Logic / Unit 6 Logic 271 worked example, 60 to 90 seconds
First Predicate Proof
The idea
The first predicate proof strips a quantifier to reach a statement about a named individual, then finishes with the propositional rules already in hand.
In practice
The premises: for all x, Hx horseshoe Mx. And Hs, where s names Socrates. The goal is Ms. Line three: Hs horseshoe Ms, from line one by universal instantiation. A universal claims something of everything, so it may be instantiated to any name, including one already in the proof. Line four: Ms, from lines three and two by modus ponens — the same rule from Unit five, unchanged. Two moves. This is the argument that Unit four could not touch: propositional logic saw three unrelated simple statements and had to call the inference invalid, because the connection lived inside the sentences rather than between them. Quantifiers reach inside, and the propositional machinery then does its work exactly as before.
Where people slip
Universal instantiation is free with names, and existential instantiation is not. Treating them alike is the error the next video is built around, and it is easy to fall into here precisely because this proof gives no reason to be careful.
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