Unit 2: The Truth-Table Method
Table of Contents
Construction of Truth Tables for Statement Forms
A truth table is a complete list of all possible truth values for a statement form. It is a "decision procedure" because it can mechanically decide (in a finite number of steps) if a statement form is a tautology or if an argument is valid.
How to build a truth table:
- Determine the number of rows: The number of rows is 2n, where 'n' is the number of distinct simple variables.
- 1 variable (p): 21 = 2 rows
- 2 variables (p, q): 22 = 4 rows
- 3 variables (p, q, r): 23 = 8 rows
- Create "guide columns" for the simple variables.
- For 2 variables: The 'q' column alternates T, F. The 'p' column alternates T, T, F, F.
- For 3 variables: 'r' alternates T, F. 'q' alternates T, T, F, F. 'p' alternates T, T, T, T, F, F, F, F.
- Calculate the truth values for each connective, working from the "inside out" (connectives inside parentheses first) to the "main connective."
Example: Construction for (p ⊃ q) • p
| p | q | (p ⊃ q) | (p ⊃ q) • p |
|---|---|---|---|
| T | T | T | T |
| T | F | F | F |
| F | T | T | F |
| F | F | T | F |
Classifying Statement Forms
By looking at the main connective's column in a completed truth table, we can classify any statement form into one of three categories:
- Tautology (or Tautologous): A statement form that is always true, regardless of the truth values of its components. The main column is all T's.
- Example: p v ~p (Law of Excluded Middle)
- Contradiction (or Contradictory): A statement form that is always false. The main column is all F's.
- Example: p • ~p (Law of Non-Contradiction)
- Contingent (or Contingency): A statement form that is sometimes true and sometimes false. The main column has at least one T and at least one F.
- Example: p ⊃ q
Decision Procedure: Testing for Validity/Invalidity
A valid deductive argument is one where it is impossible for the premises to be true and the conclusion to be false at the same time.
The truth-table method tests for validity by checking every possible scenario (every row).
The Test:
- Symbolize the argument.
- Create a single truth table that includes columns for all premises and the conclusion.
- Examine every row of the completed table.
- Search for a "counterexample" row: a row where all premises are TRUE (T) and the conclusion is FALSE (F).
If you find even ONE such row (T, T... / ∴ F), the argument is INVALID.
If no such row exists, the argument is VALID.
Example 1: A Valid Argument (Modus Ponens)
Argument: p ⊃ q, p / ∴ q
| p | q | Premise 1 (p ⊃ q) | Premise 2 (p) | Conclusion (q) | Notes |
|---|---|---|---|---|---|
| T | T | T | T | T | Premises T, Concl T. (OK) |
| T | F | F | T | F | |
| F | T | T | F | T | |
| F | F | T | F | F |
Result: We looked at all 4 rows. There is no row where the premises (P1, P2) are both T and the conclusion (C) is F. Therefore, the argument is VALID.
Example 2: An Invalid Argument (Fallacy of Affirming the Consequent)
Argument: p ⊃ q, q / ∴ p
| p | q | Premise 1 (p ⊃ q) | Premise 2 (q) | Conclusion (p) | Notes |
|---|---|---|---|---|---|
| T | T | T | T | T | Premises T, Concl T. (OK) |
| T | F | F | T | T | |
| F | T | T | T | F | INVALIDATING ROW! |
| F | F | T | F | F |
Result: In row 3, the premises (P1, P2) are both TRUE, but the conclusion (C) is FALSE. This one counterexample is enough to prove the argument is INVALID.