Question

Does the above mean that a complex dilemma has the following form: P | Q, (P->R) & (Q->S) |- R | S ? When trying to justify the form, I got some contradictory results in two ways. Before moving to the justification, let me introducethe following two inference rules. Are they both correct? To proveP, Q1 & Q2 |- S, i.e. with conjunctive in antecedent, we

can eitherP, Q1 |- S
--------------------------
P, Q1 & Q2 |- SorP, Q2 |- S
--------------------------
P, Q1 & Q2 |- STo proveP, Q1 | Q2 |- S, i.e. with disjunctive in antecedent, we
canP, Q1 |- S,
P, Q2 |- S
--------------------------
P, Q1 | Q2 |- S Now to proveP | Q, (P->R) & (Q->S) |- R | S, the form of a complex dilemma, let me try to approach it in two ways. Consider the first way. According to the first inference rule above,
it suffices to show that eitherP | Q, P->R |- R | SorP | Q, Q->S |- R | STo showP | Q, P->R |- R | S, according to the second inference
rule above, it suffices to showP, P->R |- R | SandQ, P->R |- R | Sthe second of which doesn't hold, neither doesP | Q, P->R |- R | S.Similarly, we can't showP | Q, Q->S |- R | S. Thus, we can't
showP | Q, (P->R) & (Q->S) |- R | Sin this way.Consider the second way to proveP | Q, (P->R) & (Q->S) |- R | S. According to the second inference rule above, it suffices to
show that eitherP, (P->R) & (Q->S) |- R | SandQ, (P->R) & (Q->S) |- R | STo showP, (P->R) & (Q->S) |- R | S, according to the first
inference rule above, it suffices to showP, P->R |- R | SorP, Q->S |- R | Sthe first of which holds, so doesP, (P->R) & (Q->S) |- R | S.Similarly, we can showQ, (P->R) & (Q->S) |- R | S. Thus,P | Q, (P->R) & (Q->S) |- R | Sholds. Thanks. You have the correct representation of the form of the argument. It is also sometimes called constructive dilemma. Your first inference rule states a sufficient condition for proving S, but not a necessary one. In order to prove P, Q1 & Q2 ⊢ S we may need both Q1 and Q2. If we can prove S from either one alone, well and good, but in general we may need both together. This is why your attempt to derive a proof under 1 fails. When asking for a justification of a rule in logic, it makes sense to ask what you would count as a proof. You could just construct a truth table. You could write a proof using some natural deduction rules, but you might object that these are just as much in need of justification. Here is a proof, writing ∨ for disjunction, ∧ for conjunction, and → for the material conditional. 1. P ∨ Q Premise
2. P → R Premise
3. Q → S Premise
4. | ¬(R ∨ S) Assumption, for the purpose of reductio
5. | ¬R ∧ ¬S From 4, by de Morgan's rules
6. | ¬R From 5, ∧-elimination
7. | ¬P From 2, 6, modus tollens
8. | ¬S From 5, ∧-elimination
9. | ¬Q From 3, 8, modus tollens
10. | Q From 1, 7, disjunctive syllogism
11. | Q ∧ ¬Q From 9, 10, ∧-introduction
12. ¬¬(R ∨ S) From 4, 11, by reductio
13. R ∨ S From 12, double negation elimination

Answer 1

The complex dilemma represented by P | Q, (P->R) & (Q->S) |- R | S is justified, denoted as a constructive dilemma.

Explanation:

The argument demonstrates the form of a constructive dilemma, denoting that P or Q holds, alongside their implications (P->R) and (Q->S), thereby leading to the conclusions R or S. The attempted proofs highlighted two inference rules to establish implications involving conjunctive and disjunctive antecedents.

However, the first approach failed due to the insufficiency of demonstrating the implications separately. The second approach succeeded by showcasing that P and (P->R) together lead to R, and similarly for Q and (Q->S), validating the overall complex dilemma. The subsequent provided proof employs a reductio ad absurdum strategy, assuming the negation of the conclusion and deriving a contradiction to show that the initial assumption must be false, thereby confirming the validity of the argument.

The proof's logical steps systematically demonstrate how assuming the negation of R or S leads to contradictions, affirming the validity of the constructive dilemma.

This explanation elucidates the reasoning behind the final answer, emphasizing the failure and success of the approaches used in demonstrating the inference rules and the subsequent reductio ad absurdum proof to validate the complex dilemma. ""