"PROPOSITION1 A proposition is any statement in propositional logic, that is, either a propositional letter or a compound formula built up from propositional letters and logical con- nectives."
"PROPOSITION2 A proposition is the object that is the meaning, or content, of a statement or assertion. While there is vast dis- agreement amongst philosophers and logicians as to whether or not propositions exist, and over the nature of propositions if they do exist, believers in propositions typically take them to be the objects of propositonal attitudes such as belief and desire – that is, on this view, when we believe something, or desire for something to be true, it is the proposition that we believe, or desire to become true."
(Cook, Roy T.: Proposition. In: A Dictionary of Philosophical Logic, 2009)
"Propositions, sentences and statements
A sentence is a string of words formed according to the syntactic rules of a language. But a sentence has semantic as well as syntactic properties: the words and the whole sentence have meaning. Philosophers have tended to focus on the semantic properties of indicative sentences, in particular on their being true or false. They have called the meanings of such sentences ‘propositions’, and have tied the notion of proposition to the truth-conditions of the associated sentence.
The term ‘proposition’ is sometimes assimilated to the sentence itself; sometimes to the linguistic meaning of a sentence; sometimes to ‘what is said’; sometimes to the contents of beliefs or other ‘propositional’ attitudes. But however propositions are defined, they must have two features: the capacity to be true or false; and compositional structure (being composed of elements which determine their semantic properties).
One reason for distinguishing a sentence from ‘what the sentence says’ is that a sentence may be meaningless, and hence say nothing, yet still be a sentence. But perhaps the main reason is that two people, A and B, may utter the same sentence, for example, ‘I am hot’, and say the same thing in one sense but not in another. The sense in which they say the same thing is that they use the same words with the same linguistic ‘meaning’. The sense in which they say something different is that they put the same words to different ‘uses’: A uses ‘I’ to refer to A, while B uses it to refer to B. Hence what A says may be true although what B says is false. If what is said can be true in one case and false in another, A and B have not made the same ‘statement’. On the other hand, if B utters instead ‘You are hot’ when A utters ‘I am hot’, they put different sentences to the same use and make the same statement. On this view (see Strawson 1952), we must distinguish the ‘sentence’, the ‘use’ of the sentence, and the ‘statement’ made by using a sentence in a context of utterance. According to Strawson, it is not sentences but statements that are true or false.
Logicians usually abstract from the context of utterance of sentences in actual communication and talk of the propositions expressed as abstract entities. The main modern proponent of this conception is Frege. For Frege (1918), a proposition is a ‘thought’, which is both the cognitive meaning expressed by a sentence and the content of a propositional attitude such as belief or desire; thoughts are the ‘senses’ of sentences. Thoughts are distinguished according to the following principle: if it is possible rationally to believe that p and not to believe that q then the thought that p and the thought that q are distinct (see Frege, G. §§3-4; Sense and reference §2).
Under a different, but related, conception, proposed by Carnap (1947), the proposition expressed by a sentence S is the set of possible worlds in which S is true (see Semantics, possible worlds §9-10). This view violates the compositional structure requirement, since such propositions are not grasped by grasping their components (see Compositionality). It also seems unable to differentiate between some distinct propositions. For example, there would only be one necessarily false proposition, since there is only one empty set of worlds, but intuitively there are many different necessarily false propositions.
An alternative view of propositions as entities is the (neo-)Russellian view that they are collections of actual entities making up ‘facts’ or ‘states of affairs’ (see Facts §1). Thus the proposition expressed by ‘Socrates is mortal’ is the ordered pair hSocrates, being mortalix, composed of the individual Socrates and the property of being mortal. Three questions for this view are: (1) Are the contents of false sentences negative facts, and are there such facts? (2) What is the criterion of identity for facts? (3) How can propositions as facts be the contents of propositional attitudes? The last of these can be seen as a version of the problem which led Frege to postulate senses: for the fact that a = b must be the same fact as a = a, although someone who believes that a = a does not thereby believe that a = b, and hence the propositions are not the same (see Sense and reference §1). Fregeans conclude that this shows that we cannot dispense with the notion of sense. If Russellians deny this, they must complicate their account of facts or their account of propositional attitudes (see Propositional attitude statements).
Whether one defines the intuitive notion of ‘what is said’ as a context-independent entity or by recourse to the notion of a statement or proposition expressed by an utterance in a particular context, one must say what it means for two sentences to express ‘the same proposition’ or make ‘the same statement’. Both phrases rely on the notions of meaning and synonymy, criticized by Quine (1968; see Quine, W.V. §8). But if we dispense altogether with propositions, statements or any other notion of the content of what is said, and choose instead to take sentences as truth bearers, we face two problems. The first is that sentences are unlikely candidates for the role of contents of propositional attitudes: if ‘I believe I am stupid’ introduces a relation between me and the English sentence ‘I am stupid’, it should be translated into German (say) as ‘Ich glaube "I am stupid"’, although the correct translation is ‘Ich glaube daß ich dumm bin’, which says nothing about any English sentence (Church 1950). The second problem is that, in spite of our ontological scruples about admitting propositions as entities, we still need such a notion to express the content of what is said or believed, and to account for the intentionality of thought in general. In that respect, however vague or ill-defined the notion of a proposition, it cannot be dispensed with."
(Engel, Pascal: Propositions, sentences and statements. In: Routlegde Encyclopedia of Philosophy, 1998)
"Propositions, then, as standardly understood, are contents of utter- ances and of propositional attitudes, they are not indi- viduated in terms of any particular language, they are tightly bound up with, or identical to, their truth conditions, they are unchanging bearers of truth, falsity, contingency, or necessity, and they stand in relations of entailment, exclusion, and compatibility."
(Crimmins, M.: Proposition. In: Lamarque (Ed.): Concise Encyclopedia of Philosophy of Language, 1997).