{{Short description|Concept in propositional logic}} In propositional logic, '''tautological consequence''' is a strict form of logical consequence<ref>Barwise and Etchemendy 1999, p. 110</ref> 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>Q</math> is said to be a tautological consequence of one or more other propositions (<math>P_1</math>, <math>P_2</math>, ..., <math>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>P_1</math>, <math>P_2</math>, ..., <math>P_n</math>) are true, the proposition <math>Q</math> also is true.
Another way to express this preservation of tautologousness is by using truth tables. A proposition <math>Q</math> is said to be a tautological consequence of one or more other propositions (<math>P_1</math>, <math>P_2</math>, ..., <math>P_n</math>) if and only if in every row of a joint truth table that assigns "T" to all propositions (<math>P_1</math>, <math>P_2</math>, ..., <math>P_n</math>) the truth table also assigns "T" to <math>Q</math>.
==Example== {{mvar|a}} = "Socrates is a man." {{mvar|b}} = "All men are mortal." {{mvar|c}} = "Socrates is mortal."
:{{mvar|a}} :{{mvar|b}} :<math>{\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.
{| class="wikitable" style="margin:1em auto; text-align:center;" |+Joint Truth Table for ''a'' ∧ ''b'' and ''c'' ! style="width:35px; background:#aaa;"| ''a'' ! style="width:35px; background:#aaa;"| ''b'' ! style="width:35px; background:#aaa;"| ''c'' ! style="width:80px; | ''a'' ∧ ''b'' ! style="width:35px" | ''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>P_1</math> ∧ <math>P_2</math> ∧ ... ∧ <math>P_n</math> → <math>Q</math> is a substitution instance of a tautology, with the same effect.<ref name="Causey2006">{{cite book | author = Robert L. Causey | date = 2006 | title = Logic, Sets, and Recursion | publisher = Jones & Bartlett Learning | pages = 51–52 | isbn = 978-0-7637-3784-9 | oclc = 62093042 | url = https://books.google.com/books?id=NlgwptagGoEC}}</ref>
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== {{Reflist}}
==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}}.
Category:Logical consequence