Tautological consequence
In logic, a proposition q is a tautological consequence of a set of propositions p1, p2, ..., pn if and only if every row of a truth table that assigns T to all of the propositions p1, p2, ..., pn also assigns T to q. Tautological consequence is a type of logical consequence. Not all logical consequences are tautological consequences.
For example, consider the following argument:
a = b ∧ b = c
___________
∴ a = c
The conclusion of this argument is a logical consequence of the premise because it is impossible for the premise to be true while the conclusion false.[1]
Now construct a joint truth table.
| a = b | b = c | a = c | a = b ∧ b = c | a = c |
|---|---|---|---|---|
| T | T | T | T | T |
| T | T | F | T | F |
| T | F | T | F | T |
| T | F | F | F | F |
| F | T | T | F | T |
| F | T | F | F | F |
| F | F | T | F | T |
| F | F | F | F | F |
Reviewing the truth table, it turns out the conclusion of the argument is not a tautological consequence of the premise. Not every row that assigns T to the premise also assigns T to the conclusion. In particular, it is the second row that assigns T to "a = b ∧ b = c," but does not assign T to "a = c."
Notes
- ^ Barwise and Etchemendy 1999, p. 111
References
- Barwise, Jon, and John Etchemendy. Language, Proof and Logic. Stanford: CSLI (Center for the Study of Language and Information) Publications, 1999. Print.