Tautologies and Contradictions in Logic: Truth Tables Guide

A tautology is a compound statement that is true in every row of its truth table, no matter what truth values its parts have. The classic example is p ∨ ¬p: if p is true the first half is true, and if p is false the second half is true, so the whole thing can never be false.

A contradiction is the mirror image: false in every row, such as p ∧ ¬p. Statements true in some rows and false in others are called contingencies. Proofs constantly aim to produce tautologies, and proof by contradiction aims to produce a contradiction.

Key Concepts

• Tautology: final truth-table column is all true. Example: p ∨ ¬p (law of excluded middle).
• Contradiction: final column is all false. Example: p ∧ ¬p.
• Contingency: final column mixes true and false. Example: p → q.

How It Works

To classify a statement, build its truth table and inspect the final column:

1. Write all 2n rows for the variables in standard order.
2. Compute intermediate columns inside-out, then the final column.
3. If every entry in the final column is true, the statement is a tautology.
4. If every entry is false, it is a contradiction; if the entries are mixed, it is a contingency.

Worked Example

Classify (p → q) ∨ (q → p) by computing all four rows:

• Row 1 (T, T): p → q = T and q → p = T, so T.
• Row 2 (T, F): p → q = F, q → p = T, so T.
• Row 3 (F, T): p → q = T, q → p = F, so T.
• Row 4 (F, F): both implications are T (vacuously), so T.

The final column is T, T, T, T, so (p → q) ∨ (q → p) is a tautology: for any two statements, at least one of the two implications holds.

Common Misconceptions

• "A tautology is just something obviously true." Obviousness is about psychology; a tautology is about form. (p → q) ∨ (q → p) is a genuine tautology even though it surprises people. The truth table is the judge, not your intuition.
• "A contradiction is a statement that sounds impossible." "This circle is a square" is simply a false proposition. A contradiction like p ∧ ¬p is false in every row, which is a stronger structural claim.
• "If a statement is not a tautology, it is a contradiction." No: most statements are contingencies, true sometimes and false sometimes. Only the all-true and all-false extremes get the special names.

Real-World Applications

• Proof by contradiction: you assume the negation of what you want to prove and derive a contradiction (p ∧ ¬p), which is impossible, so the assumption must be false.
• Circuit design: an all-true or all-false output column is a constant signal; spotting this saves hardware.
• Automated reasoning: SAT solvers and theorem provers constantly test whether formulas are tautologies or contradictions.

Tautologies are true in every row, contradictions false in every row, and everything else a contingency. Compute the final column and read the answer off directly.