geometry that Kant did his thinking, of course.

Kant knew of Aristotle’s general doctrines on science. And via Leibniz, Kant was still hankering after them and to some extent resisting the scheme for making science brought on by Newton.

Aristotle had appealed to a mental faculty in describing how a logical principle, specifically PNC, is ascertained. That, as we have seen, was what in our time has been known as a power of intuitive induction or abstractive induction. That posit of faculty was quite opaque, and its (fallible) attachment to formal character in the world by the human mind’s proposed assimilation of said form—mind itself becoming external formalities—and Aristotle’s form-matter aspect of metaphysics were pretty roundly judged false in the modern era of philosophy (outside the preservation of Scholasticism by Catholic scholars).

I’ll close this installment by getting before us Kant’s basic treatment of the pure/applied distinction in logic.

“A logic that is general but also pure deals with nothing but a priori principles. Such a logic is a canon of understanding and of reason, but only as regards what is formal in our use of then—i.e., we disregard what the content may be (whether is is empirical or transcendental). A general logic is called applied, on the other hand, if it is concerned with the rules of the understanding as used under the subjective empirical conditions taught us by psychology. Hence such a logic empirical principles, although it is general insofar as it deals with our use of the understanding without distinguishing the understanding’s objects. . . . In general logic, therefore, the part that is to constitute the pure doctrine of reason must be separated entirely from the part that is to constitute applied (though still general) logic. Only the first of these parts is, properly speaking, a science . . . . In such pure general logic, therefore, the logicians must always have in mind two rules:

  1. As general logic, it abstracts from all content of the cognition of understanding and from the difference among the objects of that cognition, and deals with nothing but the mere form of thought.
  2. As pure logic, it has no empirical principles. Hence it does not (as people have sometimes come to be persuaded) take anything from psychology; and therefore psychology has no influence whatever on the canon of the understanding. Pure general logic is demonstrated doctrine, and everything in it must be certain completely a priori.

“What I call applied logic is a presentation of the understanding and of the rules governing its necessary use in concreto, viz., its use under the contingent conditions attaching to the subject {the mind–SB}, conditions that can impede or promote this use and that are, one and all, given only empirically. . . . Pure general logic relates to applied general logic as pure morality relates to the doctrine proper of virtue. Pure morality contains merely the moral laws of a free will as such; the doctrine of virtue examines these laws as impeded by the feelings, inclinations, and passions to which human beings are more or less subject. The doctrine of virtue can never serve as true and demonstrated science; for, just like applied logic, it requires empirical and psychological principles.”

(KrV A53–55 B77–79 – Werner Pluhar translation).

To be continued.

(This post and all those preceding it in this thread were restored from my word-processing file for the thread today because I accidentally deleted the entire thread this morning. I apologize to all who had posted in the thread, which posts were not entered into my word-processing file. At the time I deleted the thread, by the way, the number of hits on it had been 7038.)

Edited by Boydstun

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6 hours ago, Boydstun said:

To set myself the task “weed this patch of periwinkles” I may need to use language.  The two popular weeds there at this season are dog violets and a native vine I don’t know the name of. Getting to the nub of that weed-vine among the thicket of periwinkle vines and pulling out the former without pulling out the latter is a challenge. Names and language do not seem to be enlisted in executing the task; they enable only my report of this work. The weed-vine and the periwinkle are of different leaf shape and color. Tug gently on the end of the weed-vine reaching for the sun. You won’t be able to see the weed-vine you’re tugging but a few inches before it disappears (leafless in this portion of it) among the thicket of periwinkle vines hugging the earth and putting down their roots continually along their way. But as you tug on the weed-vine, you’ll be able to find with your other hand that single vine being tugged. It is tightly tensed and in synchrony with any rhythm of tugs you apply with the other hand. Repeat from there, and eventually you arrive at the nub of the weed-vine and pull out that vine by the root.

Pause at a step in which you have the single obscured weed-vine in each hand. Pull with the one hand, feel the pull in the other. That is a perceived connection between two distinct events. At this point, philosophers from Plato and Aristotle to Hume and Kant stick up their noses. Not Locke.

Been here, done this. The particular of the weed/plant is not as relevant as the ability to discern the difference. More interestingly is the identification of the Aristotelian tendency to stick up its nose at it, and the omission of such recollection from the previous recollection.

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PNC Ground Shifts to the Side of the Subject – Kant IV-c

Let’s test that analogy Kant floated (see last quotes from KrV in this post) between his distinction Pure Logic v. Applied Logic and his distinction Pure Morality v. Proper Virtue. Among the things he writes concerning the latter distinction are the following.

“[Moral laws] hold as laws only insofar as they can be seen to have an a priori basis and be necessary” (1797 6:215). Kant does not here mean necessary as in necessary for some purpose, though such laws are required for some purpose; he means just plain necessary as when we say something is necessarily so. By the proposed analogy, we have: Pure general logical laws hold only insofar as they can be seen to have an a priori basis and be necessary. 

By its moral laws, “reason commands how we are to act even though no example of this could be found, and it takes no account of the advantages we thereby gain, which only experience could teach us. For although reason allows us to seek our advantage in every way possible to us . . . still the authority of its precepts as commands is not based on these considerations. Instead it uses them (as counsels) only as a counterweight against inducements to the contrary, to offset in advance the error of biased scales in practical appraisal [for what best to do], and only then to insure that the weight of practical reason’s a priori grounds will turn the scales in favor of the authority of its precepts.” (1797 6:216)

Rather like F. Bacon and Locke, Kant thought of errors we make in applied logic as largely from prejudices (Jäsche 1800, §§75–81; Lu-Adler 2018, 67, 102, 114), and this tunes pretty well with the role of prejudice for advantage that Kant sees in the origin of moral errors. Learning applied logic—which like pure logic, applies to any topic whatever—can free one from prejudices that come onto the scene in the workings of empirical psychology as logic is put to concrete occasions of thinking. Transcendence of prejudices in the arena of applied logic frees one to think for oneself (Lu-Adler 2018, 114). All analogies break down in some ways, but I take seriously for Kant’s account of logic the conception of commanding by principles from pure logic as analogous with commanding by pure morality.

Kant characterized pure general logic as well as pure morality as objective, in contrast to subjectivity in the arena of applied general logic and in the arena of proper virtue. In pure morality, there is found a necessary law for all rational beings connected a priori with the concept of a rational will as such.


The subjective ground of desire is an incentive; the objective ground of volition is a motive; hence the distinction between subjective ends, which rest on incentives, and objective ends, which depend on motives, which hold for every rational being. Practical principles [moral principles] are formal if they abstract from all subjective ends, whereas they are material if they have put these, and consequently certain incentives, as their base. The ends that a rational being proposes at his discretion . . . (material ends) are all only relative; for only their mere relation to a specifically constituted faculty of desire on the part of the subject gives them their worth, which can therefore furnish no universal principles, no principles valid and necessary for all rational beings and also for every volition . . . Hence all these relative ends are only the ground of hypothetical imperatives.

But suppose there were something the existence of which in itself has an absolute worth, something which as an end in itself could be a ground of determinate laws . . . .

Now I say that the human being and in general every rational being exists as an end in itself, not merely as a means to be used by this or that will at its discretion; instead he must in all his actions, whether directed to himself or also to other rational beings, always be regarded at the same time as an end. . . . Rational beings are called persons because their nature already marks them out as an end in itself, that is, as something that may not be used merely as a means, and so far limits all choice (and is an object of respect). These, therefore, are not merely subjective ends, the existence of which as an effect of our action has a worth for us, but rather objective ends, that is, beings the existence of which is in itself an end . . . .

The human being necessarily represents his own existence in this way; so far it is thus a subjective principle of human actions. But every other rational being also represents his existence in this way consequent on just the same rational ground that also holds for me; thus it is at the same time an objective principle from which, as a supreme practical [moral] ground, it must be possible to derive all laws of the will. (Kant 1785, 4:427–29; also 1788, 5:86–88)

Kant does not apply the sort of argument in that last paragraph to support his contention that pure general logic is objective. Kant uses objective in various senses. He does not reach the sense of objectivity to be recognized in pure general logic by recognizing merely that this logic is common to all agents of human cognition. Indeed, applied logic and the errors it diagnoses and remedies are also at hand in all such agents. Applied logic is subjective, in one of Kant’s senses of the term, simply because it treats not purely a priori, necessary norms for how we ought to think, but partly how we actually think. In my estimate, however, Kant’s contention that principles of pure logic have normative character at all is nothing known independently of concrete operations of thought. 

It was not  unreasonable to consider a prior, necessary logical norms as objective, but I submit that their objective character we discern comes not at all from their purportedly a priori normative element, from their character as “commanding,” but from the a priori standing and necessity in the formal structures of logic, as in those two features of the formal structures of pure mathematics. They are objective in the sense that you cannot get around them. They are stubborn, as Whitehead would say. Pure geometry of Kant’s era, preeminently Euclid’s geometry, provides norms for all sorts of practical constructions and discernments, but normative import of pure geometry is incidental to it.

In contrast to the character of pure geometry, normative character, in the view of Kant, is essential to pure general logic and, as well, to applied general logic. The distinction Kant drew under those titles of general logic do not hold so sharply as he had thought in his general pronouncements about the distinction. He drew other sorts of divisions of what may reasonably be called logic, but this particular distinction is a failure. In his lectures on logic, Kant would count the prescriptions that we now call formal fallacies of inference as belonging to pure general logic, for they are norms purely of form, and they treat no psychological impediments; the informal fallacies of inference would be glossed as applied general logic, for they concern partly how we actually think, not only how we ought to think, and they deal with contingent, psychological impediments. 

Among logical prescriptions for concepts, Kant has rules for definitions of concepts. Examples would be that (i) species must parse the members under the concept such that each member belongs exclusively under one of the concept’s species or another and (ii) definitions should be in terms of internal characteristics of things under the concept being defined. These rules are guides at work unconsciously in untutored concrete reason, according to Kant’s lectures, and logic only makes them abstract, explicit formulas. The standing Kant conceived for the logical rules of judgment and rules of inference is quite different from the standing he gives to those logical rules for concepts. All these rules of general logic are for perfection of cognition as to form, but the premier rules for judgments and inferences are as commands coming externally to concrete operations of reason, tutored or untutored. It will be noticed that a definition of a concept is a judgment, so theory of the perfection of concepts as to form is brought to completion by logical rules for judgments. This entire scheme of externality and command-character with respect to concrete judgment and deduction is a forced contrivance, not, I say, a sound characterization of logic that is in our elementary texts and in Kant’s own logic lectures.

“The proposition, No thing can have a predicate that contradicts it, is called the principle of contradiction” (A151 B190). Kant rightly held that the premier fundamental principle of formal rightness in judgments is the principle of (non)contradiction (further, Pluhar 1987, lxxxixn90). He rightly held that the fundamental principle in deductive inference is not PNC, but this: whatever holds of the genus or species holds also of all things under the genus or species (Kant 1762, §2; 1792, 773; cf. Peikoff 1964, 134–35*).

“A judgment is the way that concepts belong to one consciousness universally, objectively” (Kant c.1780, 928). To say “God is moral and is not moral” is a contradiction. Is avoidance of such perfectly plain error of contradiction the great command from pure general logic concerning judgments? I hardly think so. What I see Kant wielding in his logic lectures is the likes of “God is moral and evil is not punished.” That is what was traditionally called contradictio in adjecto. In the example, one is forgetting the elementary traits of God and contradicts that concept of God in contending that evil is not punished. I want to stress that not forgetting elementary aspects of what one is talking about is touching on contingent, empirical psychological processes and hardly a purely a priori rule.

In the next installment, I intend to end the long study that is this thread. So we’ll end with the finish of these Kant issues, and I’ll not then be doubling back for more on conventionalist theories of logic. In that next, final installment, I’ll show the untenability of Kant’s sharp divide of pure general logic and applied general logic as it pertains to deductive inference and the failure of Kant’s attempted escape from the criticism (e.g. from Peikoff 1964) that Kant cannot have both his way of norm-setting by general logic yet its necessity of law straightly descriptive of human thought (as Kant appeared to maintain, on the plain face of it, in the Jäsche Logic [and in the Wiener Logic, unavailable for Peikoff 1964]). 


Gregor, M. J., trans. 1996. Immanuel Kant – Practical Philosophy. Cambridge.

Jäsche, G. B. 1800. Kant’s Logic. In Young 1992.

Kant, I. 1762. The False Subtlety of the Four Syllogistic Figures. In Walford and Meerbote 1992.

——. c.1780. The Heschel Logic. In Young 1992.

——. 1781. Critique of Pure Reason. W. S. Pluhar, trans. 1996. Hackett.

——. 1785. Groundwork of the Metaphysics of Morals. In Gregor 1996.

——. 1788. Critique of Practical Reason. In Gregor 1996.

——. 1790. Critique of Judgment. W. S. Pluhar, trans. 1987. Hackett.

——. 1792. The Dohna-Wundlacken Logic. In Young 1992.

——. 1797. The Metaphysics of Morals. In Gregor 1996.

Lu-Adler, H. 2018. Kant and the Science of Logic. Oxford.

Peikoff, L. 1964. The Status of the Law of Contradiction in Classical Ontologism. Ph.D. dissertation, New York University.

Pluhar, W. S. 1987. Introduction to Kant 1790.

Walford, D. and R. Meerbote. 1992. Immanuel Kant – Theoretical Philosophy, 1755 – 1770. 

Young, J. M., trans. 1992. Immanuel Kant – Lectures on Logic. Cambridge.

Edited by Boydstun

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PNC Ground Shifts to the Side of the Subject – Kant IV-d

In elementary logic, we have forms of inference that are valid inferences and forms of inference that are erroneous. An example of the latter is: “If a raccoon got the bird feeder down, then some raccoons are clever. Some raccoons are clever. Therefore, a raccoon got the bird feeder down.” This is the formal fallacy of Affirming the Consequent. Its general form is “If p, then q / q / therefore p.” I say that in no way is the invalidity of an argument of this form in general the cause of its invalidity in “application” to reasoning about raccoons and downed bird feeders. Our witness of the failure of right reasoning in the “application” is no less immediate than it is in the “purely formal” rendition.

In fact the cloistering from worldly particulars we have in the general form of the fallacy could reasonably leave one wondering why it has been called out as a fallacy. After all, in its general form, it is a mystery why anyone would give such an argument. Only in an “application” does the reason emerge. For there one’s genuine belief (gotten from outside this argument) of the truth of the second premise gives that premise a shininess that can for a split second loosen one’s fix on the train of the argument at hand. The point of the formality of the formal fallacy (wrong-in-form) Affirming the Consequent is only that it is a general no-no across any substitution sentences for p and q. The formal fallacy, which one might reasonably associate with Kant’s pure general logic, derives its normatively from its “applications”. This formal fallacy is no commander, contrary the current Kant insisted.

(If one found upon taking the formal case in which q just is p that p is not-p, one might give the formal exhibition of the fallacy Affirming the Consequent some serious normative sense merely of itself. However, that is not what one finds upon entering p for each occurrence of q in the layout.)

We have fallacies of deduction that are called informal, for they derail necessary conveyance of truth from premises to conclusion, although they have no crisp exhibition of general form such as is available for the fallacy of Affirming the Consequent. Example of an informal fallacy is the Fallacy of Accident. “Everything has a reason for being this way, not that. Therefore, there is a reason anything at all exists, rather than there being nothing at all.” The premise is true over a very wide range, and it is reasonably stated as a general truth in a wide range of scenes of thought. But there are some few settings in which the premise might well be false, so when affirmed as universally true, it should, strictly speaking, be affirmed only with the qualification that such-and-such setting is out of consideration. Otherwise, strictly speaking, the premise is false. Then truth is not being conveyed from the premise to the conclusion in my example. In our standard elementary formal logic today, that circumstance would not invalidate the inference, really, because the first proposition of my example for this fallacy is seen as only a conditional for the second proposition. Whereas, for Kant as for Aristotle, the two propositions are seen as  something more: they are seen as an argument. Anyway, formal logic seen as about correct and incorrect argumentative form, would for Kant count as his “pure general logic” and, fact is, that form does not source the norm had by raising up in “applied general logic” the fallacy we call Fallacy of Accident.

Formal logic is not autonomously commanding norms into logical inferences. It is not, freely from all character of the world, determining fallacies as fallacies, formal or informal. Then too, logically valid forms in their instances do not derive their correctness from the correctness also plain in the crisp exhibition of that general form of deduction. “All animals are mortal. Ralph is an animal. Therefore, Ralph is mortal.” That the conclusion is necessarily true if those premises are true does not derive from the denuded form “All A are B / this is an A / therefore, this is a B” giving high-altitude directions to lower-altitude craft.

Does Kant’s “pure general logic” amount only to formal logic as in our elementary textbooks? Is the formality of his pure general logic only that formality, schema-forms of correctness in concepts and judgements for any subject matter and any sorts of objects, a formality in logic embraced by all logic lovers from Aristotle to us? No and no.

(To be continued.)


[1] Kelley 2014, 120–21. Aristotle’s fallacy of accident in Sophistical Refutations and Rhetoric is ancestor to this fallacy, though it bears only a pale resemblance. Fallacy of Accident appears in §404 of Meier 1752, the logic text from which Kant lectured. It makes no appearance in the Jäsche Logic, though it shows up in one set of student notes of Kant’s lectures, the Hechsel. According to those notes, Kant claimed that if one “tried to clothe this [fallacy] in the form of a syllogism, the vitium would at once strike the eye” (110). This seems to say that one would fail to get it into a valid syllogistic form and that that failure would show the formal root of the logical error. The long tradition of supposing that what we call informal fallacies were somehow rooted in or reducible to formal errors may have, I suggest, bolstered Kant’s thinking that form (in an amplified version peculiar to him—subsequently influential—to be sorted in the next installment) is the bottom line of all norms of general logic. Some scholars today maintain that Aristotle supposed all the errors he compiled (which is a subset) of the lot we list as informal fallacies were reducible to errors of syllogism, which is to say errors of form. See Woods 2012, 523–24, 531, 570.


Kelley, D. 2014. The Art of Reasoning. 4th ed. Norton: New York.

Woods, J. 2012. A History of the Fallacies in Western Logic. In Handbook of the History of Logic. Gabbay, D. M., F. J. Pelletier, and J. Woods, editors. North Holland: Amsterdam.

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PNC Ground Shifts to the Side of the Subject – Kant IV-e

I have obtained the dissertation of John MacFarlane titled What Does It Mean to Say that Logic Is Formal? (2000). This work includes engagement with the problem for Kant highlighted by Peikoff 1964 of how a logical principle such as PNC can be necessary to thought being thought at all, yet a norm for thought, which implies there are thoughts that can violate the logical principle. MacFarlane has a surprising resolution for Kant, an interpretation against which I shall argue. I shall give my bottom-line assessment of Kant’s attempt to resolve his bind by appeal to his pure/applied distinction for general logic.

There is gold for Peikoff 1964 and for us in MacFarlane 2000. The latter sets out conceptions of form original with Kant that have influenced subsequent thought about logic, subsequent philosophy of logic. (Cf. Kant’s revision of the concept of the a priori and the effects of that in subsequent philosophy: Tait 1992; Peikoff 2012, 27–28, 33–34, 46–51, 56–61.)

Firstly, we should get refreshed on Kant’s views on logic as expressed in Critique of Pure Reason. Then I want to get before us more extensively than before text from the Jäsche Logic, text over which turns the contest of Kant-interpretation bearing on Kant’s purported deep failure on the descriptive-yet-normative character of formal logic: MacFarlane v. Peikoff, Putnam, me


In the perspective of Kant, reasoning in accordance with logic can falter due to various empirical circumstances of the reasoning mind. Knowing those pitfalls and how to avoid them is what Kant called applied general logic. Principles of applied logic are partly from empirical principles. As for the principles of pure general logic, logic apart from applications, “it has no empirical principles” (A54 B78). Matter of fact, pure general logic abstracts away “from all reference to its object” (A55 B79).

“Pure general logic is a science that provides nothing but a comprehensive exposition and strict proof of the formal rules of all thought” (Bxiii). The office of logic is “to abstract from all objects of cognition and their differences; hence in logic the understanding deals with nothing more than itself and its form” (Bix; A131 B170).

Knowledge requires the joint operation of a receptivity of the mind and a spontaneity of the mind. In our receptivity, sensible objects are given to us. In our spontaneity of conceptualization and judgment, those objects are thought (A15 B29). Sensory presentations are givens. The spontaneity of cognition is the ability to produce presentations ourselves. Kant calls understanding the faculty for bringing given sensible objects under concepts and therewith thinking those objects (A50–51 B74–75). Pure logic is “the science of the rules of the understanding as such” (A52 B76). These are “the absolutely necessary rules of thought without which the understanding cannot be used at all” (A52 B76).

Kant distinguishes the faculty of understanding from its superintendent, the faculty of reason. The understanding can arrive at universal propositions by induction. Correct syllogistic inferences among propositions are from reason (A130 B169; A303–4 B359–60). By its formal principles, reason provides unity to the rules of understanding (A302 B359). Kant regarded pure logic as “a canon of understanding and of reason” (A53 B77; A131 B170). A canon is a standard or rule to be followed.

Kant thought that our receptivity of given sensory presentation is not cognitive and requires conceptualization in order to become experience (A50–51 B74–75). “All experience, besides containing the senses’ intuition through which something is given, does also contain a concept of an object that is given in intuition, or that appears. Accordingly, concepts of objects as such presumably underlie all experiential cognition as its a priori conditions” (A93 B126). The sensory given presentation contains particular and specific information about the object that can be thought in concepts and judgments concerning the object. But the most general and necessary forms of objects in experience is not information supplied by the sensory given presentations (sensory intuitions), but by the understanding itself for agreement with itself (B114–16, B133n).

Without the general form of objects supplied by the understanding, there is no cognitive experience of an object. “Understanding is required for all experience and for its possibility. And the first thing that understanding does for these is not that of making the presentation of objects distinct, but that of making the presentation of an object possible at all” (A199 B244).

According to Kant, we could have no experience of objects without invoking concepts bearing, independently of experience, certain of the general forms had by any object whatsoever. The unity-act of the understanding that is the conceptual act, which gives a unified content, an object, to given sensory presentations is also the very unity-act that unifies the various concepts in a judgment (A78–79 B104–5). 

An additional power Kant gives to the understanding is the power of immediate inference. From a single premise, certain conclusions can be rightly drawn. “The proposition All human beings are mortal already contains the propositions that human beings are mortal, that some mortals are human beings, and that nothing that is immortal is a human being” (A303  B360). In these inferences, all of the material concepts, human being and mortal, appearing in conclusions were in the premise. Such inferences can be made out to be the mediate inferences of a syllogism, but only by adding a premise that is a tautology such as Some mortals are mortal (D-W Logic 769; Jäsche Logic 115).

Mediate inferences require addition of a second judgment, a second premise, in order to bring about the conclusion from a given premise. The proposition All scholars are mortal is not contained in the basic judgment All men are mortal since the concept scholar does not appear in the latter. The intermediate judgment All scholars are men must be introduced to draw the conclusion (A304 B360).

The basic judgment—the major premise of the syllogism—is thought by the understanding. This is the thinking of a rule. Under condition of that rule, the minor premise of the syllogism is subsumed, by the power of judgment. Lastly, reason makes determinate cognition by the predicate of the basic rule the new judgment, which is the conclusion (A304 B360–61).

“What usually happens is that the conclusion has been assigned as a judgment in order to see whether it does not issue from judgments already given, viz., judgments through which a quite different object is thought. When this is the task set for me, then I locate the assertion of this conclusion in the understanding, in order to see whether it does not occur in it under certain conditions according to a universal rule. If I then find such a condition, and if the object of the conclusion can be subsumed under the given condition, then the conclusion is inferred from the rule which holds also for other objects of cognition. We see from this that reason in making inferences seeks to reduce the great manifoldness of understanding’s cognition to the smallest number of principles (universal conditions) and thereby to bring about the highest unity of this cognition.” (A304–5 B361)

The faculty of reason, in contradistinction from understanding, does not deal with given sensory presentations, but with concepts and judgments. “Just as the understanding brings the manifold of intuition under concepts and thereby brings the intuition into connection,” so does reason “bring the understanding into thoroughgoing coherence with itself” (A305–6 B362).


From the Jäsche Logic:

“Everything in nature, both in the lifeless and in the living world, takes place according to rules, although we are not always acquainted with these rules. — Water falls according to laws of gravity, and with animals locomotion also takes place according to rules. . . . The whole of nature in general is really nothing but a connection of appearances according to rules; and there is no absences of rules anywhere. . . . 

“The exercise of our powers also takes place according to certain rules that we follow, unconscious of them at first, until we gradually arrive at cognition of them through experiments and lengthy use of our powers, indeed, until we finally become so familiar with them that it costs us much effort to think them in abstracto. Thus universal grammar is the form of a language in general, for example. One speaks even without being acquainted with grammar, however; and he who speaks without being acquainted with it does actually have a grammar and speaks according to rules, but ones of which he himself is not conscious.

“Like all our powers, the understanding in particular is bound in its actions to rules, which we can investigate. . . .

“We cannot think, we cannot use our understanding, except according to certain rules. But now we can in turn think these rules for themselves, i.e., we can think them apart from their application or in abstracto.” (11–12)

“A principal perfection of cognition, indeed, the essential and inseparable condition of all its perfection, is truth. Truth, it is said, consists in the agreement of cognition with its object. In consequence of this mere nominal explanation, my cognition is supposed to confirm itself, which is by cognizing it. Hence my cognition, to count as true, is supposed to agree with its object. Now I can compare the object with my cognition, however, only by cognizing it. Hence my cognition is supposed to confirm itself, which is far short of being sufficient for truth. For since the object is outside me, the cognition in me, all I can ever pass judgment on is whether my cognition of the object agrees with my cognition of the object. . . .

“The question here is . . . whether and to what extent there is a criterion of truth that is certain, universal, and useful in application. For this is what the question, What is truth?, ought to mean.

“To be able to decide this important question we must distinguish that which belongs to the matter in our cognition and is related to the object from that which concerns its mere form, as that condition without which a cognition would in general never be a cognition. . . .

". . . [There can be no criterion of material truth common to all sciences, Kant here maintains.]

“If the question is about universal formal criteria of truth, however, then here it is easy to decide that of course there can be such a thing. For formal truth consists merely in the agreement of cognition with itself, in complete abstraction from all objects whatsoever and from all difference among them. And the universal formal criteria of truth are accordingly nothing other than universal logical marks of the agreement of cognition with itself or—what is one and the same—with the universal laws of the understanding and of reason.

“These formal, universal criteria are of course not sufficient for objective truth, but they are nonetheless to be regarded as its conditio sine qua non.

“For the question of whether cognition agrees with its objects must be preceded by the question of whether it agrees with itself (as to form). And this is a matter for logic.” (49–51)


As an aside, I’d like to point out that that talk of preceding, genetic or analytic, is false. For the same reason, Kant’s view that there can be no experience of objects without application of concepts to objects (and concept of an object in general) of sensory percepts is false. That flies in the face of experience. I can be thinking of the concepts I’m typing, yet be aware of objects in my surroundings without engaging concepts or names of them. Then too, prelinguistic children as well as the higher animals plainly perceive the things they navigate and use. The falsity of both of those claims by Kant is exposed by realizing they are cases of one of his favorite logical fallacies displayed in his logic lectures, namely, the contradictio in adjecto. Conceptual operations presuppose manual operations, physical commerce. Awareness of agreement in one’s thought presupposes agreement in physical acts and awareness of them. Consciousness of mental operations presupposes consciousness of acts and objects in the world, consciousness of the world. Hat tip to Ayn Rand on much of that.

Additionally, aside, I dissent on Kant’s presumption that there is nothing of the formalities of pure logic or pure mathematics obtaining in perceived things, formalities independent of our perception or higher cognition of them. (On that see my forthcoming paper in JARS on basic metaphysics.) Rand dissented on a parallel, common view—empiricist, rationalist, or Kantian—concerning similarity relations (ITOE 14). Allan Gotthelf: “So, ‘similarity’ is an epistemological concept, and a formulation of the metaphysical base of that would be: quantitative difference within a range.” Rand: “That’s right” (ITOE App. 140).

(To be continued.)


Conant, J., editor, 1994. Words and Life – Hilary Putam. Cambridge: Harvard University Press.

Detlefsen, M., editor, 1992. Proof and Knowledge in Mathematics. New York: Routledge.

Jäsche, G. B. 1800. Kant’s Logic. In Young 1992.

Kant, I. 1781, 1787. Critique of Pure Reason. W. S. Pluhar, translator. 1996. Indianapolis: Hackett.

——. 1792. The Dohna-Wundlacken Logic. In Young 1992.

MacFarlane, J. G. 2000. What Does It Mean to Say that Logic Is Formal? Ph.D. dissertation. University of Pittsburgh.

Peikoff, L. 1964. The Status of the Law of Contradiction in Classical Logical Ontologism. Ph.D. dissertation. New York University.

——. 2012. The DIM Hypothesis. New York: New American Library.

Putnam, H. 1994. Rethinking Mathematical Necessity. In Conant 1994.

Rand, A. 1990 [1966–67]. Introduction to Objectivist Epistemology. Expanded 2nd edition. New York: Meridian.

Tait, W. W. 1992. Reflections on the Concept of A Priori Truth and Its Corruption by Kant. In Detlefsen 1992.

Young, J. M., translator, 1992. Immanuel Kant – Lectures on Logic. Cambridge: Cambridge University Press.

Edited by Boydstun

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PNC Ground Shifts to the Side of the Subject – Kant IV-e (cont.)


In the course of Hilary Putnam’s paper “Rethinking Mathematical Necessity,” he reflects on Kant’s conception of logic 

“from a desire to understand an intuition of Wittgenstein’s that I never shared. For the early Wittgenstein it was somehow clear that logical truths do not really say anything, that they are empty of sense (which is not the same thing as saying they are nonsense), sinnlos if not unsinnig. .  . . I felt dissatisfaction . . . with my ability to put myself in Wittgenstein’s shoes . . . and to even imagine the state of mind in which one would hold that truths of logic are ‘tautologies’, that they are sinnlos. It was then I thought of Kant.

“. . . What interests me here . . . is [Kant’s] repeated insistence that illogical thought is not, properly speaking, thought at all.” (246)

We have seen Kant speak, according to the Jäsche Logic, of pure general logic as “a science of the necessary laws of thought, without which no use of the understanding or of reason takes place at all, laws which are consequently conditions under which the understanding can and ought to agree with itself alone. . . .” (13) This is just the point at which Kant shows the conflict between his characterization of logical rules as the defining operating rules of thought and as norms for successful thought. That same dual inconsistent character had come up in Critique of Pure Reason. “All our judgments as such . . . must not contradict themselves. For otherwise these judgments, even in themselves (i.e., even with their object left out of account), are nothing” (A150 B189–90).

MacFarlane (2000) does not take Kant to be ever ascribing to pure general logic necessary rules of thought such that violation would make the mental proceeding not thinking at all, which of course has the unacceptable consequence that it is impossible for us to make logical errors (54). MacFarlane rejects out of hand that Kant was saying, as Putnam and most any reader of Kant all along thought Kant was saying, that the logical laws as descriptive of thought are necessary to it being thought. Rather, MacFarlane takes Kant’s conception of pure general logic as only normative. There can be no conflict of N with D if there simply is no D being put forth.

I do not think the thesis that Kant did not put forth D is sustainable. As we have seen, the Jäsche Logic, which was compiled from Kant’s set of notes for his lectures, begins with characterizing logic as having lawful determination just as there is the law of gravity or laws for animal locomotion. Student notes of the lectures indicate as well that Kant opened with similitude of laws of logic with other laws. In the Bloomberg notes (early 1770’s), from the Precritical period, Kant focuses on the laws of walking for comparison. This is a nice comparison, for there can be principles for better walking rather like principles for better thinking. In his Critical period, Kant will not settle for only that in the normative character of logic. In the Wiener notes (c. 1780) and the Dohna-Wundlacken ones (early 1790’s), Kant has gone over to dwelling on comparison of the rules of pure logic with supposed universal grammatical rules for all language. Violation of such deep grammatical rules would seem to evict one’s string of words from the bounds of language in Kant’s intended sense of language; violation of the rules of pure logic would seem to evict one from the bounds of thinking. These student notes were not part of what scholars of Kant, even German ones, had available until more recent times. But they further support the idea that Kant in his Critical period, as exhibited in KrV and in Jäsche, took pure general logic in its normative, commanding character for thought as also definitive for thought.

Furthermore, unless we hold with the interpretation of Peikoff and Putnam, I do not see how to explain Kant, according to Jäsche, posing this problem: “It is hard to comprehend how error in the formal sense of the word, i.e., how the form of thought contrary to the understanding is possible, just as we cannot in general comprehend how any power should deviate from its own essential laws” (53). We have seen how Kant tried to solve this problem by dividing general logic into pure and applied, ascribing to the former the discernment of formal, logical rules and the handing down of those rules to the latter, to applied logic, where alone is the possibility of error. Our own contemporary distinction among logical norms sorts fallacies into formal ones and informal ones. Kant could well place our formal-fallacy proscriptions into his bin of pure general logic and our informal-fallacy proscriptions—together with specific concrete instances of general, formal ones—into his bin of applied general logic. Against Kant’s pure/applied divide and his use of it in characterizing logical norms and possibility of logical errors, I have argued: that rightness and wrongness in “applications” of the purely formal has no sense of being less directly discerned in the application; that some pure norms have gotten their normativity at least in significant part from application (e.g., Affirming the Consequent); and that some applied logical norms are not on account of wrong form (e.g., Fallacy of Accident). Kant’s theory of the zone and means of logical error is a failure. His theory of the source of logical norms is inconsistent with the fact that we make logical errors.

One might try to help Kant by making a parallel with the definition of human. A human is a rational animal. Lacking rationality, an animal is not a human. Yet humans are not always rational. Transporting this pattern to Kant’s conception of logic (in his Critical period, for which he is famous) would not fit with Kant’s picture that thought ceases to be thought if it ceases to be logical. It would not fit with his view that logic is a norm with absolute necessity, not only necessity for an aim. It would not fit with his view that such absolute necessity cannot be on account of the world. Rationality does not stand to human as logic stands to thought in Kant’s outlook. 

MacFarlane 2000 construes Kant to have been saying “that no activity that is not held accountable to [logical] rules can count as thought, and not that there cannot be thought that does not conform to these rules” (87). I contend that MacFarlane’s phrase “count as thought” comes to “count as successful thought.” This returns Kant’s Critical account back to his account as of the Pre-Critical logic lectures according to the Bloomberg notes. That is, it returns Kant’s account from analogy with a universal grammar to analogy with walking. This will not do for Kant’s mature view.

In the next post (hopefully the last for this study), I’ll turn to MacFarlane’s boost in viewing how Kant’s innovations in conceptions of form influenced subsequent philosophy of logic. Let us notice just now additional vista from Putnam 1994.

“[Kant distinguished] the truths of logic not only from empirical truths, but also from synthetic a priori truths. In the case of a synthetic a priori judgment, say, ‘Every event has a cause’, Kant tells us that what makes the judgment true is not the way the world is—that is, not the way the world is ‘in itself’—but the way our reason functions; but this talk of the function and constitution of human reason has to be distinguished (by Kant) from talk of the nature of thought . . . . There is, according to Kant, such a thought as the thought that there is an event with no cause; but I can know a priori that that thought is false, because the very constitution of my reason ensures that the data of the senses, as those data are represented by my mind, will fit into a certain structure of objects in space and time related by causality. There is a sense in which the negations of synthetic a priori truths are no more descriptions of a way the world could be than are the negations of logical truths. The negation of a synthetic a priori truth is thinkable . . . . The negation of a logical truth is, in a sense unthinkable; and it is unthinkable precisely because it is the negation of a logical truth. Explanation goes no further. ‘Logical truth’ is, as it were, itself an ultimate metaphysical category.

“. . . Frege prepares the way for Wittgenstein by identifying the Kantian idea of the nature of thought with the structure of an ideal language. . . . For the Wittgenstein of the Tractatus, the opposition is between logical truths and empirical truths, not between logical truths and synthetic truths in the Kantian sense. The problem of distinguishing the way in which the structure of thought (which, as just remarked becomes the structure of the ideal language) guarantees the unrevisability of logic from the way in which the structure of reason guarantees the unrevisability of the synthetic a priori no longer arises, because either a judgment is about the world, in which case its negation is not only thinkable, but, in certain possible circumstances, confirmable, or it is not about the world, in which case it is sinnloss.” (255)

(To be continued.)


Conant, J., editor, 1994. Words and Life – Hilary Putnam. Cambridge: Harvard University Press.

Jäsche, G. B. 1800. Kant’s Logic. In Young 1992.

Kant, I. 1781, 1787. Critique of Pure Reason. W. S. Pluhar, translator. 1996. Indianapolis: Hackett.

MacFarlane, J. G. 2000. What Does It Mean to Say that Logic Is Formal? Ph.D. dissertation. University of Pittsburgh.

Putnam, H. 1994. Rethinking Mathematical Necessity. In Conant 1994.

Young, J. M., translator, 1992. Immanuel Kant – Lectures on Logic. Cambridge: Cambridge University Press.

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PNC Ground Shifts to the Side of the Subject – Kant IV-e


Peikoff writes:

“For the Platonic essentialism, the Law of Contradiction is primarily and irreducibly a Law of Divine Thought; in consequence of this, it is a law in a number of further senses: Because God’s Intellect regulates the structure of the created world, it is a Law of Existence (and thus, in the descriptive sense, a Law of Human Thought as one of those created existents); and because God, by an act of direct illumination, communicates His knowledge of the Law to finite minds, it is therefore known by human minds, and thus acquires its status as a regulative or prescriptive Law of Human Thought.” (1964, 163)

In the Platonic line, as in others, the descriptive sense in which PNC is a law for the human mind is not the sense in which it is a norm for the human mind. The descriptive sense is afoot upon saying such things as “geometrical proofs are one thing, and dialectic is another” or “proof of the irrationality of the length of a diagonal of a square to its side is one thing, and the Pythagorean theorem is another” or “twelve is one thing, and five plus seven is the same thing.” How is PNC given to the human mind as a norm, in the Platonic line? It is not a norm in God’s mind; that mind cannot violate the law or make any other sort of error. How does conferring PNC as descriptive law on all existence, including the human mind, confer PNC as a norm for the human mind?

Enduring distinctness of what is and is not included in a class of things is taken to be a good thing by Platonists. Constant, distinct ideas are desired over ideas Platonists think could arise based only on the mutable realm of sensory experience. As Peikoff observes, Plato’s dialogues are explicit only about the human innateness and normative aspect of universal Forms. Later Platonists would add the innateness and normative aspect of subject-predicate relations between essences or classes (1964, 5n10, 30n80, 50). Platonists have it that PNC is clearly the way of things we cognize only in the intellectual realm of Forms and their relations; in the mixed, human realm of matter, sense, and intellect, PNC becomes a standard of inquiry, an aspiration.[1]

For the Platonist, that standard can be available to us only by being innate with us. “How . . . can sense-perceptions suggest standards to which they do not themselves conform unless, as I should maintain Plato himself argues, a knowledge of those standards is innately within the mind? In virtue of what can a mind recognize the deficiency of the objects of its perceptions and transcend them, if it has no knowledge—even in unconscious form—prior to or apart from its perceptions?” (33n82).

Platonists such as Leibniz and Cudsworth maintained that PNC must be operational in us to some degree in order for us to be thinking at all (10–12, 207–8). This does not land them, I say,  in the predicament of Kant of having principles such as PNC necessary for thought proceeding at all, yet violation of the principle possible in our thought. The pre-Kantian Platonists have a hard-and-fast division of (i) the realm of eternal essences and eternal truths and (ii) the material, mutable world, in which we must bumble along consciously trying to adhere to the principles such as PNC in our thinking. By contrast, Kant has only his anemic Platonic gesture to a division of pure general logic and applied general logic joined with his untenable assertion that the latter with its empirical factors in thought is the sole place and factor of any logical errors.

(To be continued.)


[1] Peikoff 1964, 43–45, 50–59, 138, 146–47, 154, 163, 188–211; 1967, 95; 1982, 17–18, 110; 1991, 29, 145–46, 158; 2012, 23–24, 27.


Peikoff, L. 1964. The Status of the Law of Contradiction in Classical Logical Ontologism. Ph.D. dissertation. New York University.

——. 1967. The Analytic-Synthetic Dichotomy. In Rand 1990.

——. 1982. The Ominous Parallels. New York: Stein and Day.

——. 1991. Objectivism: The Philosophy of Ayn Rand. New York: Dutton.

——. 2012. The DIM Hypothesis. New York: New American Library.

Rand, A. 1990 [1966–67]. Introduction to Objectivist Epistemology. Expanded 2nd edition. New York: Meridian.

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