Tautological consequence

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 ($P_{1}$, $P_{2}$, ..., $P_{n}$) 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 ($P_{1}$, $P_{2}$, ..., $P_{n}$) 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 ($P_{1}$, $P_{2}$, ..., $P_{n}$) if and only if in every row of a joint truth table that assigns "T" to all propositions ($P_{1}$, $P_{2}$, ..., $P_{n}$) the truth table also assigns "T" to $Q$.

Example

a = "Socrates is a man." b = "All men are mortal." c = "Socrates is mortal."

a

b

${\therefore c}$

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 a ∧ b and c

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

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

• Logical consequence
• Tautology (logic)
• Truth table

Notes

1. Barwise and Etchemendy 1999, p. 110

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.