Logic / Unit 6 Logic 281 worked example, 60 to 90 seconds
Proving Invalidity
The idea
Proving a predicate argument invalid means building a finite universe and assigning the predicates within it so that every premise comes out true and the conclusion comes out false.
In practice
The argument left open by the previous video: for all x, there exists a y such that Lxy, therefore there exists a y such that for all x, Lxy. Take a universe of two members, a and b. Assign the relation: a loves b, b loves a, and nothing else. Now expand the premise. For each member, does someone exist that it loves? For a, yes — b. For b, yes — a. The premise is true in this universe. Expand the conclusion. It requires one single member loved by everyone. Is a loved by everyone? b loves a, but a does not love a, so no. Is b loved by everyone? a loves b, but b does not love b, so no. The conclusion is false. Premise true, conclusion false, in a universe of two. The argument is invalid, and the model certifies it. Everyone loves someone; no one is loved by all.
Where people slip
A model refutes an argument only when every premise is genuinely true in it. Checking the conclusion first and stopping there is the common shortcut, and it produces confident invalidity verdicts for arguments whose premises the model quietly falsified.
Your place on the trail
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