Constructive dilemma
Transformation rules |
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Propositional calculus |
Rules of inference |
Rules of replacement |
Predicate logic |
Constructive dilemma^{[1]}^{[2]}^{[3]} is a valid rule of inference of 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 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 disjunctive version of modus ponens, whereas, destructive dilemma is the disjunctive version of modus tollens. The constructive dilemma rule can be stated:
where the rule is that whenever instances of "", "", and "" appear on lines of a proof, "" can be placed on a subsequent line.
Formal notation[edit]
The constructive dilemma rule may be written in sequent notation:
where is a metalogical symbol meaning that is a syntactic consequence of , , and in some logical system;
and expressed as a truth-functional tautology or theorem of propositional logic:
where , , and are propositions expressed in some formal system.
Proof ^{[4]}[edit]
Step | Proposition | Derivation |
---|---|---|
1 | Given | |
2 | Given | |
3 | Given | |
4 | Double negation (3) | |
5 | Material implication (4) | |
6 | Transposition (1) | |
7 | Hypothetical syllogism (6,5) | |
8 | Hypothetical syllogism (7,2) | |
9 | Material implication (8) | |
10 | Double negation (9) |
Natural language example[edit]
- 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[edit]
- ^ Hurley, Patrick. A Concise Introduction to Logic With Ilrn Printed Access Card. Wadsworth Pub Co, 2008. Page 361
- ^ Moore and Parker
- ^ Copi and Cohen
- ^ https://math.stackexchange.com/questions/3637543/prove-constructive-dilemma-without-using-additional-assumptions