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Of course, I reject naive notion that syllogistic logic isn't part of mathematics because it doesn't deal with numbers. There is more to mathematics than working with numbers.

Also syllogistic logic seems to lack birthmark of philosophical doctrine, when everything frequently uncertain, hard to define and questionable. It feels to be more in spirit of a mathematical dicsipline, rather than philosophical one

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  • Syllogistic logic is formal logic. Formal logic can be studied with "mathematical tools" and we have mathematical logic. Commented Aug 9, 2021 at 12:14
  • @MauroALLEGRANZA I don't see how it answers my question. Commented Aug 9, 2021 at 12:16
  • Mathematical Logic is mathematics. Formal Logic is... Logic and this is usually considered part of philosophy. Commented Aug 9, 2021 at 12:21
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    There are aspects of logic that are mathematical and those that are philosophical (why choose this or that system, how it reflects common reasoning, interpreting the epistemological status of logical laws, etc.). Syllogistic is no different in this regard. Philosophy overlaps with all sorts of inquiry, including mathematics, why bother with artificial dichotomies?
    – Conifold
    Commented Aug 9, 2021 at 19:23
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    I think the answer is simple: those tools are underpowered for the modern mathematician.
    – J Kusin
    Commented May 7, 2022 at 15:34

3 Answers 3

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Why would there be pros and cons? What difference does it make whether we consider syllogistic logic to be part of mathematics instead of philosophy? In many cases the lines drawn between academic disciplines are somewhat arbitrary and influenced by history. Historically, the study of logic arose in the context of philosophy because philosophers are concerned with what distinguishes good arguments from bad. The fact that such a study can be pursued with rigour does not disqualify it from being philosophical.

Syllogistic logic is formal, since it follows formal rules and figures. Also, it may be represented using symbols, e.g. "all M is P; all S is M; therefore, all S is P". Aristotle himself used symbolic representations of his logic this way. So, syllogistic logic is part of formal logic and symbolic logic. Is it mathematical? It depends on exactly what you count as mathematics. Mathematics has no generally accepted definition: it is a rather disparate collection of theories and methods that are concerned in an abstract way with patterns and structures.

Mathematics and logic are related, because in one direction we can use mathematical methods to make logic more rigorous and to examine its properties, and in the other direction we can use logic to study the foundations of mathematics. But I would be reluctant to say that logic of any kind is simply part of mathematics. Logic and mathematics just overlap. Logic and philosophy also overlap.

Syllogistic logic is no different from other kinds of logic in this respect. It is a theory about how some combinations of propositions have other propositions as their logical consequence. It has no special claim upon being about how humans reason. How humans reason is the subject of cognitive psychology, not logic. Learning to distinguish logic from psychology was one of the important insights of Frege.

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  • The field of Epistemology is also concerned with who people present arguments. There is consideration of the content matter of the propositions & the form of the argument. In this way validity is NOT ENOUGH for real world reasoning. Thus, philosophy in some fields desires SOUNDNESS in an argument over just validity alone.
    – Logikal
    Commented Aug 9, 2021 at 23:37
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It is a mistake to assume that academic fields are disjoint. For syllogistic logic, as for much of logic in general, the only sensible answer to "Is it philosophy or mathematics" is "It's both."

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What are pro and contra arguments for considering syllogistic logic to be part of mathematics, instead of philosophy?

Aristotle's syllogistic is formal logic, formal logic is the study of logic, where logic is, in this context, the logic of human deductive reasoning.

Relation to mathematics?

Mathematics as a discipline is the application, using a symbolic scheme, of logical reasoning—i.e., human deductive reasoning—to systems of axioms. What is crucial in this connection, is that an axiom is a definition freely adopted by the mathematician. It specifies logical relations between similarly arbitrary concepts—e.g. how the concept of triangle relates logically to the concepts of point and line. Mathematical axioms work exactly like dictionary definitions. Crucially, they are assumed true rather than proved true, as would be required if mathematics was an empirical science.

And mathematical logic is a branch of mathematics. It is mathematics.

Hence, the main difference between mathematical logic and formal logic, including Aristotle's syllogistic, is that formal logic is an empirical science while mathematical logic is a non-empirical science.

One of the most spectacular consequences of this difference is that although there is in the real world only one true logic—the logic of human deductive reasoning—there are in mathematical logic several independent systems of axioms each one considered by mathematicians working with them a logic. Thus, each system is a logic, but none is a model of the logic of human deductive reasoning.

That being said, there is no good reason that we could not develop and use a mathematical model of the logic of human deductive reasoning. It has not been done yet, but this is certainly possible just as mathematical models have been used in the sciences to represent the relations between real-world things and the logic of human deductive reasoning is one real-world thing.

Philosophy?

Philosophy is very similar to mathematics. Philosophers are free to invent systems of concepts without the need or the constraint of having to prove that these systems are proper models of things in the real world. However, like the use of mathematical models in empirical sciences, philosophers can always discuss logical questions, and can investigate the logic of human deductive reasoning, and they have certainly done so over the last three millennia, though admittedly with poor results.

However, Aristotle didn't just discuss logical questions, he actually developed a formal model of the logic of human deductive reasoning, a model grounded on the empirical data available to him at the time.

The value of Aristotle's syllogistic is not so much in its practical applications, which are somewhat limited, than in the fact that it proved to be a sturdy and precise identification of logic as the logic of human deductive reasoning. This identification has had a crucial role throughout history. Logicians who contributed to formal logic, such as Stoic logicians, the Scholastic, the logicians from the Port-Royal school in France, all understood logic to be the logic of human deductive reasoning. Thus, although Aristotle's syllogistic is limited in many respects, it is nonetheless the reference point for all logicians.

So, no, formal logic is neither a branch of mathematics nor a branch of philosophy. It is an independent empirical science, like physics, psychology or the science of probabilities. The fact that few people if any are today working on a scientific model of the logic of human deductive reasoning should not confuse anyone. Good science will have to be produced before humanity can take full advantage of its most impressive asset, what is arguably the core of human intelligence.


EDIT

Another answer here makes a claim I have to dispute:

Syllogistic logic is no different from other kinds of logic in this respect. It is a theory about how some combinations of propositions have other propositions as their logical consequence. It has no special claim upon being about how humans reason.

Please take the time to read Prior Analytics carefully. The text is essentially human reasoning in action. Human reasoning put forward to explain the logic of syllogisms. It is essentially 50 pages or so of strings of conditionals, from beginning to end, if-then expressions, expressions of human reasoning.

A syllogism is a formal argument. It is the formal expression of the reasoning which is implicit in the mostly informal arguments we use in everyday conversations. As such, syllogisms are descriptive of human deductive reasoning.

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