Tautology, Contradiction and Contingency

Tautology, Contradiction and Contingency

Once we have the truth table of a statement pattern, we can look at its final column as a whole and classify the pattern into one of three kinds, purely on the basis of that column.

A statement pattern is called a tautology (denoted $t$) when its final column is T in every single row — its truth value is True no matter what truth values its prime components take. The pattern $p \vee \sim p$ is a simple example: whatever truth value $p$ has, either $p$ itself is true or its negation $\sim p$ is true, so $p \vee \sim p$ can never come out false.

A statement pattern is called a contradiction (denoted $c$) when its final column is F in every single row — it is impossible for the pattern to be true under any assignment of truth values. The pattern $p \wedge \sim p$ illustrates this: $p$ and $\sim p$ can never both be true at once, so their conjunction is always false.

A statement pattern that is neither a tautology nor a contradiction — that is, its final column contains a mixture of T's and F's — is called a contingency. The pattern $p \wedge q$ is a contingency, since it is true only when both $p$ and $q$ are true and false in the other three cases.

Important table for all the connectives

Before working through the solved examples it helps to have the truth values of every basic connective collected in one place, since every larger pattern is just built up column by column from these:

           **Solved Examples**

Ex. 1 — Construct the truth table for each of the following statement patterns.

When several connectives appear together without brackets to guide the order, they are evaluated in the following order of priority (highest to lowest): $\sim$, then $\vee$, then $\wedge$, then $\rightarrow$, then $\leftrightarrow$.

(i) $p \rightarrow (q \rightarrow p)$

We first build the inner column $q \rightarrow p$, then apply the outer $\rightarrow$ with $p$ as antecedent.

(Table 1.7) — notice the final column is T throughout; we return to what that means once tautologies are formally discussed, but for this example we are only asked to construct the table.

(ii) $(\sim p \vee q) \leftrightarrow \sim (p \wedge q)$

Build $\sim p$, then $\sim p \vee q$ on the left branch; separately build $p \wedge q$ then negate it for the right branch; finally combine the two branches with $\leftrightarrow$.

(Table 1.8)

(iii) $\sim (\sim p \wedge \sim q) \vee q$

Build $\sim p$ and $\sim q$, conjoin them, negate the conjunction, then disjoin with $q$.

(Table 1.9)

(iv) $[(p \wedge q) \vee r] \wedge [\sim r \vee (p \wedge q)]$

With three prime components $p, q, r$ there are $2^3 = 8$ rows to work through, taken in the standard order TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF. Build $\sim r$ and $p \wedge q$ first, then the two bracketed halves, then conjoin them.

(Table 1.10)

(v) $[(\sim p \vee q) \wedge (q \rightarrow r)] \rightarrow (p \rightarrow r)$

Again 8 rows. Build $\sim p$, then $\sim p \vee q$; separately build $q \rightarrow r$ and $p \rightarrow r$; conjoin the first two intermediate columns, then apply the final $\rightarrow$ using $p \rightarrow r$ as the consequent.

(Table 1.11)

Ex. 2 — Using truth tables, prove the following logical equivalences.

To prove $A \equiv B$ we build the truth table containing both $A$'s column and $B$'s column side by side and check that the two columns are identical row for row.

(i) $(p \wedge q) \equiv \sim (p \rightarrow \sim q)$

(Table 1.12) — the column for $p \wedge q$ and the column for $\sim(p \rightarrow \sim q)$ read T, F, F, F in exactly the same order, so the two columns are identical. Hence $(p \wedge q) \equiv \sim (p \rightarrow \sim q)$.

(ii) $(p \leftrightarrow q) \equiv (p \wedge q) \vee (\sim p \wedge \sim q)$

(Table 1.13) — the column for $p \leftrightarrow q$ and the column for $(p \wedge q) \vee (\sim p \wedge \sim q)$ both read T, F, F, T, so they are identical. Hence $(p \leftrightarrow q) \equiv (p \wedge q) \vee (\sim p \wedge \sim q)$.

(iii) $(p \wedge q) \rightarrow r \equiv p \rightarrow (q \rightarrow r)$

(Table 1.14) — the column for $(p \wedge q) \rightarrow r$ and the column for $p \rightarrow (q \rightarrow r)$ both read T, F, T, T, T, T, T, T, so they match in every row. Hence $(p \wedge q) \rightarrow r \equiv p \rightarrow (q \rightarrow r)$.

(iv) $p \rightarrow (q \vee r) \equiv (p \rightarrow q) \vee (p \rightarrow r)$

(Table 1.15) — the column for $p \rightarrow (q \vee r)$ and the column for $(p \rightarrow q) \vee (p \rightarrow r)$ both read T, T, T, F, T, T, T, T, matching row for row. Hence $p \rightarrow (q \vee r) \equiv (p \rightarrow q) \vee (p \rightarrow r)$.

Ex. 3 — Using truth tables, examine whether each of the following statement patterns is a tautology, a contradiction, or a contingency.

Here we build the full truth table as before, then look only at the final column: all-T means tautology, all-F means contradiction, and a mix of T and F means contingency.

(i) $(p \wedge q) \wedge (\sim p \vee \sim q)$

(Table 1.16) — the final column is F, F, F, F: false in every row. This pattern is a contradiction. (This makes sense: $p \wedge q$ demands both true, while $\sim p \vee \sim q$ demands at least one false — the two halves can never be satisfied together.)

(ii) $[p \wedge (p \rightarrow \sim q)] \rightarrow q$