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."
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