Doesn't it contradict the whole point of having a dilemma? Even if the argument is sound it allows us to take both options, instead of just one. I don't think you're supposed to be able to do this, when faced with a dilemma.
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1Even if you replace OR with XOR in the premise the conclusion cannot be strengthened to XOR (propositions on the other side of implications may overlap). So logicians opted for a rule with a basic connective and a weaker premise, from which the dilemma proper trivially follows.– ConifoldCommented Sep 9, 2021 at 4:57
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@Conifold "replace OR with XOR in the antecedent" What are you talking about? There is no disjunction in conditionals of Constructive dilemma. "propositions on the other side of implications may overlap" I have no idea what it means. "basic connective" What is it?– KarmaPeasantCommented Sep 9, 2021 at 5:04
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@Conifold If you meant that we can't have XOR in the conclusion, I'm aware about it. But I know that XOR in the disjunctive premise leads to valid argument form, I checked it with truth table.– KarmaPeasantCommented Sep 9, 2021 at 5:06
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The dilemma is P → Q, R → S, P∨R ⊢ Q∨S. You can replace P∨R by P⊕R, but what's the point? It makes the premise stronger than it needs to be, uses a connective ⊕ that many logical systems do not include as basic, and does nothing for the conclusion (because Q and S may overlap even if P and R do not).– ConifoldCommented Sep 9, 2021 at 5:15
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@Conifold "It makes the premise stronger than it needs to be" What does it mean? " but what's the point?" It restricts ways in which CD can be true. It also can be seen as more precise translation of particular argument in logical form. In this form it's really a dilemma. And finally, we can use (P&~R)v(~P&R) if we can't use operator ⊕– KarmaPeasantCommented Sep 9, 2021 at 5:28
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1 Answer
Let us discuss the question making some terminological remarks which may be useful also in broader contexts:
Constructive dilemma: P → Q, R → S, P ∨ R ⊢ Q ∨ S
The argument transfers the dilemma to the conclusion. It is constructive, for it posits the antecedents.
Destructive dilemma: P → Q, R → S, ¬Q ∨ ¬S ⊢ ¬P ∨ ¬R
The argument transfers the dilemma to the conclusion. It is destructive, for it sublates the consequents.
Notice that neither argument attempts to resolve the dilemma, but translates it into a new form.
We can carry on with disjunctive syllogism to resolve the dilemma. There are two interpretations of disjunction:
Exclusive (proper) disjunction (disjuncts are mutually exclusive; signified by 'either . . . or . . .' clause).
Inclusive (improper) disjunction (disjunctions are not mutually exclusive; signified by '. . . or . . .' without 'either').
Let us represent inclusive disjunction with ∨ as usual, and exclusive disjunction with ~. Then, we have
(1) Q ∨ S, ¬S ⊢ Q which is valid for both interpretations.
(2) Q ~ S, Q ⊢ ¬S which is valid only for the exclusive interpretation.
We can dispel one of the horns of a dilemma either by negating one or by affirming the other. In the first case, we get a valid inference for both interpretations. In the second case, we have already posited one, say Q, and resolved the dilemma, regardless of the the other horn whatsoever and whether the argument (2) is valid or not.
Therefore, from the point of the idea of dilemma, its resolution does not distinguish the interpretations.
answered Sep 12, 2021 at 13:45
