Two tempting mistakes

Two arguments look like modus ponens and modus tollens, and neither one is valid. One board shows why.

TCube(a)→Small(a)

Premise: if a is a cube, a is small. True here, because a is not a cube.

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How it works

An argument is invalid if there is even one world where every premise is true and the conclusion is false. That world is a counterexample to the argument.

Affirming the consequent goes P → Q, Q, ∴ P. Denying the antecedent goes P → Q, ¬P, ∴ ¬Q. Both run the promise backwards. P → Q says what happens when P is true. It says nothing about why Q might be true.

A single small tetrahedron breaks both.

Worth memorizing

P → Q and Q do not give P.
This is affirming the consequent.
P → Q and ¬P do not give ¬Q.
This is denying the antecedent.
To show an argument is invalid, build one counterexample world.
Every premise true, conclusion false.

Practice

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Question 1

Premises: Tet(a) → Large(a). Large(a). What follows about a’s shape?

Question 2

Premises: Dodec(b) → LeftOf(b, c). ¬Dodec(b). Does ¬LeftOf(b, c) follow?

Question 3

Premises: Small(c) → Cube(c). ¬Cube(c). ∴ ¬Small(c). What is this argument?

Question 4

Which board is a counterexample to “Large(a) → Cube(a). Cube(a). ∴ Large(a).”?

Question 5

((Cube(a)→Small(a))∧Small(a))→Cube(a)

Question 6

((Cube(a)→Small(a))∧Small(a))→Cube(a)

Where you’ll use this

WorldsBuild the board that makes every sentence true.How to solve a WorldTap a false sentence to find the piece that breaks it.For everyNext level: sentences about every piece at once

Next: For every∀x says something about every piece on the board. One counterexample makes it false.