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The Status of the Law of Contradiction in Classical Logical Ontologism

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

Leonard Peikoff first met Ayn Rand when he was seventeen. That was in 1951. His cousin Barbara Wiedman (later Branden) had become a friend of Rand’s in the preceding year. The young friends of Rand had read and been greatly moved by her novel The Fountainhead, and they were greatly impressed with Rand and her philosophical ideas as conveyed to them in conversation with her. In 1953 Peikoff moved to New York from his native Canada (where he had completed a pre-med program) and entered New York University to study philosophy, which was his passion. He was able to read Atlas Shrugged in manuscript form prior to its publication and to converse with its author. He continued at NYU for his Ph.D. in Philosophy, which he completed in 1964. That was the year Allan Gotthelf entered graduate school in Philosophy.

Ayn Rand and her distinctive ideas on metaphysics and logic, as published in 1957 in Atlas Shrugged, do not appear in Peikoff’s dissertation. Except for one modest point, his treatment of his topic is consistent with Rand’s views on metaphysics and logic, as well as with her thought on universals (ITOE 1966–67) and her broad-brush arc of the history of philosophy. His dissertation is worthy of study, certainly by me, for what have been many of the positions and arguments concerning the ontological status and epistemological origin of the Principle of Noncontradiction (PNC) in Western philosophy from Plato to mid-twentieth century. It is valuable as well for a picture of what Peikoff could bring to the discussions with Rand and her close circle, as well as to their recorded lectures and published essays (including his own “Analytic-Synthetic Dichotomy” published by Rand as an immediate follow-on to her ITOE) in the ten years or so after 1957.

A speculative sidebar: Beyond Rand’s philosophy, I doubt that Leonard Peikoff ever had anything to learn from Nathaniel Branden in philosophy. The flow of learning in philosophy not Objectivism was likely entirely the other way. That goes for the flow of reliable information in that domain as well between Peikoff and Rand. By the late ‘60’s, Peikoff, and Rand too, could of course learn from the studies of Gotthelf in Greek philosophy.

I’ll sketch and comment on the course of the intellectual adventure that is Peikoff’s dissertation in a separate thread in Books to Mind. I’ll do that shortly. In the present thread, I want to just state his broad thesis (i–viii, 239–49), then turn (i) to the Kant resources Peikoff had available and relied upon in his story and (ii) to setting out from my own available resources, these decades later, what were Kant’s views and teachings on logic, what was always available in German, and what now in English.

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 contemporary conventionalist approaches to logical truth. Peikoff argues that infirmities in all the varieties of classical logical ontologism open the option of conventionalism. He mentions that his own sympathies are with logical ontologism. Alas, repair of its failures lies beyond the inquiry of his dissertation.

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Over a period of forty years (1756–96), Kant taught logic at least thirty-two times. Lecturers at the University of Konigsberg at that time were required to proceed upon a textbook recognized by the Prussian authorities. For his classes on logic, Kant used George Friedrich Meier’s Excerpt from the Doctrine of Reason (1752). The term excerpt (Auszug) here means that it treats its subject briefly, in contrast to a larger treatment. It does not mean the shorter treatment is a snippet from a larger work. Meier was a leading figure in the German Enlightenment. He had been a student of Christian Wolff and of Alexander Baumgarten. He studied John Locke in depth and helped introduce English philosophy in German lands. He straddled the rationalist and empiricist traditions. Auszug is not a text in formal logic, though it enters what we should be learning in an elementary logic text today by Meier’s bits on concepts, judgments, good definitions, proper inferences, and informal fallacies. Auszug touches on of all sorts of things that make the various types of human knowledge possible and the expression of knowledge excellent.

I set down here some sections from Auszug: §§109–11 (which bear on Peikoff’s remarks on p. 185) and §§43, 98, 103 referred to in the 109–11 stretch. The translation is by Aaron Bunch (Bloomsbury 2016).

§43. The ignorance of a human being is (1) an absolutely necessary and unavoidable ignorance, which he cannot avoid owing to the bounds of his power of cognition; and (2) a voluntary ignorance, whose contrary cognition he could attain if he wanted to.

§98. We must not assume: (1) that a cognition is true, just because we are aware of no internal impossibility in it; (2) that it is false, just because we are aware of no internal possibility in it; (3) that a cognition is true, the groundlessness and false grounds and consequences of which we are unaware; (4) that a cognition is false, of which we cognize no correct grounds and consequences. For we human beings are not all-knowing.

§103. Learned cognition can be false in a threefold way: (1) if the cognition of the things is false, although the cognition of the grounds is correct; (2) if the cognition of the grounds is false, although the cognition of the things is correct; (3) if the representation of the connection between the true grounds and consequences is incorrect . . . . Thus, a true learned cognition must be at the same time a correct cognition of the things, of the grounds, and of their connection . . . .

§109. Error consists in our taking false cognition to be true, and true cognition to be false. Consequently, (1) every erroneous cognition is false . . . (2) not every false cognition is erroneous, namely if we cognize that it is false; (3) error arises from false cognition. Had we no false cognition at all, we could also have no errors. Error is worse than merely false cognition, for error is a secret poison. Learned cognition can therefore be erroneous in a threefold way §103.

§110. Error arises §109 if we break the rules of the 98th paragraph. The first source or all errors is thus ignorance . . . , if it is accompanied  by haste, whereby we deny that of which we have no cognition.

§111. Error is either avoidable or unavoidable. The former arises from an avoidable ignorance, and the latter from an unavoidable ignorance §43. The former is nothing but a blameworthy disgrace to learned cognition, but the latter cannot and may not be avoided.

It’s safe to wake up now.

I should make two points concerning §110. The focus on haste in the production of error is likely simply a Cartesian hand-me-down analysis of error: human will outrunning human understanding. The focus on denials concerning things of which we have no cognition (also in §98) is mainly a bowing of the head to religious mysteries. Sealing obeisance to mysticism into theories of rational cognition was not an innovation of Immanuel Kant.

Kant lectured for his logic classes, cued from Meier’s Auszug, but Kant was allowed to register and did register objections and to use points in this approved text as springboards to state his own views concerning those points and their neighborhoods. We have a few sets of class notes taken by students in Kant’s logic lectures. One set was taken in the early 1770’s, so that would be after his Inaugural Dissertation (1770) and during the period in which he was turning his thinking around to full vista of his Critical philosophy, as would be brought to press in 1781 in Critique of Pure Reason (KrV). A couple of sets of logic-lectures student notes are from around 1780, when Kant was completing KrV. Another set is from the early 1790’s. We have English translations of all these sets of class notes, issued by Cambridge in 1992. It is interesting to follow the student notes from the Kant lectures corresponding to §§109–11 across those different years of notes. (Any one of them is more interesting than Auszug itself.)

One of Kant’s students was Gottlob Benjamin Jäsche. He had courses under Kant in 1791, and by end of the century, he had become a professor at Königsberg and an exponent of Kant’s philosophy. At Kant’s request, Jäsche composed a manual. It was issued in 1800, and it is titled Immanuel Kant’s Logic – A Manual for Lectures. Kant gave his lecture copy of Auszug, with all its margin notes and interleaved papers with notes that Kant had relied on at some time or other across his forty years of teaching logic. Jäsche tried to decipher the notes and include in the manual the notes he estimated to be in the later portion of Kant’s career. Kant never saw or approved the finished product. This manual remained available continuously in its original German to this day. It is given some wary weight by scholars trying to represent Kant’s views on logic. Today, Anglophone scholars know this work as The Jäsche Logic, and it is included with the other sets of student notes of Kant’s logic lectures translated into English in Immanual Kant – Lectures on Logic (Cambridge 1992).

In his dissertation, Peikoff had to rely on the portion of The Jäsche Logic that had been translated by T. K. Abbott into English in 1885 under the title Kant’s Introduction to Logic. The contemporary translator for the Cambridge collection mentions that “Abbott’s translation, though not bad, is so loose and so old-fashioned in its terminology that I have not made any use of it.” Peikoff cites the Abbott translation of the Jäsche production as simply Kant, not Jäsche. That seems to have been customary in the era of Peikoff’s dissertation. William and Martha Kneale’s monumental The Development of Logic issued in 1962. They too cite the Jäsche production as Kant, not Jäsche. Incidentally, Peikoff in his dissertation does not mention anything from this book. Perhaps he had not studied it in time. Information in this text sometimes improved on points advanced by Peikoff from his older resources, and I shall mention some of these (not related to Kant) in the next thread.

Writings of Kant himself would be the primary source in any representation of his view (which of course does not have any fog of translation for the German reader or scholar). I’d rate the original, German version of Jäsche’s manual as somewhere between a primary and a secondary source, and I’d rate that portion translated by Abbott (“not bad”) still between primary and secondary. It was unfortunate for Peikoff 1964 that Abbott did not include in his translation the Preface to the Jäsche Logic, that Abbott also did not include the logic-proper portion of the Jäsche Logic, and that Peikoff did not have, apparently, the placement of Kant in the history of logic lain out by Kneale and Kneale (1962).

Peikoff relied also on a squarely secondary source for Kant’s picture, a source originating in English in 1860. I’ll look at that source in another post on this thread another day, hopefully soon. In my next post today in this thread, I’ll copy from an earlier study of mine, what one can say of Kant’s views on logic drawing simply from KrV. I wrote that piece years before I had gotten hold of Peikoff’s dissertation, and my contrast therein to Kant is not anyone among predecessors of Kant, but to me and to my contemporary Ayn Rand. Peikoff had KrV in hand, in English, back in his dissertation days, but he makes no mention with citation of these elements of KrV in his dissertation. What can be gleaned from KrV should be posted in this thread as part of the orienting preparation for the next thread which will tackle Peikoff’s dissertation and its offspring square on and which will engage many philosophers besides Kant.

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Normativity of Logic – Kant v. Rand

Stephen Boydstun 2009

In the perspective of Immanuel 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 would call applied logic. Principles of applied logic are partly from empirical principles. As for the principles of pure logic itself, logic apart from such applications, “it has no empirical principles” (B78 A54). The principles of logic are not principles of empirical psychology, and their ultimate authority stems from something deeper than empirical necessities of thought.

Logic for Kant was Aristotelian logic. [Or so Kant thought. Stoic logic had been mixed into what he took for simply logic and credited to Aristotle.] He thought this discipline to have been set out completely by Aristotle, and he thought such finality of the discipline was due to the distinctive character of the discipline that is logic. “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; B170 A131). Logic is “vestibule” of the sciences in which we acquire knowledge. Logic is presupposed in all judgments constituting knowledge (Bix).

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 (B29 A15). 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 (B74–75 A50–51). Logic is “the science of the rules of the understanding as such” (B76 A52). These are “the absolutely necessary rules of thought without which the understanding cannot be used at all” (B76 A52).

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 (B169 A130; B359–60 A303–4). By its formal principles, reason provides unity to the rules of understanding (B359 A302). I should mention that it is not the role of reason (or of understanding) in logic that Kant tries to curb in his Critique of Pure Reason (B=1787 A=1781). This role Kant takes as within the proper jurisdiction of reason.

Kant regards logic as “a canon of understanding and of reason” (B77 A53; B170 A131). A canon is a standard or rule to be followed. How can rules of logic be rules to be followed by the understanding if they are the rules that characterize what is the form of all thought? How can the rules prescribe for X if they are descriptive of what X is? Let X be alternatively the faculty of understanding or the faculty of reason, the question arises.

Kant calls such logic general logic, and this he takes as abstracting away “from all reference of cognition to its object” (B79 A55). This conception of logic is significantly different from that of Rand: Logic is an art of identification, regimented by and towards the fact of existence and the fact that existence is identity.

Over a period of forty years, Kant taught logic at least thirty-two times. Syllogistic inference and non-contradiction were the rules for formal logic. Kant took these rules to concern some of the requirements for truth. They do not amount to all of the requirements for truth, “for even if a cognition accorded completely with logical form, i.e., even if it did not contradict itself, it could still contradict its object” (B84 A59). That much is correct, and Kant is correct too in saying that “whatever contradicts these rules is false” (B84 A59). Why? “Because the understanding is then in conflict with its own universal rules of thought, and hence with itself” (B84 A59).

How can the normativity of logic be accounted for if its principles are taken for correct independently of any relations they might have to existence and any of the most general structure of existence? Kant needs to explain how general logical norms for our thinking can be norms without taking their standard from the world and how such norms can be rules restricting what is possibly true in the world.

Might the source of norms for the construction of concepts be the source of norms for inferences when concepts are working in judgments? Can the normativity of forms of inference among judgments be tied to normativity in forms of judgments and normativity in the general forms of concepts composing those forms of judgment?

What requirements must concepts meet if they are to be concepts comprehending particulars in true ways? From the side of the understanding itself, the fundamental forms concepts may take are required to be systematically interconnected to satisfy the circumstance that the understanding “is an absolute unity” (B92 A67). Considered apart from their content, concepts rest on functions. “By function I mean the unity of the act of arranging various presentations under one common presentation” (B93 A68). So far, so good, but then Kant’s account stumbles badly.

Concepts are employed in the understanding to make judgments. In judgments, according to Kant, “a concept is never referred directly to an object” (B93 A68). Concepts, when not referring to other concepts, refer to sensory or otherwise given presentations (B177 A138–42). This is part of Kant’s systematic rejection of what he called intellectual intuition. That rejection is not entirely wrongheaded, but this facet of the rejection is one of Kant’s really bad errors. I say as follows: the fact that concepts relate perceptually given particulars does not mean that concepts do not refer directly to the particulars of which we have perceptual experience. It simply does not square with the phenomenology of thought to say that when we are using a concept we are not referring directly to the existents (or the possibility of them) falling under the concept.

Kant will have cut himself off from an existential source of normativity in judgment through concepts, thence a possible source of normativity for inferences among judgments, unless that normativity can be gotten through his indirect reference for concepts to existents through given presentations of existents. For Kant, as for most every epistemologist, concepts are unities we contrive among diverse things according to their common characteristics (B39 A25, B377 A320). The problem for Kant is that the diverse things unified are diverse given presentations in consciousness that become objects of consciousness only at the moments of conceptualization and judgment themselves (A103–6, A113–14, A119–23, B519–25 A491–97, B141–46). (Kant’s empirical realism, in A367–77, B274–79, and B232–47 A189–202, is subordinate to his transcendental idealism; but see Abela 2002 and Westphal 2004.)

The concept body can be used as a logical subject or in the predicate of a judgment. As subject in “Bodies are divisible,” body refers directly to certain given presentations of objects, but body does not refer to those objects unless in use in a judgment. In use for predicate in “Metals are bodies,” body refers to the subject concept metal, which in turn refers to certain given presentations of objects (B94 A69).

“The only use that the understanding can make of . . . concepts is to judge by means of them” (B93 A68). According to Kant, we cannot begin to understand the concept body otherwise than as in judgments. Right understanding of body means only knowledge of its particular right uses as the logical subject or in the logical predicate.

Kant observes that judgments, like concepts, are unities. It is the faculty of understanding that supplies those unities by its acts. The logical forms of judgment are not conformed to identity structures in the world or in given sensory presentations. Kant conceived those presentations as having their limitations set by relations of part to whole. He thought they could not also, in their state as givens, have relations of class inclusion (B39 A25, B377 A320). This is a facet of his overly sharp divide between sensibility and understanding. I have long held that relations of class inclusion are not concrete relations, unlike the relations of part-whole, containment, proximity, or perceptual similarity. That does not conflict, however, with the idea that what should be placed in which classes should be actively conformed to particular concrete relations found in the world.

Kant thought that our receptivity of given sensory presentation is not cognitive and requires conceptualization in order to become experience (B74–75 A50–51). “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” (B126 A93). 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” (B244 A199).

Kant is concerned to show that there are general patterns of necessity found in experience that are seamless with logical necessities. He errs in supposing that that seamlessness comes about because the general forms for any possible experience of objects logically precedes any actual experience of objects. That a percipient subject must have organization capable of perception if it is to perceive is surely so. Consider, however, that a river needs channels in order to flow, yet that does not rule out the possibility (and actual truth) that the compatibility of a valley and a river was the result of the flow of water.

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 (B104–5 A78–79).

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” (B360 A303). 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; J 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 (B360 A304).

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 (B360–61 A304).

“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.” (B361 A304–5)

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” (B362 A305–6). Reason provides cognition with logical form a priori, independently of experience. The principles of the understanding may be said to be immanent “because they have as their subject only the possibility of experience” (B365 A309). The principles of reason may be said to be transcendent in regard to all empirical givens.

The spontaneity of thought is unifying activity, whether in conceiving, judging, or inferring. Readers here will have probably noticed in Kant the themes of integration and economy, which are major in Rand’s analyses of cognition. However, for Kant the unifying activity of the understanding and of reason is not “an insight into anything like the ‘intelligible’ structure of the world” (Pippin 1982, 93).

Kant represents understanding and reason as working together as a purposive system. I maintain, in step with Rand, that all purposive systems are living systems or artifacts of those living systems. We hold that only life is an ultimate end in itself; life is the ultimate setter of all needs. The purposive system that is the human mind is the information-and-control system having its own dynamic needs derivative to serving the needs of the human individual and species for continued existence. Life has rules set by its needs for further life.

Life requires not only coherent work among its subsystems, but fitness with its environment. Rules of life pertain to both. Rules of mind pertain to both (cf. Peikoff 1991, 117-19, 147-48). Rules of logic do indeed enable coherent work of the mind, but they also yield effective comprehension of the world. Identity and unity are structure in the world, and, in their organic elaboration, they are structure of the viable organism (cf. ibid., 125–26). The normativity of logic arises from the need of the human being for life in the world as it is.



Abela, P. 2002. Kant’s Empirical Realism. Oxford.

Kant, I. 1992. Lectures on Logic. J. M. Young, translator and editor. Cambridge.

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

Peikoff, L. 1991. Objectivism: The Philosophy of Ayn Rand. Dutton.


Pippin, R. B. 1982. Kant’s Theory of Form. Yale.

Westphal, K. R. 2004. Kant’s Transcendental Proof of Realism. Cambridge.

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Peikoff’s squarely secondary source on Kant (d. 1804), relied upon in his dissertation, is Prolegomena Logica: An Inquiry into the Psychological Character of Logical Processes (1860, 2nd ed.). The author is Henry Longueville Mansel (1820–1871), an English philosopher, theologian, and priest. In philosophy he esteemed and followed broadly the Scottish thinker Sir William Hamilton (1788–1856). That is not the Sir William Hamilton we adore today. Our hero, the bringer of Hamiltonian Mechanics and inventor of quaternions was not from Edinburgh, but Dublin (1805–1865). William Hamilton of Edinburgh was a barrister, but with interest in research in formal logic and interest in German philosophy. He introduced Kant to English readers and tried to show harmony between Kant and the Scottish philosophers of Common Sense, such as Thomas Reid.

In Prolegomena to any Future Metaphysics (1783, 4:257–60, 369–71), Kant had cut down the possibility of success of Reid, Oswald, and Beattie in their appeals, however popular, to common sense for answer to Hume’s request for a warranty of thinking there to be any necessary connection between distinct events. Common sense, necessary and welcome as it is in the domain of its natural function, can only sensibly bear on that domain of perceptually manifest patterns (including applied mathematics), and it cannot be used to warrant any a priori principles, such as causality, contained in those patterns. Common sense cannot sensibly run on beyond its right domain and answer metaphysical questions. To the considerable fortune of our time, I should note the scholarship of Manfred Kuehn (1987) in tracking down the many ingresses of Scottish Common Sense philosophy to German lights, including Kant, in Scottish Common Sense in Germany 1768–1800.

Henry Mansel tells us in Prolegomena Logica that his own views are greatly indebted to Kant and to William Hamilton of Edinburgh, although, his own views do not coincide perfectly with either of theirs. One shared aim of Hamilton and Mansel was to concoct a bound to confound secular reason in its all-too-common vistas of materialism, atheism, and clockmaker deism. In 1858 Mansel gave a controversial lecture “The Limits of Religious Thoughts” plotting the old line that because we have no positive notions of God, the unconditioned, or the absolute (absolute in the senses: free from any relation as a condition of its existence or [ii] out of any relation to human knowledge), reason must be supplemented with faith, and no criticisms of theology based on human conceptions are valid. Follow-on treatises were lobed between J. S. Mill and Mansel.

C. S. Peirce (b.1839) was very familiar with the writings of William Hamilton of Edinburgh and the writings of Henry Mansel. In a personal manuscript of 1864: “I hold the Doctrine of Common Sense to be well fitted to Reid’s philosophical caliber and about as effective against any of the honored systems of philosophy as a potato-pop-gun’s contents might be against Gibraltar” (I.153). Peirce criticized Hamilton’s organization of syllogistic, with Hamilton’s quantification over predicates, in an 1865 lecture at Harvard (I.294–98). In 1886 Peirce would write:

“The quantification of the predicate, an idea originating with the botanist George Bentham, was for some years in vogue in this country and in England. It was developed with a singular defect of clear thought by Sir William Hamilton, and more ably by Stanley Jevons. . . . If the theory of the syllogism had been extended to relative terms, the quantification of the predicate might have been useful; but in the hands of Sir William Hamilton it only led to a complicated syllogistic, full of blunders; in the hands of Jevons, it stood in the way of a better development of a meritorious system of formal logic.” (V.352–53).

Peirce was not impressed with Hamilton’s efforts of innovation in formal logic, which were a dim prospect anyway for Hamilton, who was short on mathematics. The spring of formal logic with reaches far beyond its medieval spring would first sing, in 1847, in George Boole. I shall be returning to this nineteenth-century chorus in my other thread of straight-off survey and discussion of Peikoff’s dissertation.

Peirce wrote a paragraph during the winter of 1869–70 remarking on Henry Mansel’s Prolegomena Logica:

“Now if you examine Hamilton’s logic or any of those logics which are the immediate product of pure Kantianism as his was (—not his [Hamilton’s] peculiar system but his lectures in which his system does not appear as it was worked up later) you will find logic defined as the Science of Thought as Thought—or something of that sort. . . . Take for example Mr. Mansel’s admirable Prolegomena Logica where the Kantian conception of logic is developed in the most consistent and beautiful manner.” (II.349)

I’ll just see about that.

We are ready now for the Peikoff dissertation itself, for that other thread.



The citations for Peirce in the preceding are from Writings of Charles S. Peirce – A Chronological Edition (Indiana). By the way, Hamilton’s Lectures on Logic and Metaphysics [1859], an entry in Peikoff’s bibliography, is available for free online at archive.org.

Edited by Boydstun

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The installments (in the other thread “Peikoff’s Dissertation”) of my representation of and commentary on Peikoff’s dissertation that I have completed and posted are:  

Plato – 3/17/17

Aristotle I – 5/14/17 

Aristotle II – 11/2/17 

Due to a stretch of writing my book, in some Aristotle areas, I’ve only just now resumed studies required for my next installment on Peikoff’s dissertation. In this continuation, 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 the cited 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).

Here is the Table of Contents for Peikoff’s dissertation. The three installments I mentioned of my series concerned the first 4 chapters of the dissertation. I’ll include here the detail Contents for the remaining, final chapter.

The Status of the Law of Contradiction in Classical Logical Ontologism

Table of Contents

I. Platonism: The Law’s Epistemological Status

II. Platonism: The Law’s Ontological Status

III. Aristotelianism: The Law’s Epistemological Status

IV. Aristotelianism: The Law’s Ontological Status

V. The Demise of Logical Ontologism

—Some central features of non-ontologism in logic, whether Kantian or conventionalist.

—Kantianism as intermediate between ontologism and conventionalism; some difficulties it has faced in the attempt to sustain such a position.

—Some problems for the theory of the Law of Contradiction suggested by the later Platonist view of essences as Divine thoughts.

—How the attempt to resolve such problems pointed toward a Kantian account of the Law; some signs of this in Cudworth.

—Some difficulties in the Aristotelian Form-Matter ontology; the effects of Locke’s rejection of it on his ability to defend logical ontologism.

—Suggestions of conventionalism in Locke; the relation between these and his rejection of realism in the theory of universals.


I’d like to indicate here the book with which I resume my studies for treatment of the issues in the remainder of Peikoff’s dissertation. The summary information here about this book is an addition to all my report on Kant’s ideas on logic in earlier posts in the present thread “Peikoff’s Dissertation – Prep.”

Kant and Aristotle – Epistemology, Logic, and Method

Marco Sgarbi (2016)

From the back cover:

Kant and Aristotle reassesses the prevailing understanding of Kant as an anti-Aristotelian philosopher. Taking epistemology, logic, and methodology to be the key disciplines through which Kant’s transcendental philosophy stood as an independent form of philosophy, Marco Sgarbi shows that Kant drew important elements of his logic and metaphysical doctrines from Aristotelian ideas that were absent in other philosophical traditions, such as the distinction of matter and form of knowledge, the division of transcendental logic into analytic and dialectic, the theory of categories and schema, and the methodological issues of the architectonic. Drawing from unpublished documents including lectures, catalogues, academic programs, and the Aristotelian-Scholastic handbooks that were officially adopted at Königsberg University where Kant taught, Sgarbi further demonstrates the historical and philosophical importance of Aristotle and Aristotelianism to these disciplines from the late sixteenth century to the first half of the eighteenth century.” 


Here are excerpts from the author’s prospectus for 1 and 2:

Chapter 1 – “I contextualize Kant’s facultative logic within the Aristotelian tradition. Kant denies that facultative logic can be based on the philosophical attempts of John Locke and Nicolas Malebranche, who were more concerned with psychology or metaphysics. . . . I examine facultative logic in Aristotle and the Aristotelian tradition with particular reference to Zabarella and the rise of gnostology [science concerning the mental habit that has to do with the cognizable as cognizable, i.e, the mode of knowing the object in general] and noology [study of the mind’s operation of forming subject-predicate propositions and study of the principles and axioms issuing from such propositions]. . . . I show that Kant can be considered as a part of this philosophical Aristotelian tradition from the time of his early writings up to the Critique of Pure Reason. . . . I examine Kant’s relation to the so-called discipline of physiology, characterizing his Kantian categories as a habit of the mind characteristic of the Aristotelian tradition. . . . Characterize the origin of Kant’s notion of pure concepts of understanding as acquired concepts. I compare Kant’s ideas with those of Locke and Leibniz on the polemic against innatism . . . .”

Chapter 2 – “Deals with two fundamental concepts of Kantian epistemology, namely the matter and form of knowledge, and outlines their Aristotelian origin. . . . Philosophical significance of this conception in Kant’s precritical philosophy and in the transcendental aesthetic and logic of his later years. . . . Kant’s appropriation of the Aristotelian syllogism and doctrine of categories. . . . I suggest that Kant’s reawakening from a dogmatic slumber is connected with his rediscovery of Aristotelian categories. Once having established the nature of the categories, I argue that Kant’s conception of categories and schema comes from the nominalistic interpretation of categories elaborated by Königsberg Aristotelianism, and in particular by Rabe [Paul Rabe, c.1700]. . . . I emphasize the epistemological value of analytic and dialectic for Aristotle. Then I suggest the hypothesis that, in the slipstream of the Königsberg Aristotelian tradition, the analytic of concepts corresponds to gnostology, while the analytic of principles corresponds to noology. More specifically, I demonstrate Rabe’s influence on Kant’s conception of analytic and dialectic in conceiving the former as the logic of concepts and principles and the latter as the logic of probability, or logic of illusion.”



“In the conclusion, I show how the failure of the precritical logical and metaphysical projects prompted Kant to develop the Critique of Pure Reason. I then summarize briefly the result of my research, thereby providing justification for my thesis that Kant’s work must be included within the Aristotelian tradition.” –M. Sgarbi

Edited by Boydstun

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Here is some work related to Peikoff’s treatment of Locke on philosophy of logic in his 1964 dissertation. The Peikoff 1985 cited herein is a bit of his dissertation. I wrote this paper in 2012.

Contradiction and Red/Green – Locke 

Recall Ayn Rand’s law of identity in application to attributes, such as color. “Existence is Identity, Consciousness is Identification. / Whatever you choose to consider, be it an object, an attribute or an action, the law of identity remains the same. A leaf cannot be . . . all red and all green at the same time . . .”(1957, 1016).

The logical character of the proposition that a particular surface cannot be all red and all green at the same time has been controversial. In Book 1 of his Essay concerning Human Understanding (EU), John Locke held “principles of demonstration ‘Whatever is, is’ and ‘It is impossible for the same thing to be and not be’” (EU 1.1.4) as unnecessary for the acquisition of all knowledge. “These maxims are not in the mind so early as the use of reason . . . . How many instances of the use of reason may we observe in children, a long time before they have any knowledge of this maxim, ‘That it is impossible for the same thing to be and not be’?” (EU 1.1.12).

In Locke’s view of development, one first gets ideas of particular things through sense impressions, then general ideas by abstraction from the particular ideas. Before one has learned that it is impossible for the same thing to be and not be, indeed, before one has learned to speak, one has learned that bitter is not sweet, white is not black, and red is not blue. Upon that and upon coming to speech, one learns that wormwood and sugarplums are not the same thing, that a rod and a cherry are not the same thing, that a square is not a circle, and so forth. Upon the same grounds that one came to know those differences, one later comes to know that it is impossible for the same thing to be and not be (EU 1.1.15-20; 1.3.3–4; 2.1.6, 22–26; 4.7.9–10).

Turning from development of knowledge to its structure, Locke goes on to say in Book 4 that in apprehending that white is not black or that a circle is not a triangle we do so directly. Knowledge that our ideas of these things are mutually exclusive is not known by demonstration, hence they require no principles of demonstration such as the principle of noncontradiction. We grasp the exclusivity of black and white in a self-evident perception, which Locke calls intuition. Like Plotinus, Anselm and many others before him, Locke thinks of all other knowledge as dependent on intuitive knowledge. The latter is the ultimate source of all certainty in knowledge. “A man cannot conceive himself capable of a greater certainty than to know that any idea in his mind is such as he perceives it to be; and that two ideas, wherein he perceives a difference, are different and not precisely the same” (EU 4.2.1; see also 3.8.1; 4.7.4, 19).

Is the sum of the angles of a triangle the same or variable from one triangle to another? Seeing the sameness of that sum and the sameness of that sum to the angle of a half-circle for all triangles in the Euclidean plane is not self-evident, but requires demonstration (EU 4.2.2). Each step of a demonstration, in Locke’s view, requires intuitive knowledge (EU 4.2.6). The mind can perceive immediately the agreement or disagreement of each step in the demonstration just as the eye can immediately perceive that black and white are distinct and whether a white paper is entirely so or contains some black marks (EU 4.2.5).

It is intuitive knowledge alone, in Locke’s sense of the concept, that is at the base of human knowledge, certain or probable (EU 4.2.8; 4.2.19). It enables the certain demonstrations in geometry, demonstrations with the ideas of “extension, figure, number, and their modes” (EU 4.2.9; a triangle is a relatively simple mode: 2.12.4; 2.31.3; 3.3.18; 3.9.19; 4.4.6; 4.7.9). 

We enjoy accuracy in making and discerning differences in our simple ideas of extension, figure, and number because they are quantitative. (On simple ideas, see EU 2.2.1; 2.3.1; 2.5; 2.7.7-9; 2.30.2; 2.31.2, 12; 2.32.9, 14–16; 3.4.11, 14; 3.8.2; 4.1–2; 4.3.1–21; 4.4.4.) Those ideas are of the primary qualities of things such as Rand’s leaf. Secondary qualities apparently of things, such qualities as colors, are ideas produced in us by the impingement of particles endowed with only primary qualities. In Locke’s day, there were no ways of measuring relations of sameness and difference in the degree of a secondary quality, though he ventures to suppose that greater intensities of primary qualities produce greater degrees of a secondary quality. There are presumably degrees of differences in whiteness that we cannot discern, degrees corresponding to fine degrees of differences in primary qualities that could produce them.

All the same, there are degrees of difference in secondary qualities we do perceive, and such intuitions suffice to found inferential knowledge beyond subjects such as geometry or mechanics. Where the difference in discerned difference in a secondary quality “is so great as to produce in the mind clearly distinct ideas, whose differences can be perfectly retained, there these ideas or colors, as we see in different kinds, as blue and red, are as capable of demonstration as ideas of number and extension” (EU 4.2.13). Where Locke has written “as capable of demonstration,” I think he means “as capable of use in demonstration” (though he could mean additionally what he argues elsewhere in the treatise: there is equally zero capability of any of these different kinds being discerned by demonstration).

Is a sheet of paper before Locke, a paper waiting for the first impression of his pen, all white? Even if it is everywhere white, is it anywhere also red (yet not pink) or black (yet not gray)? We have no direct verdict below our visual thresholds. How does Locke have us know for sure that if the sheet is all white, it is in no part also red or black?

Leaving rather faint the issue of how we know secondary qualities are always produced by primary qualities (cf. Ayers 2011, 146–51), Locke sinks intuitive knowledge concerning imperceptible secondary qualities into the bedrock of intuitive knowledge of primary qualities (EU 4.2.11–12; 4.3.11–13, 15; cf. Descartes 1632, 3–6; 1647–48, 255–56; on Locke’s distinction of primary-secondary, see EU 2.8.9–23; 2.30.2). Locke maintains that for any particular object whatsoever, at a particular time, its extension will be a particular extension, excluding all other particular extensions; its figure will be a particular figure, excluding all others; its motion will be a particular motion, excluding all others. He maintains furthermore that particles of light reflected from a definite part of an object to a particular place of a viewer cannot appear both yellow and azure. “For it is as impossible that the very same particle of any body should at the same time differently modify or reflect the rays of light, as that it should have two different figures and textures at the same time” (EU 4.3.15; see also 2.32.14; unique awareness from unique physical inputs had been embraced also by Descartes, supra; Des Chene 2001, 139–40). Locke generalizes these various sorts of particular exclusion: an object cannot have two exclusive degrees of a given quality simultaneously.

Locke realizes that the impossibility of a leaf being all red and all green is a case of the principle that it is impossible to be both A and non-A. After all, he takes the latter principle to be a generalization of such exclusions encountered in leaf color, fortified by underlying exclusivities in the characters of primary qualities (cf. EU 2.27.1, 4). To say the impossibility of a leaf being at once all red and all green is a case of the logical impossibility of being both A and non-A is not to say the impossibility of the former case derives from the impossibility of the latter general principle. That redness of entire leaf and greenness of entire same leaf are mutually exclusive is not shown by the general principle there are mutually exclusive attributes of entities in existence. Locke correctly recognizes that. “These particular instances, when well reflected on, are no less self-evident to the understanding than the general maxims [superfluously] brought to confirm them: and it was in those particular instances that the first discoverer found the truth, without the help of the general maxims: and so may any one else do, who with attention considers them” (EU 4.7.11[3]).

One weakness in Locke’s sensualist tendency to inductively warrant the necessity of noncontradiction by necessity of exclusions one encounters in sensory qualities is noted by Leonard Peikoff. Leaving the necessity of the principle of noncontradiction as only a necessity encountered in sensory qualities “seems to invite an immediate Humian type of refutation” (Peikoff 1985, 199). Locke’s theory of abstraction is inadequate to the task of delivering from perceptual bases the principle of noncontradiction, noncontradiction in particular and specific identity, noncontradiction in natured entities, their actions, and their attributes. In Locke’s view, we do not attain certain knowledge of the essential natures of physical entities, unlike the situations of our knowledge of simple sensory qualities (primary and secondary) and of mathematical entities. Staying within his view, there is no possibility of arriving at Rand’s absolute principles of identity and noncontradiction through the objects of the senses (EU 2.1.3–9, 22–26; 2.8.7–8; 2.10.6; 2.11.1–9; 2.18.6–7; 2.22.4–5; 2.25.9; 3.3.6–20, 28–38, 49; 4.4.1–6; 4.6.4–16; 4.7.4, 9–10, 16–19; 4.8; 4.11.13–14). 

It is a defect of Locke’s view of color qualities that it is set on difference and sameness in sensed qualities as secondary and their tie to qualities as primary. According to Locke’s scheme, the former, such as the qualities red and green, are in us; they are modifications of our sense organs, thence appearing in our minds, wrought by impinging primary qualities. Rand is set, rather, on the leaf and its color nature. Yes, red and green are different things. Yes, part of the story of how we attain express understanding of the principle of noncontradiction is by prior learning of sameness, difference, and exclusivity encountered in experience. But the sense of noncontradiction Rand is deploying—the sense of identity and exclusion Rand is deploying—is of natured entities. One does not need to know anything about the travel of light to the eye, the physical nature of light, the physiology of the eye, or perceptual thresholds in order to know that it is of the nature, the identity, of leaves and our visual power that no leaf can be at once all red and all green (cf. EU 4.11.2).

Thanks to a presentation of Paul Churchland’s, I have experienced afterimages that seem to be entirely of two distinct colors at once. That experience has now become available to me (more tenuously) in Churchland’s chapter on chimerical colors in Neurophilosophy at Work (2007). I follow the viewing procedure, and in figure 9.11, I get to see fading afterimage discs that are at once mauve and black or at once blue and black and so forth. The point of constructing this figure is for the experience of what Churchland calls “impossibly dark” afterimage colors (the scare quotes are his), not to show patches of afterimages that are two colors at once. I leave it for the reader to dig into this important book’s purposes and their fulfillment.

Afterimages are always fading. What I am experiencing as mauve and black “at once” is occurs while mauve is turning to black. Whether this experience is rightly a case of seeing a portion of surface, screen or page, as two colors could be reasonably disputed. Suppose it qualifies as such a case. It can be taken under wing in Rand’s picture by saying that leaves have their color nature, afterimages have theirs. The same sort of assimilation could be taken were it the case in the future we learn (i) that canonical reflectance profiles of a surface (Churchland 2007, Ch. 10) can be made two canonicals at once by some treatment of the surface and (ii) our visual system can be made to discern them distinctly when artificially and appropriately altered by, say, an electronic apparatus. Were leaves susceptible to said treatment giving them two canonical reflectance profiles, then in Rand’s conception of noncontradiction in attributes, we should say untreated leaves have their color nature, treated leaves have theirs.

In the experience of Rand or Locke, it was not only the surfaces of leaves that were not anywhere at once both red and green. The same was found of any physical surface whatever. It was found likewise for volumetric color. Locke, and Rand too, might recall a certain liquid which when poured into another produces two colors; but the orange and azure produced are regional in the volume, not both simultaneously throughout (EU 2.11.3). 

Locke, like all of us, would want to look into the particulars of the physiology that make afterimage “impossibly dark” colors possible. Understanding of this physiology indeed enabled prediction of this previously unknown effect (Churchland 2007, Ch. 9). Locke could be pleased to see that afterimage colors have their bases in some primary qualities. These colors, like all experienced colors so far as we know, are patterns of nervous activities transforming retinal patterns of activity. Afterimage effects are artifacts of a visual system adaptive in its evolution to color perception of the world. The exclusion character of different canonical surface-reflectance profiles is present in the exclusions of the pattern of nervous activities that are color. But the particulars of this transport of exclusivity shows that Locke’s reasoning from exclusions among primary qualities to exclusions among secondary qualities was spurious.

Returning to Locke’s sensualist epistemology more generally, afterimage experience of two colors everywhere in a region and my futuristic technological scenario quake Locke’s claim that we know the essence of colors just by knowing red is not green and not blue and so forth. Our knowing distinct simple ideas of qualities, secondary or primary, and our putting distinct names on these distinct ideas is insufficient to capture their exclusionary character, their essential natures as attributes of entities. Let the quake shake off that pretension of closure. It remains for Locke and everyone that “red is green” is a contradiction of experience or a contradiction of correct labels. Experience supporting “red is not green” can still support a principle of noncontradiction that recognizes there are distinct items in the world and distinct names to keep them straight. These remains are, however, not distinctive of attributes. They fall short of recognizing the radical general dependence of attributes on their entities and recognizing the full structure of noncontradiction as it applies to attributes.

There is more trouble for Locke. We have seen he held we grasp the exclusivity of red and green by self-evident perception, which he called intuition. Though we become ready to grasp the general exclusivity of A and non-A by such previous sensory experiences of exclusivity, the former is self-evident and self-evident in the same way as the latter (EU 4.7.9–10). If the intuition that nothing can be at once all red and all green were self-evident in the same way as red not being green is self-evident, then fallibility of self-evidence in the proposition that nothing is at once red and green all over shows fallibility of self-evidence for the principle that nothing is at once A and non-A. My afterimage experience and my futuristic, scientifically informed scenario indicate fallibility in the “self-evidence” of the proposition that nothing is at once red and green all over. Time to check premises, for the result that contradictions are not perfectly self-evidently false is patently false.

We may have also some trouble for Rand, but as suggested already, it can be skirted without significant alteration of her metaphysics. In saying a leaf cannot be at once all red and all green, she may have been relying on an Aristotelian sort of dynamical contrariety among colors themselves. Then regardless of what surface or medium may sport colors, the colors dynamically exclude each other at a given place. Her language is pretty strongly against this interpretation, for she speaks of leaf color, though that could by slim chance indicate merely that the dynamical contrariety of colors holds for all surfaces, including leaf surface. Any such line of thought about attributes can be omitted from Rand’s metaphysics. There remain more than enough riches of identity to get beyond Locke.



Ayers, M. 2011. Primary and Secondary Qualities in Locke’s Essay. In Primary and Secondary Qualities. L. Nolan, editor. Oxford.

Churchland, P. 2007. Neurophilosophy at Work. Cambridge.

Descartes, R. 1632. Treatise on Light. In Gaukroger 1998.

——. 1647–48. Description of the Human Body. In Gaukroger 1998. 

De Chene, D. 2001. Spirits & Clocks – Machine & Organism in Descartes. Cornell.

Gaukroger, S., editor, 1998. Descartes – The World and Other Writings. Cambridge.

Locke, J. 1690. An Essay concerning Human Understanding. A. C. Fraser, editor. 1894. Dover.

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

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

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