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In other words, how is Aristotle's logic represented in Euclid's Elements?

I have read many articles where Euclid's Elements is linked to Aristotle's logic, but I do not understand, and I can't find any examples explaining how deductive logic (i.e. syllogism) or laws of thought or any other work from Aristotle could have influenced the development of geometry and the concepts Euclid came up with when writing his book.

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  • $\begingroup$ Can you give an example of such an article? $\endgroup$ – Mikhail Katz Dec 10 '17 at 11:29
  • $\begingroup$ @MikhailKatz The article Logic and Mathematics, by Stephen G. Simpson, contains the sentence “The methodological ideas of Aristotle decisively influenced the structure and organization of Euclid's monumental treatise on geometry, the Elements.” $\endgroup$ – José Carlos Santos Dec 11 '17 at 13:15
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You are right: syllogisms are not used by Euclid. More generally: “Although Aristotle emphasized the use of syllogisms as the building blocks of logical arguments, Greek mathematitians apparently never used them.” (I am quoting from A History of Mathematics, by Victor J. Katz (3rd edition)).

The infuence of Aristotle on Euclid lies elsewhere. Quoting from the same source: “If one accepts the premises of a syllogism as true, then one must also accept the conclusion. One cannot, however, obtain every piece of knowledge as the conclusion of a syllogism. One has to begin somewhere with truths that are accepted without argument. Aristotle distinguishes between the basic truths that are peculiar to each particular science and the ones that are common to all. The former are often called postulates, and the latter are known as axioms.” I suppose that you recognise in this description the way Euclid's Elements are written.

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As Simpson clearly states in the section "Aristotelian logic" of his article, principle 3 is the law of excluded middle: $P \vee \neg P$ is true. Euclid's Elements indeed rely on classical logic rather than intuitionistic logic. There have been attempts recently to rewrite Euclid in an intuitionistic framework; see e.g., the work of Michael J. Beeson.

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Heath, the famous translator of the Elements, concludes in his introduction to vol. 1 of his translation of the Elements, §"3. First Principles: Definitions, Postulates, and Axioms", that Euclid's usage of these terms aligns most closely to Aristotle's. Heath begins that § by quoting in extenso from Aristotle's Posterior Analytics 1.10 (76a5) ("Difference between principles and non-principles, common and proper principles") and commenting upon it. Here's Heath's translation with his useful parenthetical remarks relating what Aristotle is saying to geometry:

“By first principles in each genus I mean those the truth of which it is not possible to prove. What is denoted by the first (terms) and those derived from them is assumed; but, as regards their existence, this must be assumed for the principles but proved for the rest. Thus what a unit is, what the straight (line) is, or what a triangle is (must be assumed); and the existence of the unit and of magnitude must also be assumed, but the rest must be proved. Now of the premisses used in demonstrative sciences some are peculiar to each science and others common (to all), the latter being common by analogy, for of course they are actually useful in so far as they are applied to the subject-matter included under the particular science. Instances of first principles peculiar to a science are the assumptions that a line is of such and such a character, and similarly for the straight (line); whereas it is a common principle, for instance, that, if equals be subtracted from equals, the remainders are equal. But it is enough that each of the common principles is true so far as regards the particular genus (subject-matter); for (in geometry) the effect will be the same even if the common principle be assumed to be true, not of everything, but only of magnitudes, and, in arithmetic, of numbers.

“Now that which is per se necessarily true, and must necessarily be thought so, is not a hypothesis nor yet a postulate. For demonstration has not to do with reasoning from outside but with the reason dwelling in the soul, just as is the case with the syllogism. It is always possible to raise objection to reasoning from outside, but to contradict the reason within us is not always possible. Now anything that the teacher assumes, though it is matter of proof, without proving it himself, is a hypothesis if the thing assumed is believed by the learner, and it is moreover a hypothesis, not absolutely, but relatively to the particular pupil; but, if the same thing is assumed when the learner either has no opinion on the subject or is of a contrary opinion, it is a postulate. This is the difference between a hypothesis and a postulate; for a postulate is that which is rather contrary than otherwise to the opinion of the learner, or whatever is assumed and used without being proved, although matter for demonstration. Now definitions are not hypotheses, for they do not assert the existence or non-existence of anything, while hypotheses are among propositions. Definitions only require to be understood: a definition is therefore not a hypothesis, unless indeed it be asserted that any audible speech is a hypothesis. A hypothesis is that from the truth of which, if assumed, a conclusion can be established. Nor are the geometer’s hypotheses false, as some have said: I mean those who say that ’you should not make use of what is false, and yet the geometer falsely calls the line which he has drawn a foot long when it is not, or straight when it is not straight.’ The geometer bases no conclusion on the particular line which he has drawn being that which he has described, but (he refers to) what is illustrated by the figures. Further, the postulate and every hypothesis are either universal or particular statements; definitions are neither” (because the subject is of equal extent with what is predicated of it).

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