In syllogistic, a proposition in which the predicate is affirmed or denied of all or part of the subject. Thus, categorical propositions are of four basic forms: “Every S is P,” “No S is P,” “Some S is P,” and “Some S is not P.” These are designated by the letters A, E, I, and O, respectively; thus, “Every man is mortal” is an A-proposition. Categorical propositions are to be distinguished from compound and complex propositions, into which they can enter as integral terms. In particular, they contrast especially with hypothetical propositions, such as “If every man is mortal, then Socrates is mortal.”
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In logic and philosophy, proposition refers to either (a) the content or meaning of a meaningful declarative sentence or (b) the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence. Propositions in either case are intended to be truth-bearers, that is, they are either true or false.
The existence of propositions in the former sense, as well as the existence of "meanings", is disputed. Where the concept of a "meaning" is admitted, its nature is controversial. In earlier texts writers have not always made it sufficiently clear whether they are using the term proposition in sense of the words or the "meaning" expressed by the words. To avoid the controversies and ontological implications, the term sentence is often now used instead of proposition or statement to refer to just those strings of symbols that are truth-bearers, being either true or false under an interpretation.
Philosophy requires more careful definitions. The above definition, for example, allows "Is snow white?" and "Ist Schnee weiß?" to express the same proposition if they have the same meaning, although neither of them, being questions, could be either true or false. One such more careful definition might be that
Two meaningful declarative sentence-tokens express the same proposition if and only if they mean the same thing.thus defining proposition in terms of synonymity. Unfortunately, the above definition has the result that two sentences which have the same meaning and thus express the same proposition, could have different truth-values, e.g "I am Spartacus" said by Spartacus and said by John Smith; and e.g. "It is Wednesday" said on a Wednesday and on a Thursday.
Some philosophers hold that other kinds of speech or actions also assert propositions. Yes-no questions are an inquiry into a proposition's truth value. Traffic signs express propositions without using speech or written language. It is also possible to use a declarative sentence to express a proposition without asserting it, as when a teacher asks a student to comment on a quote; the quote is a proposition (that is, it has a meaning) but the teacher is not asserting it. "Snow is white" expresses the proposition that snow is white without asserting it (i.e. claiming snow is white).
Propositions are also spoken of as the content of beliefs and similar intentional attitudes such as desires, preferences, and hopes. For example, "I desire that I have a new car," or "I wonder whether it will snow" (or, whether it is the case "that it will snow"). Desire, belief, and so on, are thus called propositional attitudes when they take this sort of content.
In mathematical logic, propositions, also called "propositional formulas" or "statement forms", are statements that do not contain quantifiers. They are composed of well-formed formulas consisting entirely of atomic formulas, the five logical connective, and symbols of grouping. Propositional logic is one of the few areas of mathematics that is totally solved, in the sense that it has been proven internally consistent, every theorem is true, and every true statement can be proved. (From this fact, and Gödel's Theorem, it is easy to see that propositional logic is not sufficient to construct the set of integers.) The most common extension of propositional logic is called predicate logic, which adds variables and quantifiers.