Logic / Unit 4 Logic 148 worked example, 60 to 90 seconds
Short Method Walkthrough
The idea
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.
In practice
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.
Where people slip
A contradiction has to be forced, not merely stumbled into. Choosing a value freely when the premises left it open, then finding a conflict, proves nothing about the argument — only that one arbitrary choice happened not to work.
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