Short Method Walkthrough

The short method assumes the argument is invalid, then tries to build the assignment that would prove it. A forced contradiction means no such assignment exists and the argument is valid; a completed assignment is the counterexample.

First argument: p horseshoe q, and p, therefore q. Assume invalidity — set the conclusion q false and both premises true. The second premise makes p true. Now look at the first premise with p true and q false: a conditional with a true antecedent and a false consequent is false, but it was assumed true. The same letter is forced both ways, so the assumption cannot be carried out. No counterexample exists, and the argument is valid. Second argument: p horseshoe q, and q, therefore p. Assume invalidity again — conclusion p false, premises true. The second premise makes q true. Check the first premise with p false and q true: a conditional with a false antecedent is true, which is what was needed. Nothing is forced twice. The assignment completes, and it is the counterexample: p false, q true. The argument is invalid.