"A is A" - Whose Formula?


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To Peter’s connection in the preceding post and to earlier remarks in the thread on identity and the Scholastics and Leibniz, I’d like to record that the fundamental “three laws of logic” or “three laws of thought” were current in logic books by the time of Kant. An example is a text of Johann Maaß (1793), whose remarks on the laws are quoted and discussed by Bernard Bolzano (1837, vol. 1, p. 150, in the 2014 translation into English).

Also, to this thread, we should have:

. . .

. . . But what innovation did she really do in the field? Logic as "the art of non-contradictory identification"? (Did I quote her right?) Aristotle must have said something like that. . . .

Pertinent points to your suggestion about Aristotle, excerpts from my book:

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. 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)

. . .

By “greatest of your philosophers,” Rand meant Aristotle. Unlike moderns such as Leibniz, Baumgarten, Kant, and Rand, Aristotle did not connect a “law of identity,” in so many words, with his principle of noncontradiction.[1] And he 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.[2]

Aristotle was the founder of logic, and his great contribution thereto was his theory of correct inference, which is 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 extend 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 (first mood of the second figure).[3]

There are places in which Aristotle connects (what we call the law of identity) “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” (Metaphys. 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).[4] Aristotle realized too that any existent not only is, but is a what.[5]

Rand acknowledges the greatness of Aristotle particularly for his laws of logic, as they are called in elementary logic texts today and the last few centuries: the laws of noncontradiction,[6] excluded middle,[7] and identity. Those are important principles of logic, though, as we have seen, Aristotle was not securely on board with that last one. It is not clear that Rand was cognizant of the even greater importance for logic of the theory of correct inference that Aristotle invented with his theory of syllogism. . . .


[1] Leibniz 1678; Baumgarten 1757 [1739], §11; Kant 1755, 1:389; 1764, 2:294.

[2] Aristotle, Metaph. 1041a10–24.

[3] Kneale and Kneale 1962, 235–36; see also Kant 1800, §44n2.

[4] See also Aristotle, Metaph. 1006b26–27, 1007a26–27.

[5] Metaph. 1030a20–24.

[6] De Int. 17a33–35; Metaph. 1011b26–27; Plato, Rep. 436b.

[7] De Int. 17b27–29; Metaph. 996b26–30.

It would take some explaining to Aristotle for him to see what was meant by defining logic as "the art of noncontradictory identification." He certainly viewed the principle of noncontradiction as a pervasive rule of right thinking because it is a fundamental truth of anything real. But he would not accept it as the fundamental principle of inference (contra Harriman/Peikoff on Aristotle). For that he would point to one form of syllogism* in which we grasp directly the necessary correctness of the inference and with reference to which the correctness of all the other forms of syllogism can be shown to depend (see further and more precisely in Jonathan Lear's Aristotle and Logical Theory). With the more general view that logic is the art of some sort of identification, Aristotle should come along, for it is identity in Rand's specific-nature sense of the concept that is basis of the rightness of inference in that premier syllogism (identity, but not causality, Roger).

I’d like to add some further history, among Objectivist writers, on ascription of the “three laws of logic” to Aristotle. Firstly is Rand’s remark, in the “About the Author” pages following the text of Atlas Shrugged, that the titles of the three parts of the novel are her tribute to Aristotle: Non-Contradiction / Either-Or / A is A. In that note she also praised Aristotle for his “definition of the laws of logic,” and on page 1016, she had defined logic as “the art of non-contradictory identification.”

In her 1960 essay “For the New Intellectual,” Rand credited Aristotle with conceiving the world as she conceived it, as a world in which A is A, independently of wishes or feelings. In his 1960’s lectures The Basic Principles of Objectivism, Nathaniel Branden remarked: “Atlas Shrugged is a hymn to logic, to the power and importance of logic. You will observe that the three parts of Atlas Shrugged are named after the three Aristotelian laws of logic: ‘Non-Contradiction’—‘Either/Or’—‘A is A’.” A few paragraphs later: “The three Aristotelian laws of logic are: the Law of Identity, the Law of Contradiction, the Law of Excluded Middle. The last two are merely corollaries or restatements of the first” (Vision, p.66). (That assertion of priority is concurred in by Leibniz and by Bolzano, although, almost as poorly as Aristotle, they did not have “A is A” firmly set as addressing not only that-ness of A, but what-ness of A.)

In his 1991 Objectivism: The Philosophy of Ayn Rand, Leonard Peikoff musters the following in his defense of Rand’s definition of logic. He ends up over-emphasizing the role of non-contradiction in comparison to straight identity (in Rand’s sense) in deduction, in my view, as in my quoted material above in this post. “A simple example [and most fundamental example of all syllogistic inference, I’d like to add] from the field of deductive reasoning is the Socrates syllogism: ‘All men are mortal. Socrates is a man. Therefore, Socrates is mortal’. The conclusion follows, because to deny it would be to contradict the premises. . .’” (119). No, that particular and elementary inference is directly by identity; mediation by non-contradiction is not required.

In his 2014 How We Know, Harry Binswanger has a section on “The Three Laws of Logic.” He refers to Aristotle as being the one who “identified the Law of Non-Contradiction, stating that it is the basic principle of all knowledge” (192). He does not claim Aristotle saw it as the basic principle of deductive inference. He notes of the Law of Non-Contradiction and the Law of Excluded Middle, as had Branden and Peikoff, that they stem from the law of identity, and he indicates that this is thought beyond the vista reached by Aristotle. “Later Aristotelians recognized that both these laws stem from the axiom of identity: ‘A is A.’ A thing is what it is” (193). Yes, but until you Objectivists, I’m not sure any really got there all the way. Dr. Binswanger exhibits in this section a fine appreciation of the ways in which identity, in Rand’s full sense, enters into that elementary syllogistic inference to the mortality of Socrates.

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I hope Fred Seddon isn't reading this, or he'll remind me about his theory that Plato is a lot closer to Aristotle than most people think.


Other way around. Aristotle was a student of Plato.

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But Bob, proximity (and similarity) are symmetric relations. Animal life in the era of the dinosaurs is close in kind to animal life in our era (in comparison to the proximity in kind of animal life in the Cambrian era to animal life in our era). Closeness of ideas of teacher to ideas of student equals closeness of ideas of student to ideas of teacher. Right?

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  • 4 months later...


I have noted that “Every A is A” was used by Robert Kilwardby.* This predates use by Antonius, whom Peikoff had come across and noted in his dissertation (1964).* Internet searches on Antonius should use for his second name this spelling: Andreae. The following is good on his life and works (about 2 minutes download):

Antonius Andreae – Scotism’s Best Supporting Auctor by Marek Gensler

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