Tautological consequence

Concept in propositional logic

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 {\displaystyle Q} is said to be a tautological consequence of one or more other propositions ( P 1 {\displaystyle P_{1}} ,  P 2 {\displaystyle P_{2}} , ...,  P n {\displaystyle 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 ( P 1 {\displaystyle P_{1}} ,  P 2 {\displaystyle P_{2}} , ...,  P n {\displaystyle P_{n}} ) are true, the proposition  Q {\displaystyle Q} also is true.

Another way to express this preservation of tautologousness is by using truth tables. A proposition  Q {\displaystyle Q} is said to be a tautological consequence of one or more other propositions ( P 1 {\displaystyle P_{1}} ,  P 2 {\displaystyle P_{2}} , ...,  P n {\displaystyle P_{n}} ) if and only if in every row of a joint truth table that assigns "T" to all propositions ( P 1 {\displaystyle P_{1}} ,  P 2 {\displaystyle P_{2}} , ...,  P n {\displaystyle P_{n}} ) the truth table also assigns "T" to  Q {\displaystyle Q} .

## Example

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

∴ c {\displaystyle {\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

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Tautological consequence can also be defined as  P 1 {\displaystyle P_{1}} ∧  P 2 {\displaystyle P_{2}} ∧ ... ∧  P n {\displaystyle P_{n}} →  Q {\displaystyle Q} 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.

## See also

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* Logical consequence
* Tautology (logic)
* Truth table

## Notes

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1. ^ Barwise and Etchemendy 1999, p. 110
2. ^ Robert L. Causey (2006). Logic, Sets, and Recursion. Jones & Bartlett Learning. pp. 51–52\. ISBN 978-0-7637-3784-9. OCLC 62093042.

## References

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* 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.