In philosophy, a proposition is the meaning of a declarative sentence, where "meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the non-linguistic Truth-bearer of truth or falsity which makes any sentence that expresses it either true or false.
In mathematics the term proposition refers to a statement that may or may not be true, whilst the term axiom refers to a statement that is taken to be true within a domain of discourse.
While the term "proposition" may sometimes be used in everyday language to refer to a linguistic statement which can be either true or false, the technical philosophical term, which differs from the mathematical usage, refers exclusively to the non-linguistic meaning behind the statement. The term is often used very broadly and can also refer to various related concepts, both in the history of philosophy and in contemporary analytic philosophy. It can generally be used to refer to some or all of the following: The primary bearers of (such as "true" and "false"); the objects of belief and other propositional attitudes (i.e. what is believed, doubted, etc.); the of "that"-clauses (e.g. "It is true that the sky is blue" and "I believe that the sky is blue" both involve the proposition the sky is blue); and the meanings of declarative sentences.
Since propositions are defined as the sharable objects of attitudes and the primary bearers of truth and falsity, this means that the term "proposition" does not refer to particular thoughts or particular utterances (which are not sharable across different instances), nor does it refer to concrete events or facts (which cannot be false). Propositional logic deals primarily with propositions and logical relations between them.
Some philosophers argue that some (or all) kinds of speech or actions besides the declarative ones also have propositional content. For example, yes–no questions present propositions, being inquiries into the truth value of them. On the other hand, some Semiotics can be declarative assertions of propositions, without forming a sentence nor even being linguistic (e.g. traffic signs convey definite meaning which is either true or false).
Propositions are also spoken of as the content of 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, doubt, and so on, are thus called propositional attitudes when they take this sort of content.
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. For more, see the entry on internalism and externalism in philosophy of mind.
Propositions show up in modern formal logic as objects of a formal language. A formal language begins with different types of symbols. These types can include variables, operators, function symbols, predicate (or relation) symbols, quantifiers, and propositional constants. (Grouping symbols such as Delimiter are often added for convenience in using the language, but do not play a logical role.) Symbols are Concatenation together according to Recursion rules, in order to construct strings to which Truth value will be assigned. The rules specify how the operators, function and predicate symbols, and quantifiers are to be concatenated with other strings. A proposition is then a string with a specific form. The form that a proposition takes depends on the type of logic.
The type of logic called propositional, sentential, or statement logic includes only operators and propositional constants as symbols in its language. The propositions in this language are propositional constants, which are considered atomic propositions, and composite (or compound) propositions, which are composed by recursively applying operators to propositions. Application here is simply a short way of saying that the corresponding concatenation rule has been applied.
The types of logics called Predicate logic include variables, operators, predicate and function symbols, and quantifiers as symbols in their languages. The propositions in these logics are more complex. First, one typically starts by defining a term as follows:
For example, if + is a binary function symbol and x, y, and z are variables, then x+(y+z) is a term, which might be written with the symbols in various orders. Once a term is defined, a proposition can then be defined as follows:
For example, if = is a binary predicate symbol and ∀ is a quantifier, then ∀x,y,z (x is a proposition. This more complex structure of propositions allows these logics to make finer distinctions between inferences, i.e., to have greater expressive power.
In this context, propositions are also called sentences, statements, statement forms, formulas, and well-formed formulas, though these terms are usually not synonymous within a single text. This definition treats propositions as Syntax objects, as opposed to Semantics or Mental world objects. That is, propositions in this sense are meaningless, formal, abstract objects. They are assigned meaning and truth-values by mappings called interpretations and valuations, respectively.
In mathematics, propositions are often constructed and interpreted in a way similar to that in predicate logic—albeit in a more informal way. For example. an axiom can be conceived as a proposition in the loose sense of the word, though the term is usually used to refer to a proven mathematical statement whose importance is generally neutral by nature. Other similar terms in this category include:
Assuming a structured view of propositions, one can distinguish between singular propositions (also Russellian propositions, named after Bertrand Russell) which are about a particular individual, general propositions, which are not about any particular individual, and particularized propositions, which are about a particular individual but do not contain that individual as a constituent. Structured Propositions by Jeffrey C. King
Two meaningful declarative sentences express the same proposition, if and only if they mean the same thing.which defines proposition in terms of synonymity. 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 definition of proposition is:
Two meaningful declarative sentence-tokens express the same proposition, if and only if they mean the same thing.Unfortunately, the above definitions can result in two identical sentences/sentence-tokens appearing to have the same meaning, and thus expressing the same proposition and yet having different truth-values, as in "I am Spartacus" said by Spartacus and said by John Smith, and "It is Wednesday" said on a Wednesday and on a Thursday. These examples reflect the problem of ambiguity in common language, resulting in mistaken equivocation of the statements. “I am Spartacus” spoken by Spartacus is the declaration that the individual speaking is called Spartacus and it is true. When spoken by John Smith, it is a declaration about a different speaker and it is false. The term “I” means different things, so “I am Spartacus” means different things.
A related problem is when identical sentences have the same truth-value, yet express different propositions. The sentence “I am a philosopher” could have been spoken by both Socrates and Plato. In both instances, the statement is true, but means something different.
These problems are addressed in predicate logic by using a variable for the problematic term, so that “X is a philosopher” can have Socrates or Plato substituted for X, illustrating that “Socrates is a philosopher” and “Plato is a philosopher” are different propositions. Similarly, “I am Spartacus” becomes “X is Spartacus”, where X is replaced with terms representing the individuals Spartacus and John Smith.
In other words, the example problems can be averted if sentences are formulated with sufficient precision, that their terms have unambiguous meanings.
A number of philosophers and linguists claim that all definitions of a proposition are too vague to be useful. For them, it is just a misleading concept that should be removed from philosophy and semantics. W. V. Quine, who granted the existence of sets in mathematics, maintained that the indeterminacy of translation prevented any meaningful discussion of propositions, and that they should be discarded in favor of sentences. Strawson, on the other hand, advocated for the use of the term "statement".