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Question seems self explanatory. Is there anything in mathematics that can be stated to be true without using a logical syllogism?

Had a discussion with somebody about this recently. Sorry if this is the wrong category.

closed as unclear what you're asking by Conifold, Mauro ALLEGRANZA, Mark Andrews, Eliran, Jishin Noben Mar 12 at 9:19

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Yes, since logicians have not seriously used them for the last century or so. Syllogisms are only included in introductory courses as historical references. They are not a fundamental aspect of modern logical reasoning.

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    You are confusing Aristotle's system with the general idea of a syllogism. The hypothetical and disjunctive syllogisms are used all the time in both formal systems like natural deduction and every day mathematics. – Not_Here Mar 4 at 22:14
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    @Not_Here - but the "weak point" of the question (as per the first comments above) is exactly : what does th OP is meaning with "syllogism" ? Does he conflate it (worngly) with formal logic ? – Mauro ALLEGRANZA Mar 5 at 10:46
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The question is:

Is there anything in mathematics that can be stated to be true without using a logical syllogism?

According to Wikipedia George Boole's system could handle multi-term propositions that Aristotle's two-termed propositions could not:

...Boole's system could handle multi-term propositions and arguments, whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce: "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that is a rectangle is a square that is a quadrangle".

Furthermore Frege "introduced a calculus, a method of representing categorical statements (and statements that are not provided for in syllogism as well) by the use of quantifiers and variables." This allowed one to rephrase mathematics without using syllogisms.


"Syllogism" Wikipedia https://en.wikipedia.org/wiki/Syllogism

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    As in the above answer, you're conflating Aristotle's system with the general idea of a syllogism. We use hypothetical and disjunctive syllogisms all the time in formal and informal reasoning. It is true that modern logic is not centered around only studying syllogisms, but it is not true that mathematics is done "without using syllogisms". Whether or not everything can be rephrased to not include those deduction rules is the question you should be answering, but saying that those rules are not used anymore is factually incorrect. – Not_Here Mar 4 at 22:18

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