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Peikoff's Dissertation

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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.



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.



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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