A truth table evaluates a compound logical statement for every possible assignment of truth values to its basic statements. It replaces intuition about a complicated sentence with a complete, checkable enumeration.
For two independent proposition letters p and q, there are four assignments: true–true, true–false, false–true and false–false. A table must include them all. Here T and F mean logical truth and falsity, not probabilities or frequencies.
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Build the input rows systematically
With n proposition letters, there are truth assignments. For three letters, list eight rows by holding p true for four rows and false for four, q true for two at a time, and r alternating each row.
Other row orders are valid, provided every truth-value assignment appears. Omitting an awkward row can make two different statements appear equivalent or hide an invalid argument.
If a specific application restricts the possible combinations, first build the ordinary truth table and then identify the relevant restricted rows. Do not confuse equivalence under an extra assumption with equivalence for every assignment.
Use the basic connective rules
| p | q | |||||
|---|---|---|---|---|---|---|
| T | T | F | T | T | T | T |
| T | F | F | F | T | F | F |
| F | T | T | F | T | T | F |
| F | F | T | F | F | T | T |
Conjunction requires both inputs true. Inclusive disjunction requires at least one. An implication is false only in the second row, where its antecedent is true but its consequent is false. A biconditional is true when both values agree, including the false–false case.
Read each connective according to its definition rather than everyday conversational emphasis. In particular, an implication with a false antecedent is true in this truth-functional system; it does not provide evidence that its consequent is true.
Add intermediate columns for a compound expression
Consider . The outermost connective is “and”, so evaluate the two pieces and first.
| p | q | |||
|---|---|---|---|---|
| T | T | T | F | F |
| T | F | T | F | F |
| F | T | T | T | T |
| F | F | F | T | F |
Only the false–true row makes the complete expression true. The expression is equivalent to , not to q alone. When p and q are both true, q is true but the complete expression is false.
Exhaust all assignments, including false premises
Diagram explanation and exact data
The four outputs, in TT, TF, FT, FF order, are false, false, true and false. The same outputs occur for not p and q.
| p | q | (p ∨ q) ∧ ¬p |
|---|---|---|
| True | True | False |
| True | False | False |
| False | True | True |
| False | False | False |
The table and the four-case diagram carry the same result. The diagram highlights the satisfying case, while the table shows how each intermediate value was obtained.
Test equivalence by comparing complete columns
Two formulas are logically equivalent when their final columns agree on every assignment. De Morgan’s law is a useful example:
| p | q | ||
|---|---|---|---|
| T | T | F | F |
| T | F | T | T |
| F | T | T | T |
| F | F | T | T |
Because every row agrees, the formulas are equivalent in classical propositional logic. This is a finite exhaustive proof of that truth-functional equivalence. It is not the same kind of claim as verifying a few numerical examples of a theorem with infinitely many cases.
The equivalence can be checked similarly. The converse does not have the same column as .
Classify a formula by its final column
A tautology is true on every assignment, such as . A contradiction is false on every assignment, such as . A contingent formula is true on some assignments and false on others; p itself is an example.
These descriptions refer to the formula’s logical structure. An ordinary arithmetic truth such as is true, but its truth is not established merely by replacing the whole sentence with a proposition letter. Truth tables treat such basic letters as inputs; arithmetic establishes their actual values.
Use a table to test an argument
An argument is valid when no assignment makes every premise true and its conclusion false. For the argument “if p then q; p; therefore q”, the two premises are and p. Whenever both are true, q must be true.
This is modus ponens. The implication
is a tautology. Validity concerns the connection between premises and conclusion. A valid argument with false premises does not thereby establish a true real-world conclusion; establishing the premises is a separate responsibility.
Find a counterexample to an invalid pattern
The argument “if p then q; q; therefore p” is invalid. Set p false and q true. Both premises are then true, but the conclusion is false. This pattern is called affirming the consequent.
For a numerical interpretation, let p mean “n is divisible by 4” and q mean “n is even”. The integer 6 makes the consequent true without making the antecedent true. The implication itself remains valid for integers.
Test the contrapositive argument
“If p then q; not q; therefore not p” is valid. A true p would force q, contradicting the second premise. A truth table confirms that no counterexample row exists. This is modus tollens.
Know what a truth table does not decide
A truth table does not decide whether an empirical premise is accurate, whether an undefined predicate has a value, or whether a quantified statement over infinitely many integers is true. It tests the connective structure once the proposition inputs are specified.
Tables also grow rapidly: ten proposition letters require rows. Algebraic logical equivalences can simplify larger expressions, but their transformations must preserve meaning. Parentheses are preferable to relying on a remembered precedence rule, especially when implications are nested.
Practice questions
- Evaluate when p is false and q is true.
- How many input rows are needed for four proposition letters?
- Is the argument , , therefore valid? Explain the forbidden counterexample.
Check your answers
- True. The disjunction is true and is true, so their conjunction is true.
- rows. Each of the four letters has two possible truth values.
- Yes. A counterexample would need p true and q false while stayed true, but that is exactly the assignment on which the implication is false.
Related topics
Implication and Equivalence connects these tables to mathematical language. Mathematical Proof explains how valid inference works with definitions and established facts rather than only proposition letters.