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 the joint truth table that assigns T to all propositions p1, p2, ..., pn also assigns T to q. p1, p2, ..., pn are said to tautologically imply q. Tautological consequence is a type of logical consequence.[1] Not all logical consequences are tautological consequences.
Example
Consider the following argument:
a = "Socrates is a man."
b = "All men are mortal."
c = "Socrates is mortal."
a ∧ b
___________
∴ 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. Now construct a joint truth table.
| a | b | c | a ∧ b | 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," but does not assign T to c.
Denotation and properties
The double turnstile is used to denote that q is a tautological consequence of p as . q is a tautological consequence of p if and only if the material implication is a tautology.[2]
It follows from the definition that if a proposition p is a contradiction then p tautologically implies every proposition, because there is no truth valuation that causes p to be true and so the definition of tautological implication is trivially satisfied. Similarly, if p is a tautology then p is tautologically implied by every proposition.
See also
Notes
References
- Barwise, Jon, and John Etchemendy. Language, Proof and Logic. Stanford: CSLI (Center for the Study of Language and Information) Publications, 1999. Print.
- Kleene, S. C. (1967) Mathematical Logic, reprinted 2002, Dover Publications, ISBN 0-486-42533-9.