Unit 6: Truth Tables
Learning Objectives
- Construct truth tables for the four basic logical connectives (NOT, AND, OR, IF-THEN)
- Use truth tables to evaluate the truth of compound propositions
- Verify validity of argument forms using truth tables
- Identify tautologies and contradictions
- Understand how truth tables connect formal logic to propositional reasoning
What is a Truth Table?
A truth table is a systematic method for determining the truth value of a compound proposition under every possible combination of truth values for its component propositions. Truth tables provide a mechanical — and completely reliable — way to check whether an argument form is valid.
The number of rows in a truth table depends on the number of propositional variables: with n variables, there are 2n rows. Two variables give 4 rows; three variables give 8 rows; and so on.
Three variables (P, Q, R): 8 rows
Four variables: 16 rows
We evaluate one connective at a time, working from the innermost parentheses outward — just as in arithmetic where you evaluate brackets before multiplication before addition.
The Four Basic Connectives
Propositional logic uses four basic connectives to build compound propositions. Each has a precise definition captured in its truth table.
NOT (negation) — ¬P
NOT reverses the truth value of a proposition.
| P | ¬P (NOT P) |
|---|---|
| T | F |
| F | T |
AND (conjunction) — P ∧ Q
AND is true only when both propositions are true.
| P | Q | P ∧ Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
OR (disjunction) — P ∨ Q
OR (inclusive) is true when at least one proposition is true.
| P | Q | P ∨ Q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
IF-THEN (conditional) — P → Q
The conditional is false only when P is true and Q is false.
| P | Q | P → Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Evaluating Compound Propositions
To evaluate a compound proposition, work column by column. Each column represents one logical step. Consider: (P ∧ Q) → ¬P
| P | Q | ¬P | P ∧ Q | (P ∧ Q) → ¬P |
|---|---|---|---|---|
| T | T | F | T | F |
| T | F | F | F | T |
| F | T | T | F | T |
| F | F | T | F | T |
Because the final column is not all T, this formula is not a tautology. In row 1, the compound proposition is false.
Tautologies and contradictions
Tautology
A proposition that is true under every possible assignment of truth values.
P ∨ ¬P (P or not P — always true)
Contradiction
A proposition that is false under every possible assignment of truth values.
P ∧ ¬P (P and not P — always false)
Verifying Argument Validity
An argument form is valid if and only if there is no row in its truth table where all premises are true and the conclusion is false. Equivalently, the formula (P1 ∧ P2 ∧ … ∧ Pn) → C is a tautology.
Verifying Modus Ponens
Form: If P then Q; P; therefore Q. Combined formula: ((P → Q) ∧ P) → Q
| P | Q | P → Q | (P → Q) ∧ P | ((P → Q) ∧ P) → Q |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | F | T |
| F | F | T | F | T |
The final column is all T — confirming that modus ponens is a tautology, hence always valid.
Exposing an invalid form
Affirming the consequent: If P then Q; Q; therefore P. Formula: ((P → Q) ∧ Q) → P
| P | Q | P → Q | (P → Q) ∧ Q | ((P → Q) ∧ Q) → P |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | T | F |
| F | F | T | F | T |
Row 3 is F — the formula is not a tautology, confirming that affirming the consequent is invalid.
Practice: Complete These Tables
Complete the truth table for each formula. Then determine whether each is a tautology, a contradiction, or neither.
- ¬P ∨ P
- P ∧ ¬P
- P → (P ∨ Q)
- (P ∧ Q) → P
- ¬(P ∧ Q) ↔ (¬P ∨ ¬Q) (De Morgan's Law)
- ¬P ∨ P — Tautology. Always true (law of excluded middle).
- P ∧ ¬P — Contradiction. Always false (law of non-contradiction).
- P → (P ∨ Q) — Tautology. If P is true, P ∨ Q is true. If P is false, the conditional is trivially true.
- (P ∧ Q) → P — Tautology. If both P and Q are true, then certainly P is true.
- ¬(P ∧ Q) ↔ (¬P ∨ ¬Q) — Tautology. This is De Morgan's Law: "not (P and Q)" is equivalent to "(not P) or (not Q)".
Check Your Understanding
A tautology is a proposition that is:
The conjunction P ∧ Q is true when:
An argument form is valid if and only if its combined formula (premises → conclusion) is:
Review
Expand each concept to check your understanding before moving on.
A truth table lists every possible combination of truth values for the variables in a proposition, and shows the resulting truth value of the whole expression. With n variables there are 2n rows. Evaluate one connective at a time, building up from the innermost subformulas.
NOT (¬P): true when P is false; false when P is true.
AND (P ∧ Q): true only when both P and Q are true.
OR (P ∨ Q): true when at least one of P or Q is true.
IF-THEN (P → Q): false only when P is true and Q is false.
A tautology is true under every possible assignment of truth values (e.g., P ∨ ¬P). A contradiction is false under every assignment (e.g., P ∧ ¬P). Most compound propositions are neither — they are true under some assignments and false under others.
An argument is valid if the combined formula (all premises conjoined, then placed as antecedent of a conditional with the conclusion as consequent) is a tautology. If there is even one row where all premises are true but the conclusion is false, the argument is invalid. This is called a counterexample.
Two important logical equivalences:
¬(P ∧ Q) ≡ ¬P ∨ ¬Q: "not (P and Q)" = "(not P) or (not Q)"
¬(P ∨ Q) ≡ ¬P ∧ ¬Q: "not (P or Q)" = "(not P) and (not Q)"
These equivalences allow negation to be distributed through conjunctions and disjunctions.
Key concepts covered in this unit: truth table, logical connective, negation (NOT / ¬), conjunction (AND / ∧), disjunction (OR / ∨), conditional (IF-THEN / →), tautology, contradiction, counterexample, logical equivalence, De Morgan's Laws.
Proceed to Unit 7: Informal Fallacies I when ready.