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.

In mathematics, the word "proposition" is often used as a synonym for "theorem".

Common usage contrasted with philosophical usage

In common usage, different sentences express the same proposition when they have the same meaning. For example, "Snow is white" (in English) and "Schnee ist weiß" (in German) are different sentences, but they say the same thing, so they express the same proposition. Another way to express this proposition is , "Tiny crystals of frozen water are white." In common usage, this proposition is true.

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.

Historical usage

Usage in Aristotle

Aristotelian logic identifies a proposition as a sentence which affirms or denies the predicate of a subject. An Aristotelian proposition may take the form "All men are mortal" or "Socrates is a man." In the first example, which a mathematicial logician would call a quantified predicate (note the difference in usage), the subject is "men" and the predicate "all are mortal". In the second example, which a mathematicial logician would call a statement, the subject is "Socrates" and the predicate is "is a man". The second example is an atomic element in Propositional logic, the first example is a statement in predicate logic. The compound proposition, "All men are mortal and Socrates is a man," combines two atomic propositions, and is considered true if and only if both parts are true.

Usage by the Logical Positivists

Often propositions are related to closed sentences, to distinguish them from what is expressed by an open sentence, or predicate. In this sense, propositions are statements that are either true or false. This conception of a proposition was supported by the philosophical school of logical positivism.

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.

Usage by Russell

Bertrand Russell held that propositions were structured entities with objects and properties as constituents. Others have held that a proposition is the set of possible worlds/states of affairs in which it is true. One important difference between these views is that on the Russellian account, two propositions that are true in all the same states of affairs can still be differentiated. For instance, the proposition that two plus two equals four is distinct on a Russellian account from three plus three equals six. If propositions are sets of possible worlds, however, then all mathematical truths are the same set (the set of all possible worlds).

Relation to the mind

In relation to the mind, propositions are discussed primarily as they fit into propositional attitudes. Propositional attitudes are simply attitudes characteristic of folk psychology (belief, desire, etc.) that one can take toward a proposition (e.g. 'it is raining', 'snow is white', etc.). In English, propositions usually follow folk psychological attitudes by a "that clause" (e.g. "Jane believes that it is raining"). In philosophy of mind and psychology, mental states are often taken to primarily consist in propositional attitudes. The propositions are usually said to be the "mental content" of the attitude. For example, if Jane has a mental state of believing that it is raining, her mental content is the proposition 'it is raining'. Furthermore, since such mental states are about something (namely propositions), they are said to be intentional mental states. Philosophical debates surrounding propositions as they relate to propositional attitudes have also recently centered on whether they are internal or external to the agent or whether they are mind-dependent or mind-independent entities (see the entry on internalism and externalism in philosophy of mind).

Treatment in logic

As noted above, in Aristotelian logic a proposition is a particular kind of sentence, one which affirms or denies a predicate of a subject. Aristotelian propositions take forms like "All men are mortal" and "Socrates is a man."

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.

Objections to propositions

A number of philosophers and linguists claim that the philosophical definition of a proposition is too vague to be useful. For them, it is just a misleading concept that should be removed from philosophy and semantics. W.V. Quine maintained that the indeterminacy of translation prevented any meaningful discussion of propositions, and that they should be discarded in favor of sentences.

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