## Can all mathematical reasoning be translated into traditional logic?

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Can all mathematical reasoning be translated into traditional (Aristotelian, syllogistic) logic?

It would seem not ∵ one cannot syllogistically establish the validity of the reasoning in the following argument:

• 3 is greater than 2.
• 2 is greater than 1.
• ∴, 3 is greater than 1.

This doesn't work because "greater than 2" ≠ "2", or 2 ≯ 2.

The form of the following syllogism is valid, but it shows how a false mathematical premise can lead to a true conclusion:

• All multiples of 5 are even.
• 80 is a multiple of 5.
• ∴, 80 is even.

Thus, it doesn't seem traditional logic can handle mathematical reasoning. Didn't Aristotle, the medieval logicians, et al. realize this?

Poincaré thought that mathematical induction consisted in an ∞ number of syllogisms. Is that true?
(cf. Pierre Duhem's article contra Poincaré: "The Nature of Mathematical Reasoning" from "La nature du raisonnement mathématique," Revue de philosophie 21 (1912): 531-543.)

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You can certainly create a Hilbert system with modus ponens as the only rule of inference, and describe quite a bit of math within it (certain automated proof checkers do this). But all of mathematics within one formal system...easier said than done. Certainly proofs involving commutative diagrams might be a bit inelegant...

– Alex Nelson – 2016-05-19T15:25:46.180

As you say, the traditional syllogistic logic is modernly translated into monadic predicate logic which is a fragment of first-order logic. To "generate" math, of course, you need - in addition to logical language and the "inference mechanism" provided by logical axioms and rules - also mathematical specific axioms.

– Mauro ALLEGRANZA – 2016-05-19T15:56:40.970

What exactly do you mean by "reduce"? In your question you seem to talk about translation to formal language, and not reduction (e.g. in the sense of logicism). – Eliran – 2016-05-19T16:52:56.740

@EliranH corrected. thanks. Yes, I mean "translate." (Yes, "reduction" is a separate logical term, not what I meant.) – Geremia – 2016-05-19T17:30:58.977

2all numbers greater than some number are greater than all numbers less than that number. 3 > 2, 1 < 2, therefore 3 > 1. The problem with traditional syllogism is quantification and anaphora (bound vars). we can't give a precise referent for "that number", since "some number is indeterminate". by contrast a modern version of same would bind x y and z by saying "for all x, y, z, x>y and y>z implies x>z", which covers the case of 3, 2, 1. you cannot translate traditional syllogism into modern FOL without changing it, since it lacks key conceptual innovations of the latter. – None – 2016-05-19T19:08:16.313

1ps. what counts as "mathematical reasoning" changes. mathematicians were perfectly content with Aristotlean logic, until they weren't. – None – 2016-05-19T19:11:36.613

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• Prior, A. N. "Logic, Traditional." Encyclopedia of Philosophy. Ed. Donald M. Borchert. 2nd ed. Vol. 5. Detroit: Macmillan Reference USA, 2006. 493-506. Gale Virtual Reference Library. Web. 20 May 2016.

mentions

Smiley's abstract:

Anyone who reads Aristotle, knowing something about modern logic and nothing about its history, must ask himself why the syllogistic cannot be translated as it stands into the logic of quantification. It is now more than twenty years since the invention of the requisite framework, the logic of many-sorted quantification.

He concludes:

If the Aristotelian logic, after a long pre-eminence and a shorter period of disrepute, is now more temperately regarded, the change is surely due to Lukasiewicz' formalisation of the traditional syllogistic in the 1930's, and his bringing modern techniques and ideas to bear on the resulting system. But the price paid for a rehabilitation of the traditional logic through an algebra of the Łukasiewicz type is a certain divorce from the main current of modern logic: Łukasiewicz was even led to conclude (op. cit. [Łukasiewicz, Jan. 1957. Aristotle's syllogistic from the standpoint of modern formal logic. Oxford: Clarendon Press. ], p. 130) that the syllogistic of Aristotle "exists apart from other deductive systems, having its own axiomatic and its own problems." The result is a certain ambivalence in the current attitude towards the old logic - when we compile our World Team of logicians we tend to include Aristotle as (non-playing) captain. This attitude, at once admiring and dismissive, is well illustrated in Łukasiewicz' conclusion that "The syllogistic of Aristotle is a system the exactness of which surpasses even the exactness of a mathematical theory, and this is its everlasting merit. But it is a narrow system and cannot be applied to all kinds of reasoning, for instance to mathematical arguments. … The logic of the Stoics, the inventors of the ancient form of the propositional calculus, was much more important than all the syllogisms of Aristotle. We realize today that the theory of deduction and the theory of quantifiers are the most fundamental branches of logic." (p. 131.)

It would of course be absurd and anachronistic for me to try to vindicate Aristotle's choice of subject-matter by suggesting that he was consciously guided by anything like the modern idea of quantification. But without committing this mistake there are two observations which I think may properly be made. One is that if it is anachronistic to suggest that Aristotle's logic is 'really' a theory of quantification then it is equally anachronistic to suggest that it is 'really' a theory of primitive functors A, I, etc. As Łukasiewicz himself remarks in his book, "the logic of Aristotle is formal without being formalistic"; and what I have for the sake of convenience called the 'traditional' theory in § 2 is, both in its conscious conception as an algebra of non-empty classes and in its formalistic vocabulary and axiomatisation, as distinctively 'modern' as the logic of quantification. The other remark to be made is that the logic of many-sorted quantification is in no sense something existing "apart from other deductive systems". Not only is it formally no more than a systematic reduplication of the standard single-sorted logic, but it is also the obvious framework for the formalisation of a whole range of mathematical theories: any branch of geometry will furnish one example and Russell's or von Neumann's set theories another. I should like therefore to think that the translations introduced above would help to counter the suggestion of even a residual incompatibility between the modern and the Aristotelian formal logic.

Thomas Greenwood answers "Yes" to the OP's question; cf. his “*The Unity of Logic*,” Thomist : A Speculative Quarterly Review 8 (January 1, 1945): 457–470.

– Geremia – 2016-08-23T21:34:47.243

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In a word, no.

I am going to take the most charitable version of your question.

1) I am presuming that mathematical entities are grounded somehow, and by reasoning you are referring to manipulating mathematical entities. The reason I'm making this presumption is because syllogistic logic says nothing about what's being reasoned about (the content of thought as it were, it could be people and attributes, it could be numbers and properties) so I presume you mean they are generated/given elsewhere.

2) Are there ways to reason about mathematical entities that exceed syllogistic logic? Very simply, yes. Notice that even though syllogistic logic says nothing about the content of thought it actually does make some assumptions. Let us call these assumptions operational or even epistemological assumptions. It assumes you can always give a true or false answer to the conclusions. It also assumes that even though the objects of cognition are unspecified they are discrete in nature.

3) So syllogistic logic fails to be able to talk about undefined notions like division by zero. It fails to be able to talk about unending procedures. It fails to be able to talk about the continuum rather than discrete objects/sets so it fails to be able to talk about the objects that are in Robinson's non-standard analysis and actual infinities and infinitesimals.

I recommend Susan Haack's Philosophy of Logics to understand more.

nice answer. thanks. Haack's book looks interesting. thank you – Geremia – 2016-05-20T00:52:48.667

you are more than welcome. feel free to ping me if you have any questions. – igravious – 2016-05-20T00:54:04.060

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You may want to check out Gödel, Escher, Bach by Hofstadter. It is in fact possible to encode all the Peano axioms together with first-order logic into a purely "typographic" system as Hofstadter calls it. Theoretically that captures all of number theory (although the proof of commutativity of addition given in the book shows how intractable a project that would be). The proof of your first example would go something like:

Ex:SS0+Sx=SSS0
Ey:S0+Sy=SS0
...various steps including transitivity of equality
Ez:S0+Sz=SSS0


Of course x and y above are 0, and z above is S0 a.k.a. 1.

So it can be done.

Although maybe the answer you were looking for is: only if you add in some mathematical axioms as well, as @Mauro's comment above also says.

Those are FOL formulas, but they are not Aristotelian syllogisms as he narrowly construed them. – Fizz – 2021-02-26T00:46:39.023

Yes, I've read GEB (years ago). I'm familiar with Gödel. – Geremia – 2016-05-20T17:52:16.980