They are nothing but certain rules which determine the use of language, i. e. of expression by combination of “signs.” They are concerned with symbols, not with reality. They are nevertheless in a certain sense applicable to the world of facts, because symbols (words, letters, etc.) are used in speaking about facts. In short, they speak of reality, but they do not say anything about it. This can best be made clear by referring to simple examples. I shall first take a mathematical example, and afterwards a logical one.
You will all admit that there is absolutely no difference of meaning between the two sentences: “I have in my pocket twelve dollars” and “I have in my pocket seven plus five dollars.” Well, the famous “proposition” that seven plus five equals twelve is nothing but the rule which tells us that we may transform the one sentence into the other without changing the meaning.
You observe this is not the expression of any fact in the universe. It does not assert that there are twelve objects in the world, or seven or five; it does not assert that any one can count or has counted any objects: it just gives us to understand that a man who says “Here are twelve things” and another who says “Here are five plus seven things” have not said anything different, but have only used different words in order to express the same meaning (provided, of course, that they used the words five, seven and twelve in their ordinary sense; and when a little while ago I spoke of the great difficulties involved in this issue I had in mind chiefly the difficulty of clarifying the “ordinary sense”). That our arithmetical rule “applies” to any objects whatever is nothing remarkable or wonderful, for it does not say anything about any objects.
This point comes out perhaps even with greater clarity when we examine a purely logical example, as for example, the Principle of the Excluded Middle. When I say: “My friend will either come or not come tomorrow,” the logical principle just referred to assures me that this statement is always true, whenever and wherever it is made — but is that statement really a proposition? does it assert anything about my friend or his coming, or indeed about any other fact in the world? Evidently not. It speaks of my friend and his coming, but does not say anything about them, it asserts nothing whatever. After I have heard the sentence I know absolutely no more about the world than I did before; the sentence has communicated no fact to me.
