Elementary propositions, the simplest kind of proposition, consist of names (4.22) and depict a possible state of affairs (4.21). Just as the existence or non-existence of any possible state of affairs has no bearing on the existence or non-existence of any other possible state of affairs, so does the truth or falsity of any elementary proposition have no bearing on the truth or falsity of any other elementary proposition. And just as the totality of all existent states of affairs is the world, so the totality of all true elementary propositions is a complete description of the world (4.26).
Any given elementary proposition is either true or false. Combining the two elementary propositions, p and q, produces four separate truth- possibilities: (1) both p and q are true, (2) p is true and q is false, (3) p is false and q is true, and (4) both p and q are false. We can express the truth-conditions of a proposition that joins p and q—say, "if p then q—in terms of these four truth- possibilities in a table, thus:
p | q | T | T | T T | F | T F | T | F F | F | T
This table is a propositional sign for "if p then q." The results of this table can be expressed linearly, thus: "(TTFT)(p,q)" (4.442). From this notation it becomes clear that there are no "logical objects," such as a sign expressing the "if then" conditional (4.441).
A proposition that is true no matter what (e.g. "(TTTT)(p,q)") is called a "tautology" and a proposition that is false no matter what (e.g. "(FFFF)(p,q)") is called a "contradiction" (4.46). Tautologies and contradictions lack sense in that they do not represent any possible situations, but they are not nonsense, either. A tautology is true and a contradiction is false no matter how things stand in the world, whereas nonsense is neither true nor false.
Propositions are built up as truth-functions of elementary propositions (5). The "truth-grounds" of a proposition are the truth-possibilities under which the proposition comes out true (5.101). A proposition that shares all the truth- grounds of one or several other propositions is said to follow from those propositions (5.11). If one proposition follows from another, we can say the sense of the former is contained in the sense of the latter (5.122). For instance, the truth-grounds for "p" are contained in the truth-grounds for "p.q" ("p" is true in all those cases where "p.q" is true), so we can say that "p" follows from "p.q" and that the sense of "p" is contained in the sense of "p.q."
We can infer whether one proposition follows from another from the structure of the propositions themselves: there is no need for "laws of inference" to tell us how we can and cannot proceed in logical deduction (5.132). We must also recognize, however, that we can only infer propositions from one another if they are logically connected: we cannot infer one state of affairs from a totally distinct state of affairs. Thus, Wittgenstein concludes, there is no logical justification for inferring future events from those of the present (5.1361).
We say that "p" says less than "p.q" because it follows from "p.q." Consequently, a tautology says nothing at all, since it follows from all propositions and no further propositions follow from it.
The logic of inference is the basis for probability. Let us take as an example the two propositions "(TFFF)(p,q)" ("p and q") and "(TTTF)(p,q)" ("p or q"). We can say that the former proposition gives a probability of one/3 to the latter proposition, because—excluding all external considerations—if the former is true, then there is a one in three chance that the latter will be true as well. Wittgenstein emphasizes that this is only a theoretical procedure; in reality there are no degrees of probability: propositions are either true or false (5.153).
Truth tables are tables we can draw up to schematize a proposition and determine its truth-conditions. Wittgenstein does this at 4.31 and 4.442. Wittgenstein did not invent truth tables, but their use in modern logic is usually traced to his introduction of them in the Tractatus. Wittgenstein was also the first philosopher to recognize that they could be wielded as a significant philosophical tool.
The assumption that underlies Wittgenstein's work here is that the sense of a proposition is given if its truth conditions are given. If we know under what circumstances a proposition is true and under what circumstances it is false, then we know all there is to know about that proposition. On reflection, this assumption is perfectly reasonable. If I know what would have to be the case for "Your dog is eating my hat" to be true, and if I know what would have to be the case for it to be false, then I can be said to know what that proposition means. An exhaustive list of the truth-possibilities of a proposition, coupled with an indication of which truth-possibilities make the proposition come out true and which false, will tell us all we need to know about that proposition.
This is exactly what truth tables do. Any proposition, according to Wittgenstein, consists of one or more elementary proposition, each of which can be true or false independently of any other. If we put all the elementary propositions that constitute a given proposition into a truth table that lists all the possible combinations of true or false that can hold between them, we will have an exhaustive list of the truth-conditions of the given proposition. Thus, a truth table can show us the sense of the proposition. The proposition "p.q" ("p and q") can equally well be expressed as a truth table, or as "(TFFF)(p,q)."
The great advantage of this notation is that it expresses the sense of a proposition without any of the connectives we normally find in logical notation, such as "and," "or," and "if then." Clearly, none of these connectives are essential to the sense of the proposition, thus giving credence to Wittgenstein's "fundamental idea" (4.0312) that "the 'logical constants' are not representatives." In a truth table, the connections between elementary propositions "show" themselves, and so need not be said.
Wittgenstein also explains that this method can "show" the workings of logical inference, thus rendering unnecessary the "laws of inference" that both Frege and Russell had built into their axiomatic systems. One proposition follows from a second proposition if the first is true whenever the second is true. If we express "p or q" as "(TTTF)(p,q)" and "p and q" as "(TFFF)(p,q)" we can see that the former follows from the latter by comparing their truth-grounds: where there is a "T" in the latter proposition, there is a corresponding "T" in the former proposition. We don't need a law of inference to tell us this: it shows itself plainly in the truth-grounds of the two propositions.
The limiting cases of propositions are tautologies and contradictions. Wittgenstein uses the German word sinnloss ("senseless") to describe the peculiar status of tautologies and contradictions, in contrast to unsinnig, or "nonsensical." They are not nonsense because they consist of elementary propositions and are held together in a logical way. However, these elementary propositions are held together in such a way that they do not represent any possible state of affairs. Tautologies, as necessarily true and not representative of any particular fact, are particularly interesting to Wittgenstein. As we shall see, he will claim at 6.1 that the propositions of logic are tautologies.