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Boydstun

Peikoff's Dissertation

12 posts in this topic

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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 noncontradition (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.

 

References

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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My remarks in this post concerned issues undertaken by Peikoff 1964 (the first two of his five chapters) on Platonist perspectives on the epistemological and the ontological standing of PNC. My next post will concern Peikoff’s third and fourth chapters, on Aristotelian perspectives on those standings. In a third post, I’ll address Peikoff’s fifth chapter, on the demise of classical logical ontologism and some alternatives to it that were adopted.

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(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:

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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:

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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 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 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:

Quote

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.

Notes

[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, tinotiz); 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; 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.”

References

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 G, D, and E. 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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On 5/14/2017 at 7:06 AM, Boydstun said:

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.

The second sentence is pretty awkward.

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Posted (edited)

.

Peikoff's Dissertation

Prep

Plato

Aristotle I

To be continued.

~~~~~~~~~~~~~~~~

In the 'Aristotle I' post, I had written that “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 exists’” (1007b19–26). The translation I had quoted was by Ross. I made an error in my transcription. It should read ‘. . . so that nothing really exists.”

That translation of 1007b26 very possibly should be otherwise. These other ways squash the suggestion that here Aristotle is virtually stating Rand’s “Existence is Identity.” The translations of Kirwan 1993 and of Reeve 2016 do not say “. . . so that nothing really exists.” Rather, they say “. . . so that nothing is truly one” and “. . . so that there is nothing that is truly one.” If these later translations are truer to Aristotle’s text here, then the connection between existence and identity is here rather more indirect, turning on rigid attachment of oneness to existence and depending on the fullness of Rand’s Identity being covered by the variety of Aristotle’s ways of oneness.

New Reference

Reeve, C. D. C., translator, 2016. Aristotle, METAPHYSICS. Hackett.

http://ndpr.nd.edu/news/metaphysics/

Edited by Boydstun

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Posted (edited)

On 5/14/2017 at 7:06 AM, Boydstun said:

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

 

On 5/14/2017 at 7:06 AM, Boydstun said:

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]

 

The following is from ITOE, p. 177.  I think it may address the issue in the above two quotes?

Prof. F:  My question is about the relationship between concepts and propositions.  Concepts are logically prior, aren't they?

AR:  Yes.

Prof. F:  If every concept is based upon a definition, isn't that definition a proposition?

AR: Oh yes.

Prof. F:  Well then, the concept is in this case based on a proposition.

AR: No, but the first concepts are not.  First-level concepts, concepts of perceptual concretes, are held without definitions.  And I even mentioned in the book that most people would find it very difficult to define the commonplace, easy, first-level concepts for that very reason.  They are held first without any definitions, mainly in visual form, or through other sensory images. 

Would this apply to the constructive geometry employed by the Greeks?  That is, the "intuitive induction" mentioned above is a first-level concept that can be seen to be true prior to stating it in propositional (Euclidean Axiom) form?

Rand also states on p. 14:

Similarity is grasped perceptually; in observing  it, man is not and does not have to be aware of the fact that it involves a matter of measurement.

So beyond just "similarity," the observation and demonstration of relationships via constructive geometry are done automatically at the perceptual level?  That is, they form the basis of concepts, but are not concepts themselves until stated in propositional form.

Edited by New Buddha
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Posted (edited)

.

No, Budd. We do not have or even enter the discipline of geometry, such as that of Euclid without grasp of language stating axioms, postulates, definitions, and permitted constructions and remarking on the lettered diagrams. Prior to having language and any discursive concept, we have action- and image-schemata which of course entails some knowledge of geometrical forms (round, rollable ball / shape and motions of the dog); and without those attainments, an infant nearing its first word, its onset of first discursive concept, could not engage in the co-reference with caretakers by joint gaze or by finger pointing, which are necessary to attainment of first word. But that is not geometry in the sense of the structure Euclid lays before us, extensively and deeply illuminating character of space.

Euclid, by the way, should be a prerequisite for study of philosophy in its epistemological and metaphysical wings. Both of those wings today, of course, should also be informed by assimilation of the results of our modern science of cognitive developmental psychology* and our physics. The absolute prerequisites for approach of philosophy in its theoretical parts are an elementary course in logic and a course working through Euclid's plane geometry. (With a serious interest, one can learn those two things in one's own study, without taking a course. That geometry is offered in high school, and what a divine joy it was and is for me. I had elementary logic in college. Then on my own I was able to learn whatever more advanced geometry [synthetic, like Euclid and later, or analytic, like Descartes and later] and more advanced logic.) Not knowing Euclid's geometry is not knowing really much of what Plato or Aristotle or Kant or Peikoff (dissertation) or I are talking about in epistemology or metaphysics. Not knowing both Euclid and the syllogistic logic of Aristotle's from elementary logic is like having a flat tire when turing Aristotle's thought in theory of science, definition, and metaphysics. There is no substitute for opening Euclid and just doing it.

Euclid’s axioms (or common notions) are part of the starting points for the system and are not argued for. One is “things which are equal to the same thing are also equal to one another.” That is a relationship rather far beyond something one was simply presented in perception. It may fairly count as an intuitive induction, but we need to learn more about what that process was thought to be and what have been the problems with it. Work for the sequel of this study on LP’s dissertation.

Edited by Boydstun

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Posted (edited)

Is there a link you can provide to Peikoff's dissertation?

Edit: Woops, I see that there now is.

 

 

Edited by New Buddha

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2 hours ago, New Buddha said:

Is there a link you can provide to Peikoff's dissertation?

Edit: Woops, I see that there now is.

Where do you see this?

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30 minutes ago, KALADIN said:

Where do you see this?

Woops, I see that I'm wrong.  I thought that the 4th post provided links.  Maybe Boydstun can provide us one.

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Posted (edited)

On 7/4/2017 at 6:32 PM, Boydstun said:

Euclid, by the way, should be a prerequisite for study of philosophy in its epistemological and metaphysical wings. Both of those wings today, of course, should also be informed by assimilation of the results of our modern science of cognitive developmental psychology* and our physics.

I couldn't agree more, and after having read some links to your other works over the last couple of days, I have a better idea of where you are coming from.  I approach things from different "psychology/neurology" schools, however, and look forward to giving you some alternate ideas to ponder.  I think you are taking some things as "givens"  that aren't.

On 7/4/2017 at 6:32 PM, Boydstun said:

That geometry is offered in high school, and what a divine joy it was and is for me.

My "divine joy" was in the 2nd semester of college taking Intro to Physics and Analytical Geometry as preparation for later Engineering courses.  I was horrible at math up thru High School.  I remember thinking, "Damn, why didn't they teach these first!"

Edited by New Buddha

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