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Truth Table
A table listing every possible truth-value combination of input propositions and the resulting truth value of a compound formula.
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A table listing every possible truth-value combination of input propositions and the resulting truth value of a compound formula.
A truth table is a mathematical table used in classical propositional logic to show how the truth value of a compound formula depends on the truth values of its atomic propositions. For n atomic variables it has 2^n rows, one per assignment. Truth tables provide a mechanical decision procedure: a formula is a tautology if its final column is all true, a contradiction if all false, and contingent otherwise. They also validate arguments — an argument is valid iff no row makes all premises true and the conclusion false.
For n variables: 2^n rows; connectives: ¬, ∧, ∨, →, ↔
- p ∧ q is true only when both p and q are true
- p → q is false only when p is true and q is false
- p ∨ ¬p is a tautology (always true)
How to recognize it
- Use a truth table whenever you need to check all combinations of T/F values
- Count rows: 2 variables = 4 rows, 3 variables = 8 rows, n variables = 2^n rows
- Evaluate sub-formulas column by column, innermost first
Common mistakes
- Missing a row (forgetting one combination of inputs)
- Computing p → q as 'true when p is true' (it is false only when p true and q false)
- Treating p ↔ q as 'p and q' (biconditional requires equal truth values, including both false)