Editing Constructive dilemma

{{Short description|Rule of inference of propositional logic}}
{{Infobox mathematical statement
| name = Constructive dilemma
| type = [[Rule of inference]]
| field = [[Propositional calculus]]
| statement = If <math>P</math> implies <math>Q</math> and <math>R</math> implies <math>S</math>, and either <math>P</math> or <math>R</math> is true, then either <math>Q</math> or <math>S</math> has to be true.
| symbolic statement = <math>\frac{(P \to Q), (R \to S), P \lor R}{\therefore Q \lor S}</math>
}}
{{Transformation rules}}

'''Constructive dilemma'''<ref>Hurley, Patrick. A Concise Introduction to Logic With Ilrn Printed Access Card. Wadsworth Pub Co, 2008. Page 361</ref><ref>Moore and Parker</ref><ref>Copi and Cohen</ref> is a [[Validity (logic)|valid]] [[rule of inference]] of [[propositional calculus|propositional logic]]. It is the [[inference]] that, if ''P'' implies ''Q'' and ''R'' implies ''S'' and either ''P'' or ''R'' is true, then either ''Q or S'' has to be true. In sum, if two [[material conditional|conditionals]] are true and at least one of their antecedents is, then at least one of their consequents must be too. ''Constructive dilemma'' is the [[Logical disjunction|disjunctive]] version of [[modus ponens]], whereas [[destructive dilemma]] is the disjunctive version of ''[[modus tollens]]''. The constructive dilemma rule can be stated:

:<math>\frac{(P \to Q), (R \to S), P \lor R}{\therefore Q \lor S}</math>

where the rule is that whenever instances of "<math>P \to Q</math>", "<math>R \to S</math>", and "<math>P \lor R</math>" appear on lines of a proof, "<math>Q \lor S</math>" can be placed on a subsequent line.

== Formal notation ==
The ''constructive dilemma'' rule may be written in [[sequent]] notation:

: <math>(P \to Q), (R \to S), (P \lor R) \vdash (Q \lor S)</math>

where <math>\vdash</math> is a [[metalogic]]al symbol meaning that <math>Q \lor S</math> is a [[logical consequence|syntactic consequence]] of <math>P \to Q</math>, <math>R \to S</math>, and <math>P \lor R</math> in some [[formal system|logical system]];

and expressed as a [[truth-functional]] [[tautology (logic)|tautology]] or [[theorem]] of propositional logic:

:<math>(((P \to Q) \land (R \to S)) \land (P \lor R)) \to (Q \lor S)</math>

where <math>P</math>, <math>Q</math>, <math>R</math> and <math>S</math> are propositions expressed in some [[formal system]].

== Natural language example ==

:If I win a million dollars, I will donate it to an orphanage.
:If my friend wins a million dollars, he will donate it to a wildlife fund.
:Either I win a million dollars or my friend wins a million dollars.
:Therefore, either an orphanage will get a million dollars, or a wildlife fund will get a million dollars.

The dilemma derives its name because of the transfer of disjunctive operator.

==References==
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[[Category:Rules of inference]]
[[Category:Dilemmas]]
[[Category:Theorems in propositional logic]]