Tautological Consequence
From Handwiki 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].
Example
a = "Socrates is a man." b = "All men are mortal." c = "Socrates is mortal."
- a
- b
- [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.
| 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
- ↑ Barwise and Etchemendy 1999, p. 110
- ↑ 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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Categories: [Logical consequence]
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