Propositional Logic · Validity

Truth Tables as
a Validity Test

A truth table can test whether a propositional argument is valid by checking every possible scenario at once. If there is even one row where all the premises are true and the conclusion is false, the argument is invalid.

The central question
Is there any possible scenario — any row — where all the premises are true and the conclusion is false?
If no such row exists → the argument is valid. The truth of the premises guarantees the truth of the conclusion in every possible scenario.

If even one such row exists → the argument is invalid. That row is a counterexample — a possible situation where the premises hold but the conclusion fails.
1
How to Test Validity
1
Set up the table. List all variables, determine the number of rows (2ⁿ), and fill in all combinations of T and F systematically.
2
Add a column for each premise and the conclusion. Work column by column, building sub-formulas up to the main connective of each.
3
Scan for a critical row — any row where every premise column is T and the conclusion column is F.
4
If no such row exists: valid. If one exists: invalid — and that row tells you exactly what a counterexample looks like.
2
Worked Examples
✓ Valid
P → Q
P
∴ Q
1
Two variables (P, Q) → 4 rows. Columns: P, Q, P→Q (premise 1), P (premise 2), Q (conclusion).
2
Scan for any row where P→Q is T and P is T but Q is F.
P Q P → Q P Q
TTTTT
TFFTF
FTTFT
FFTFF
No row has both premises true and conclusion false. Row 1 is the only row where both premises are T — and conclusion is T there too. The argument is valid.
✗ Invalid
P → Q
Q
∴ P
1
Two variables → 4 rows. Same setup, but the second premise is Q and conclusion is P.
2
Scan for a row where P→Q is T and Q is T but P is F.
P Q P → Q Q P
TTTTT
TFFFT
FTTTF
FFTFF
Row 3 is a critical row — both premises are T but the conclusion is F. This is a counterexample: P is false, Q is true, and the argument breaks down. The argument is invalid.
3
What the Final Column Can Tell You
Tautology
A formula that is true in every possible row — no matter what truth values the variables take. Also called a logical truth.
Final column: all T
Example: P ∨ ~P
"It is raining or it is not raining" — always true.
Contingent
A formula that is true in some rows and false in others. Its truth value depends on the actual truth values of the variables.
Final column: mix of T and F
Example: P → Q
True sometimes, false sometimes, depending on P and Q.
Contradiction
A formula that is false in every possible row — no assignment of truth values can make it true. Also called a logical falsehood.
Final column: all F
Example: P ∧ ~P
"It is raining and it is not raining" — never true.
The connection to validity: A valid argument is one whose corresponding conditional — all premises conjoined as the antecedent, conclusion as the consequent — is a tautology. If (P₁ ∧ P₂ ∧ … ∧ Pₙ) → C is true in every row, the argument is valid. This is why validity testing and tautology testing are two sides of the same coin.