The term "proposition" is used ambiguously, which may be confusing, and indeed in three ways (statement, idea of statement, fact represented by statement).

Thus, in the second sense "it rains", "il pleut" and "es regnet" all express the same proposition, in three natural languages, while in the first sense, these are three different propositions, since they do not consist of the same letters and interpunction in the same sequence.

However, this feature makes the term 'proposition' useful, since one thereby has a means of referring both to a statement and the idea it states, and a term to indicate that different statements in different languages may express the same idea.

1. Threefold interpretation of statements

As indicated above, the need for a term like "proposition" arises from the fact that a string of words may be taken to stand for a string of words, for an idea represented by that string of words, or for a fact represented directly by an idea and indirectly by a string of words.

This suggests that it is generally useful to conceive of propositions in terms of the following schema, where the term proposition, that occurs four times in it, may be taken by any statement:

(proposition) = ("proposition", 'proposition', [proposition])

where the term (proposition) stands for the triple composed of

(1) the string used to expressed it, between double quotes, and in a definite language
(2) the idea represented by the string just mentioned, between single quotes, and itself, albeit represented by terms of some language, itself not normally an element of a language (though this may happen with statements about language)
(3) the fact represented by the idea just mentioned, between straight brackets, and itself, albeit represented by terms of some language, itself not normally an element of a language (though this may happen with statements about language)

Note that in general statements i.e. strings of words that are understood by some speaker of the language to which the words belongs have this triple aspect for the speaker, for he will know both the statement and its meaning i.e. the idea expressed by it (supposing for the moment that the statement is unambiguous), and usually will have some belief about the fact it represents, if any, that usually takes the form of a truth-value or probability.

And it is also noteworthy that very often one has much clearer ideas about what a proposition means - "the number of leaves on all the trees in the territory of the city of London on august 1, 1700 at 12.00 in the afternoon was even" - then about its truth or probability.