The sense of a proposition.

A proposition corresponds, or not, to matters of fact. These, in turn, can be the case or not. The sense of a proposition consists of this correspondence. The simplest propositions, the elemental propositions, assert that a matter of fact is the case. An indication that a proposition is elemental is that no elemental proposition can contradict it.

An elemental proposition consists of a list of names liked to each other. It is clear that when we analyze propositions, we must reach elemental propositions which consist of names in direct combination. This raises the question of how to combine them into propositions. Even if the world is infinitely complex, so that every fact consists of infinitely many matters of fact and each of them is composed of infinitely many objects of consideration, there would still have to be objects of consideration and matters of fact.

Names only occur in the context of an elemental proposition. They are simple symbols, indicated by single letters ('x', 'y', 'z'). Wittgenstein writes elemental propositions as functions of names, so that they have the form 'fx', 'O (x,y)', etc. or indicates them by the letters 'p', 'q', 'r'.

When using two signs with one and the same meaning, he expresses this by putting the sign '=' between them. So (a = b) means that the sign 'b' can be substituted for the sign 'a'. If he uses an equation to introduce a new sign 'b', deciding that it shall replace a known sign 'a', then, like Russell, he writes the equation, a definition, in the form 'a = b Def.' A definition is a rule dealing with signs. Expressions of the form 'a = b' are, therefore, mere tools of representation. They say nothing about the meaning of the signs 'a' and 'b'.

Can we understand two names without knowing whether they signify the same thing or two different things? Can we understand a proposition in which two names occur without knowing whether their meaning is the same or not? If one knows both an English word and a German word with the same meaning, then it is impossible for me not to know that. It is not possible that one cannot translate them. Expressions like 'a = a', or those derived from them, are neither elemental propositions nor significant in any other way. (This will become evident later.)

If an elemental proposition is true, the matter of fact it describes is the case; if false, then not. Giving all true elemental propositions describes the world completely. What amounts to the same thing, one can give all elemental propositions and mark them as true or false. Of these matters of fact any combination can be the case and the remainder not be.