Negating 'it is cold and foggy' is not 'it is not cold and not foggy.' Getting the opposite of a sentence exactly right is harder than it looks, and it is where De Morgan's laws, truth tables, and the strange little word 'unless' earn their keep.
Every compound sentence is built from five machines. Flip P and Q and watch all five react.
| connective | sentence | value |
|---|---|---|
| negation | ¬P | · |
| conjunction | P ∧ Q | · |
| disjunction | P ∨ Q | · |
| conditional | P → Q | · |
| biconditional | P ↔ Q | · |
Read P → Q as a promise: 'if P happens, Q happens.' The only way to break the promise is for P to happen without Q. If P never happens, the promise was never tested, so it was never broken, so it stands as true.
Are these two sentences consistent, can they both be true at once?
1. My roommate is neither tired nor has an assignment due, but she wants coffee.
2. My roommate wants coffee only if she has an assignment due or is tired.
The first two columns are done for you. Click the empty cells to cycle T / F, then check your table.
Work inside-out. First evaluate P ∨ Q (done for you), then negate it for ¬(P ∨ Q), then AND with R. For the conditional, remember: false only on rows where R is T and P ∨ Q is F.
Three lamps claim to be the negation of A ∧ B. Only one deserves the title: the true negation must light up in exactly the worlds where A ∧ B is dark. Flip the switches and find the impostor.
Now negate in plain language. One option in each set is exactly right.
¬(A ∧ B) ≡ ¬A ∨ ¬B · ¬(A ∨ B) ≡ ¬A ∧ ¬B. Negation flips ∧ into ∨ and back, pushing the ¬ inward onto each part.
For conditionals, first rewrite: P → Q ≡ ¬P ∨ Q. Then negate with De Morgan: ¬(P → Q) ≡ P ∧ ¬Q. The negation of 'if P then Q' is NOT another if-then. It is 'P happened and Q didn't.'
Two readings of the same idea, one with if, one with unless. Are they the same promise?
1. If foes cannot hurt you, then yours is the world and everything that's in it.
2. Unless foes can hurt you, yours is the world and everything that's in it.
| F | Y | ¬F → Y | F ∨ Y |
|---|
'Unless F, Y' means 'if not F, then Y', which is ¬F → Y, which is logically equivalent to F ∨ Y. When translating, the cheapest safe move is: replace 'unless' with 'or'.