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I don't have much prior knowledge on this topic, but I am looking for a fundamental difference/difference between Aristotelian logic and mathematical logic.

The difference, which I read previously somewhere, is that Aristotelian logic sets some limits that mathematical logic does not adhere to. Mathematical logic links the world together, while Aristotelian logic only links cause and effect. For example, mathematical logic can conclude that grass is green from the fact that fish swim; Is this correct?

If there are other fundamental differences, please point them out

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  • Logic is formal, and this fundamental discovery is due to A. Your final examples are maybe musundertood...the grass-fish inference was not available to A's syllogistic because it is propositoonal. Dec 17, 2023 at 10:11
  • I am just pointing out that fish swim and grass is green are two facts related to the world in which we live, and therefore one can be inferred from the other starting from the fact that they are related to this world. Actual examples can be taken from mathematics, such as deducing the Pythagorean theorem by knowing the fact that the sum of the angles of a triangle is equal to 180° Because they both stem from Euclidean geometry. Dec 17, 2023 at 15:47
  • You cannot deduce one from the other. What is the logical link between them? In geometry, we deduce theorem from axioms. The fact that grass is green is not a theorem. Dec 17, 2023 at 17:26
  • I think it is good for you to read this article cut-the-knot.org/triangle/pythpar/PTimpliesPP.shtml Dec 17, 2023 at 19:38
  • Does this answer your question? Can all mathematical reasoning be translated into traditional logic?
    – Geremia
    Mar 16 at 4:06

1 Answer 1

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The term 'mathematical logic' is used somewhat loosely to refer both to the mathematical study of logic and to particular formal systems of logic. Since you are asking to contrast it with Aristotelian logic, I shall take it that you are referring to first order classical logic. There are many other formal logics, but that is the most commonly used.

Aristotelian logic may be considered a small fragment of first order classical logic with many restrictions. The main ones being:

  1. Aristotelian logic is limited to monadic predicates, so it can express propositions about properties of things but not relations between things.

  2. Aristotelian logic limits propositions to one quantifier, so it can only express very simple propositions.

  3. Aristotelian logic cannot express the logic of propositional connectives, 'and', 'or', 'not', 'if'. The logic of these was developed separately by stoic logicians.

I list some other limitations in my answer to this question.

Aristotelian logic uses different conventions for representing sentences of the form "All ...". For example, in Aristotle's logic, "All S is P" entails "Some S is P" whereas in classical logic it does not.

Aristotle's logic can be translated into a fragment of first order logic. This was demonstrated in the 1970s by John Corcoran and Timothy Smiley. One difference is that Aristotelian logic is decidable, whereas first order classical logic in its full generality is only semidecidable. Neil Tennant has also demonstrated that Aristotelian logic is relevant and intuitionistic and so it is also a fragment of what he calls Core Logic.

Because of its limitations, Aristotelian logic is not suitable for expressing the kinds of propositions that occur within science and mathematics. This is why classical logic is so commonly used: it was developed with scientific and mathematical use in mind. Some commentators on Aristotle maintain that one cannot separate Aristotle's logic from his metaphysics and that he was engaged in a different kind of project from the modern approach to logic. However, Aristotle's logic is not limited to statements of cause and effect as you suggest.

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  • This means that the basic difference lies in terms of breadth and in terms of the use of mathematical symbols. Aristotelian logic is only part of first-class logic, and there are no differences in the sense of different results. Aristotelian logic is narrow and there are many issues that it cannot judge, but when it judges a case Certainly, it will give the same answer as first-order logic. Dec 17, 2023 at 15:51

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