Syllogistic logic

• Aristotelian logic

• Term logic

• Traditional logic

• Syllogistic logic

• Aristotelian syllogistic logic

• Two-term logic

Syllogistic logic, also known as Aristotelian logic, term logic, traditional logic is a loose name for an approach to logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late XIX century.

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

In its earliest form, defined by Aristotle, from the combination of a general statement (the major premise) and a specific statement (the minor premise), a conclusion is deduced.

A famous example:

All men are mortal.               (major premise)
Socrates is a man.                (minor premise)
Therefore, Socrates is mortal.    (conclusion)

Aristotle defines the syllogism as "a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so". Despite this very general definition, Aristotle limits himself to categorical syllogisms that consist of 3 categorical propositions.

A categorical proposition (or syllogism) is a proposition that asserts or denies that all or some of the members of one category (the subject term) are included in another (the predicate term).

Aristotle identified 4 primary distinct types of categorical propositions and gave them standard forms, called A, E, I, O. If the subject category is S and the predicate category is P they are:

syllogistic logic: | propositional logic:

• A: All S are P | ∀x. S(x) -> P(x) or ¬∃x. S(x) ∧ ¬P(x)

• E: All S are not P | ∀x. S(x) ->¬P(x) or ¬∃x. S(x) ∧ P(x)

• I: Some S are P | ∃x. S(x) ∧ P(x)

• O: Some S are not P | ∃x. S(x) ∧ ¬P(x)

Basics

The fundamental assumption behind syllogistic logic is that propositions are composed of 2 terms (hence the name two-term logic or term logic) and that the reasoning process is in turn built from propositions:

• The term is a part of speech representing something, but which is not true or false in its own right, such as "man" or "mortal".

• The proposition consists of two terms, in which one term (the "predicate") is "affirmed" or "denied" of the other (the "subject"), and which is capable of truth or falsity.

• The syllogism is an inference in which one proposition (the "conclusion") follows of necessity from two others (the "premises").

A proposition may be universal or particular, and it may be affirmative or negative.

There are 4 types of propositions:

• A: Universal and affirmative ("All philosophers are mortal")

• I: Particular and affirmative ("Some philosophers are mortal")

• E: Universal and negative ("All philosophers are not mortal")

• O: Particular and negative ("Some philosophers are not mortal")