Downsampling

2014-01-30T21:59:54Z

157.89.212.93: Changed use of the word "unity" to use of the word "one". They mean the same thing here and "one" is less confusing to someone not familiar with the jargon of the field.

{{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 name of a [[validity|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 ''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 rule can be stated:

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

where the rule is that whenever instances of "<math>P \to Q</math>", "<math>R \to S</math>", and "<math>P \or R</math>" appear on lines of a proof, "<math>Q \or 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 \or R) \vdash (Q \or S)</math>

where <math>\vdash</math> is a [[metalogic]]al symbol meaning that <math>Q \or S</math> is a [[logical consequence|syntactic consequence]] of <math>P \to Q</math>, <math>R \to S</math>, and <math>Q \or S</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) \and (R \to S)) \and (P \or R)) \to (Q \or S)</math>

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

== Variable English ==

If P then Q.
If R then S.
P or R.
Therefore, Q or S.

== 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 operants.

==References==
{{Reflist}}

{{DEFAULTSORT:Constructive Dilemma}}
[[Category:Rules of inference]]
[[Category:Dilemmas]]
[[Category:Theorems in propositional logic]]

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