How “if … then” works in logic

P → Q is a promise. It is broken in exactly one case: P is true and Q is false.

TCube(a)→Large(a)

a is a cube and it is large. The promise is kept: true.

1 of 4

How it works

Think of P → Q as a promise: “if P happens, Q happens too.” The promise is broken only when P happens and Q does not.

When P is false, the promise never comes due, so the sentence counts as true. That feels odd at first, and it is exactly what makes “every cube is small” work later on.

P	Q	P → Q
T	T	T
T	F	F
F	T	T
F	F	T

Worth memorizing

→ is false in exactly one case.
True on the left, false on the right.
A false left side makes the whole sentence true.
Logicians call this vacuously true.
→ does not run backwards.
Cube(a) → Large(a) says nothing about a large tetrahedron.

Practice

0 of 5 right

Question 1

Tet(a)→Small(a)

Question 2

Large(b)→Tet(b)

Question 3

Cube(c)→Small(c)

Question 4

Small(a)→LeftOf(a, c)

Question 5

FrontOf(c, b)→Dodec(c)

Next: Therefore: modus ponensYou know P → Q. You know P. ∴ Q. Every deduction in every puzzle is built from steps like this one.