Tautological consequence
In
Another way to express this preservation of tautologousness is by using truth tables. A proposition is said to be a tautological consequence of one or more other propositions (, , ..., ) if and only if in every row of a joint truth table that assigns "T" to all propositions (, , ..., ) the truth table also assigns "T" to .
Example
a = "Socrates is a man." b = "All men are mortal." c = "Socrates is mortal."
- a
- b
The conclusion of this argument is a logical consequence of the premises because it is impossible for all the premises to be true while the conclusion false.
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
Tautological consequence can also be defined as ∧ ∧ ... ∧ → is a substitution instance of a tautology, with the same effect. ^{[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.
- ISBN 0-486-42533-9.