Capstone: Full Predicate Analysis
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This capstone translates a prose argument with a relational premise and an only clause, proves it, then weakens one premise and refutes the result with a two-member model.
The passage: only staff members hold keys, and every key holder can open the vault, so anyone who can open the vault is a staff member. Translate. Only staff members hold keys reverses, giving for all x, Kx horseshoe Sx. Every key holder can open the vault: for all x, Kx horseshoe Ox. The conclusion: for all x, Ox horseshoe Sx. Attempt the proof. Line three: assume Oa for conditional proof, with a fresh. Line four: Ka horseshoe Oa, by universal instantiation on the second premise. Line five: Ka horseshoe Sa, by universal instantiation on the first. And now the proof stalls. Line four runs from Ka to Oa, and the assumption gives Oa, which is the consequent — affirming it licenses nothing. The argument as stated does not go through. Test that suspicion with a model. Universe of two: a and b. Let a be staff, hold a key, and open the vault. Let b open the vault, hold no key, and not be staff. First premise: every key holder is staff — a is the only key holder and a is staff, true. Second premise: every key holder opens the vault — true of a. Conclusion: everyone who opens the vault is staff — b opens it and is not staff, false. Both premises true, conclusion false. The argument is invalid, and the prose concealed it by running the conditional backwards.
A stalled proof is evidence, not a verdict. Failing to find a derivation may mean the argument is invalid or may mean the route was missed, and only a model settles which. Declaring invalidity because a proof would not come is the same error as declaring validity because a counterexample would not come.
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