In propositional logic, tautological consequence is a strict form of logical consequence 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 those one or more other propositions (, , ..., ) 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|
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
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.
- Barwise and Etchemendy 1999, p. 110