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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 Q is said to be a tautological consequence of one or more other propositions (P1, P2, ..., Pn) 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 (P1, P2, ..., Pn) are true, the proposition Q also is true.

Another way to express this preservation of tautologousness is by using truth tables. A proposition Q is said to be a tautological consequence of one or more other propositions (P1, P2, ..., Pn) if and only if in every row of a joint truth table that assigns "T" to all propositions (P1, P2, ..., Pn) the truth table also assigns "T" to Q.

Example

Consider the following argument:

a = "Socrates is a man."

b = "All men are mortal."

c = "Socrates is mortal."

abc


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 is false.

Now construct a joint truth table.

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 conclusion also assigns T to every proposition in the premise. In particular, it is the second row that assigns T to "ab," 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.

See also

Notes

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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.
  1. Barwise and Etchemendy 1999, p. 110