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 [math]\displaystyle{ Q }[/math] is said to be a tautological consequence of one or more other propositions ([math]\displaystyle{ P_1 }[/math], [math]\displaystyle{ P_2 }[/math], ..., [math]\displaystyle{ P_n }[/math]) 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 ([math]\displaystyle{ P_1 }[/math], [math]\displaystyle{ P_2 }[/math], ..., [math]\displaystyle{ P_n }[/math]) are true, the proposition [math]\displaystyle{ Q }[/math] also is true. Another way to express this preservation of tautologousness is by using truth tables. A proposition [math]\displaystyle{ Q }[/math] is said to be a tautological consequence of one or more other propositions ([math]\displaystyle{ P_1 }[/math], [math]\displaystyle{ P_2 }[/math], ..., [math]\displaystyle{ P_n }[/math]) if and only if in every row of a joint truth table that assigns "T" to all propositions ([math]\displaystyle{ P_1 }[/math], [math]\displaystyle{ P_2 }[/math], ..., [math]\displaystyle{ P_n }[/math]) the truth table also assigns "T" to [math]\displaystyle{ Q }[/math].

## Contents

* 1 Example
* 2 Denotation and properties
* 3 See also
* 4 Notes
* 5 References

## Example

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

[math]\displaystyle{ {\therefore c} }[/math]

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

Tautological consequence can also be defined as [math]\displaystyle{ P_1 }[/math] ∧ [math]\displaystyle{ P_2 }[/math] ∧ ... ∧ [math]\displaystyle{ P_n }[/math] → [math]\displaystyle{ Q }[/math] 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

* Logical consequence
* Tautology (logic)
* Truth table

## Notes

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. https://books.google.com/books?id=NlgwptagGoEC.

## 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.

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