Propositional Logic · Evaluation

Truth
Tables

A truth table systematically displays the truth value of a proposition for every possible combination of truth values its component parts could have. It is one of the most powerful tools in propositional logic.

What is a truth table?
A truth table lists every possible scenario — every combination of true and false for the variables — and shows what the compound formula evaluates to in each one.
Rows represent possible scenarios. With n variables, there are always 2ⁿ rows.
Columns represent the variables and sub-formulas, built up step by step toward the main connective.
The column under the main connective gives the truth value of the whole formula in each scenario.
Truth tables are mechanical — no judgment required, just careful application of each connective's definition.
1
How Many Rows?
2ⁿ rows
The number of rows is always 2 raised to the power of the number of variables. Each variable can be T or F, and every combination must appear exactly once.

1 variable (P) → 2¹ = 2 rows  ·  2 variables (P, Q) → 2² = 4 rows  ·  3 variables (P, Q, R) → 2³ = 8 rows  ·  4 variables → 2⁴ = 16 rows

To fill in the columns systematically: alternate T/F every 1 row for the first variable, every 2 rows for the second, every 4 rows for the third, and so on.
2
Truth Tables for the Five Connectives
Negation
~P
P ~P
TF
FT
Flips the truth value. T becomes F, F becomes T.
Conjunction
P ∧ Q
PQP ∧ Q
TTT
TFF
FTF
FFF
True only when both P and Q are true. False otherwise.
Disjunction
P ∨ Q
PQP ∨ Q
TTT
TFT
FTT
FFF
False only when both P and Q are false. True otherwise.
Conditional
P → Q
PQP → Q
TTT
TFF
FTT
FFT
False only when P is true and Q is false — a true condition that isn't met. True in all other cases, including when P is false.
Biconditional
P ↔ Q
PQP ↔ Q
TTT
TFF
FTF
FFT
True when P and Q match — both true or both false. False when they differ.
3
Extending to Complex Formulas
Worked Example
(P ∨ Q) → ~P
For a formula with more than one connective, build the table column by column, working from the smallest sub-formulas up to the main connective. Each column feeds into the next.
Column order for (P ∨ Q) → ~P
1
Start with the raw variables: P and Q
2
Compute ~P (negation of P)
3
Compute P ∨ Q (disjunction)
4
Compute the main connective: (P ∨ Q) → ~P using columns 3 and 2
P Q ~P (step 2) P ∨ Q (step 3) (P ∨ Q) → ~P main connective
T T F T F
T F F T F
F T T T T
F F T F T
Read the table row by row. In row 1: P is true, Q is true → ~P is false → P ∨ Q is true → a true antecedent with a false consequent → the conditional is false. In row 3: P is false → ~P is true → P ∨ Q is true → a true antecedent with a true consequent → true.
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Challenge Problems
Construct a complete truth table for each formula. Remember to identify the variables, determine the number of rows, and work column by column toward the main connective.
1
P → (P ∧ Q)
2
~P ∨ (P → Q)
3
(P ∧ ~Q) ↔ ~(P → Q)
4
(P ∨ Q) ∧ (~P ∨ R)