What is the difference between "Syllogism" and "Law of Syllogism"?

Question

The logical arguments are the basis for Artificial Intelligence. That is why I picked AI community to ask my question.

Reading from Wikipedia,

A syllogism is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true.

Again from Wikipedia, deductive reasoning,

is the process of reasoning from one or more statements (premises) to reach a logical conclusion.

As part of deductive reasoning, it lists Modus Ponens, Modus Tollens, and Law of syllogism where Law of syllogism is defined as

In term logic the law of syllogism takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another.

Based on these articles, is it safe to assume that Syllogism uses Modus ponens, Modus tollens, and Law of syllogism to arrive at conclusion? Does that mean "Law of syllogism" is part of "syllogism" or is it something different?

P.S.

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Answer 1

A syllogism is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true.

This definition is somewhat misleading. It suggests that there is a logical reasoning in between premises and conclusion, so to speak, but there is none. The premises and the conclusion have to be regarded as the deductive reasoning itself. It is deductive because the conclusion is understood as following necessarily from the premises. And this is all there is to it.

Based on these articles, is it safe to assume that Syllogism uses Modus ponens, Modus tollens, and Law of syllogism to arrive at conclusion? Does that mean "Law of syllogism" is part of "syllogism" or is it something different?

"Law of Syllogism" is just one particular syllogism. It is more usually called "hypothetical syllogism":

If ϕ, then ψ;

If ψ, then ξ;

So, if ϕ, then ξ.

This is the expression of the transitivity of the implication (or of the conditional).

A deductive reasoning is a sequence of logical truths so articulated with each other as to justify the claim that a conclusion follows necessarily from the premises given.

A logical truth is an implication which appears self-evidently true, at least to most people.

The modus ponens and the modus tollens are logical truths since they are implications and they appear self-evidently true, but they merely are the most prominent of all logical truths.

All syllogisms are logical truths, including the hypothetical syllogism. So you can use any syllogism as part of a deductive reasoning. Deductive reasoning can involve any logical truth as long as it helps justify the claim that the conclusion follows from the given premises.