# Categorical proposition

[https://en.wikipedia.org/wiki/Categorical\_proposition](https://en.wikipedia.org/wiki/Categorical_proposition)

_A categorical proposition_ is a statement that asserts/denies that all/some of the members of one category (the subject term) are included in another category (the predicate term). Syllogisms are categorical statements.

Aristotle had identified 4 primary types of categorical propositions and gave them standard forms: `A`, `E`, `I`, `O`.

| 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)                      |

`S` is the subject category, `P` is and the predicate category.

Aristotle's _square of opposition_ codifies the logical relations among the different forms (e.g. that an A-statement is contradictory to an O-statement). The relations of the square of opposition may allow _immediate inference_, whereby the truth or falsity of one of the forms may follow directly from the truth or falsity of a statement in another form.

Arguments consisting of 3 categorical propositions, 2 premises and a conclusion, are called _categorical syllogisms_.

Formal arguments using categorical syllogisms have fallen away due to the increased expressive power of modern logic systems.
