Tautological consequence

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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 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 .


Example[edit]

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.

Joint Truth Table for ab and c
a b c ab 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 ab, but does not assign T to c.

Denotation and properties[edit]

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[edit]

Notes[edit]

  1. ^ Barwise and Etchemendy 1999, p. 110

References[edit]

  • 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.