Question

Define predicates and prove the following using the appropriate rules of inference:

Anyone who lives in Hawaii lives close to the ocean. If you don’t know how to surf, you don’t live close to the ocean. Anyone who knows how to surf, owns a surfboard. Therefore, anyone who lives in Hawaii owns a surfboard.

Answer 1

Solution

A(x) = x lives in Hawaii
B(x) = x lives close to the ocean
C(x) = x knows how to surf
D(x) = x owns a surfboard

∀x(A(x) -> B(x))

¬∀x(C(x) -> ¬B(x))

∀x(C(x) -> D(x))

∀x(A(x) -> D(x))

Explanation

Anyone who lives in Hawaii lives close to the ocean

∀x(A(x) -> B(x))

If you don’t know how to surf, you don’t live close to the ocean

¬∀x(C(x) -> ¬B(x))

Anyone who knows how to surf, owns a surfboard

∀x(C(x) -> D(x))

Therefore, anyone who lives in Hawaii owns a surfboard.

∀x(A(x) -> D(x))

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