In propositional logic, tautological consequence is a strict form of logical consequence^{[1]} in which the tautologousness of a proposition is preserved from one line of a proof to the next. Not all logical consequences are tautological consequences. A proposition is said to be a tautological consequence of one or more other propositions (, , ..., ) in a proof with respect to some logical system if one is validly able to introduce the proposition onto a line of the proof within the rules of the system; and in all cases when each of (, , ..., ) are true, the proposition also is true.
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 .
a = "Socrates is a man." b = "All men are mortal." c = "Socrates is mortal."
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.
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.