Process of elimination
It’s P or Q. It isn’t P. ∴ Q. With more options, rule out all but one and the last is forced.
How it works
If P ∨ Q is true and P is false, the only way left for the ∨ to be true is Q. This is elimination.
It works for longer lists. If it’s one of five things and four are ruled out, it’s the fifth. A sudoku cell with one candidate left is exactly this. So is a Queens row with one open square.
Be careful going the other way. ∨ is inclusive, so P ∨ Q and P tell you nothing about Q. Both can be true.
Worth memorizing
From P ∨ Q and ¬P, conclude Q.
This is elimination.
It’s one of these, and all but one are ruled out, so it’s the last one.
P ∨ Q and P do not give ¬Q.
∨ is inclusive. Both sides can be true.
Practice
0 of 6 rightPremises: Small(a) ∨ LeftOf(a, b). ¬Small(a). Which conclusion follows?
Premises: Cube(a) ∨ Small(a). Cube(a). Does ¬Small(a) follow?
A sudoku cell can only hold 1, 4 or 7. Its row already has a 1 and its box already has a 7. What goes in the cell?