Definition
A categorical proposition is a statement that asserts a relationship of inclusion or exclusion between two categories (classes) of things. Traditional Aristotelian logic identifies exactly four standard forms, labeled by the vowels A, E, I, and O. The A-form ("All S are P") is a universal affirmative; the E-form ("No S are P") is a universal negative; the I-form ("Some S are P") is a particular affirmative; the O-form ("Some S are not P") is a particular negative.
Each proposition has a quantifier (all, no, some), a subject term, a copula (a form of "to be"), and a predicate term. Reducing natural-language sentences to one of these four canonical shapes is the prerequisite for evaluating syllogistic arguments.
Why it matters
How it works
Translation into standard form often requires rewriting. "Dogs bark" becomes "All dogs are barking-things"; "Some politicians lie" becomes "Some politicians are liars"; "Nothing matters" becomes "No things are mattering-things." The translation forces a copula and a predicate-class to appear explicitly, which is awkward in English but essential for the logical machinery to engage.
Once translated, propositions can be combined into syllogisms — three-line arguments with two premises and a conclusion, each in categorical form. The validity of the syllogism depends entirely on the pattern of A, E, I, and O propositions and the arrangement of their terms across the three statements. Aristotle catalogued exactly the patterns that yield valid inferences (Barbara, Celarent, Darii, Ferio, and so on), and the system was the dominant framework for formal reasoning in the West for over two millennia.