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I am reading parts of Euclid’s Elements and I am surprised, rightly or wrongly, to see that Euclid did not recognize that a conditional is logically equivalent to its contrapositive form. Indeed, one often sees one proposition immediately followed by another proposition giving its contrapositive and a separate proof (usually by a reductio argument on the original statement) .

The Stoics, who came just after Euclid, were the first to explicitly recognise modus tollens as valid argument form, and the equivalence of a conditional and its contrapositive follows from the application of modus tollens.

Since the equivalence of the two forms is evident from their truth tables, I was equally surprised to read that truth tables have their origins in the late 19th century. (According to the wiki entry on truth tables it was the logician C.S. Pierce who is credited with first formulating truth tables, including them in an unpublished manuscript dated 1893.)

So that “narrows” down the recognition of the equivalence of the two forms to sometime between Euclid and Pierce - a period of about 2200 years.

Q : When did mathematicians first use the contrapositive form to prove a conditional?

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    $\begingroup$ I don't know much about mathematical history this long ago, but I wonder if saying the Greeks didn't recognize the logical equivalence of a conditional and its contrapositive is very misleading. Perhaps by the (implicit and likely unstated) standards of "equivalent argumentation" that the Greeks used, these would be considered logically different by the Greeks. $\endgroup$ – Dave L Renfro Jul 18 '16 at 20:56
  • $\begingroup$ @DaveLRenfro Yes, I think that is the case - that the Greeks used different standards and considered the two forms to be logically different. Yet, given the dozens of instances where Euclid stated and proved a conditional and then immediately states and proves the contrapositive (using a reductio arguement on the original conditional), one would have thought he might have twigged that there was a logical relationship between the two forms. There is also cases where the equivalence of two conditions is stated and proved in four successive propositions - conditional,converse,contra,inverse. $\endgroup$ – Nick Jul 18 '16 at 21:33
  • $\begingroup$ I'm not very knowledgeable about Greek mathematics, so what I'm about to suggest might be a bit naive or misdirected, but I seem to recall that Euclid was written as a textbook, so perhaps the extra proofs were for pedagogical purposes, either as extra practice in proving basic statements or possibly to serve, by giving examples, of how the various forms of conditionals relate to each other. $\endgroup$ – Dave L Renfro Jul 18 '16 at 22:00
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This is one of those cases where it is easy to take modern ideas for granted, and wonder how something so "obvious" could be missed. Modern formal logic with its array of logical operations, Boolean algebra and predicates is the creation of late 19th century, only finalized after much labor by Russell and Whitehead around 1910. Logical analysis that translates natural sentences into formally manipulable ones was also a 19th century invention (especially Frege's), and so was the axiomatic method (especially Pasch's, Peano's and Hilbert's). Stoics did introduce some propositional forms in the 3rd century BC, but not in relation to mathematics, and they never took. Between them and 19th century the only person to pursue propositional logic was Leibniz, who does list contraposition as a valid form in Generales Inquisitiones (1686). But like most of his logical works it was not published until 1902.

For everybody else formal logic was identical to Aristotle's syllogistic, where one could get from "all humans are mortal" to "all immortals are non-human", but which did not have expressive means for intricate mathematical reasoning. Friedman discusses in detail (pp.475-477) how expressive poverty of syllogistic impeded the rigorization of calculus. Since it was the only established formal logic flipping conditionals "blindly" would not have been kosher. And in the natural reasoning the difference between proof by contraposition and proof by contradiction is highly artificial, in both the conclusion is negated and the negation of something in the premise is derived. So discerning the use of "contraposition" before Leibniz is a matter of one's willingness to read between the lines. This said, according to Wirth, Fermat was particularly self-conscious:

"Fermat was the first who — instead of just proving a theorem — analyzed the method of proof search. Moreover, in his letter for Huygens and in [Diophantus, 1670], Fermat was also the first to provide a correct verbalization of proofs by Descente Infinie and to overcome the presentation of induction proofs as “generalizable examples”".

"For instance, in Euclid’s Elements, Proposition VIII.7 is just the contrapositive of Proposition VIII.6, and this is just one of several cases that we find a proposition with a proof in the Elements, where today we just see a corollary. Moreover, even Fermat reported in his letter for Huygens that he had had problems to apply the Method of Descente Infinie to positive mathematical statements... Because of the work of Frege and Peano, these logical differences may be considered trivial today. Nevertheless, they were not trivial before, and to understand the history of mathematics and the fine structure in which mathematicians reasoned, the distinction between affirmative and negative theorems and between direct and apagogic methods of demonstration is important".

P.S. Despite the common reading of Hilbert into Euclid, Euclid was not interested in building a deductive system from axioms and logic, and his "deviations" from this "task" are not oversights. He was interested in systematizing geometry, which he saw as an idealized representation of the "sensibles", a kind of "theoretical physics", and explicating common assumptions and types of reasoning used by geometers. These included reasoning from construction of diagrams, which was considered legitimate as long as they were used "indeterminately", see Acerbi's Euclid's Pseudaria (4.1). When contraposition remains valid in such reasoning is not a priori obvious, in general it fails in constructivism.

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    $\begingroup$ Taking for granted our modern views was the kernel of my attempt to understand Euclid's methods. You have clearly filled in the detail here. Especially interesting is Fermat's frustrations with his own methods. You are a true polymath. Thanks also for correcting my understanding of Euclid's motivations as expressed in my comment to user Dave ( in your "ps" ). $\endgroup$ – Nick Jul 19 '16 at 1:23
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I don't have enough reputation to comment. Are you reading Euclid from the original Ancient Greek text or are you reading it from a translation? And can you give an example of two (not necessarily consecutive) propositions where you observed this? I also noticed that you observed that an equivalence is proved in four successive propositions in some cases. Can you give an example of this?

In many instances, Euclid proves and states one conditional and immediately proves the converse, not the contrapositive. But this might be difficult to catch because of the word order. I tried to translate from Ancient Greek earlier and I was surprised myself. But as an example you can see the guide for Proposition 1.18 and 1.19 by David E. Joyce.

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  • $\begingroup$ Hi. I am reading an English translation, currently looking at book 10. So, for example, X.5 is followed by the contrapositive X.6. Then X.7 give the converse and X.8 the inverse. Curiously, X.9 gives all four forms in one proposition. $\endgroup$ – Nick Jul 18 '16 at 23:58
  • $\begingroup$ Sorry, I slightly misstated the example above. X.5, X.6, X.7, and X.8 give all four in a slightly different order. $\endgroup$ – Nick Jul 19 '16 at 0:00
  • $\begingroup$ Ok. I must make clear my claim. X.5 is followed by the converse as X.6. The contrapositive of X.6 is given as X.7. The contrapositive of X.5 is given as X.8. Sorry for my sloppiness. $\endgroup$ – Nick Jul 19 '16 at 0:07
  • $\begingroup$ Thanks. Honestly, I wasn't able to go to book X before and I didn't know this example. It is curious. I thought I recall instances where I saw that the contrapositive is used in order to prove a theorem in earlier books but I can't seem to find them now. $\endgroup$ – Özgür Esentepe Jul 19 '16 at 0:15
  • $\begingroup$ I am having a look at David Joyce's guide to Euclid which you have referenced. Joyce also claims that Euclid always proves the contrapositive separately. $\endgroup$ – Nick Jul 19 '16 at 0:25

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