Statement Patterns: Tautology, Contradiction and Contingency

When simple statements are replaced by variables $p, q, r, \ldots$ and combined by connectives, the resulting expression — for example $(p \wedge q) \rightarrow p$ — is called a statement pattern. A statement pattern is not itself true or false; it takes a truth value only once the variables are assigned truth values. Classifying a statement pattern by the shape of its final truth-table column is one of the most frequently examined ideas in this chapter.

Every statement pattern falls into exactly one of three classes according to its last column:

• Tautology — the pattern is true for every assignment of truth values (the final column is all $T$). A tautology is a logical certainty. The simplest example is $p \vee \sim p$ ("either $p$ or not $p$"), which is true whatever $p$ is.

• Contradiction — the pattern is false for every assignment (the final column is all $F$). The standard example is $p \wedge \sim p$ ("$p$ and not $p$"), which can never be true.

• Contingency — the pattern is true for some assignments and false for others (the final column contains at least one $T$ and at least one $F$). Most everyday compound statements, such as $p \wedge q$, are contingencies.

A useful link: a statement pattern is a tautology if and only if its negation is a contradiction, and vice versa. This mirrors the everyday sense that the denial of a certainty is an impossibility. …