Truth tables, negations and islanders
Truth tables, negating statements with the quantifiers “for all” and “there exists”, and knights and knaves puzzles.
How to solve it
Logic checks reasoning the way arithmetic checks numbers. A formula in $n$ variables has $2^n$ assignments, and it can be tested on all of them. A negation flips the quantifiers: “for all” becomes “there exists” and vice versa, and the inner part is negated.
Step by step
- A truth table: write out all $2^n$ assignments and evaluate the formula on each, from the inner brackets outwards.
- The implication $A \to B$ is false in only one case: $A$ true, $B$ false.
- Negating with quantifiers: replace every $\forall$ with $\exists$ and back, and negate the final condition: $<$ becomes $\ge$.
- Knights and knaves: turn each statement into the condition “if the speaker is a knight, the statement is true, otherwise false” and try the cases.
Common mistakes
- Calling an implication with a false premise false. With $A$ false, $A \to B$ is true.
- Negating only the quantifiers and keeping the condition, or the other way round.
- Negating “$< \varepsilon$” as “$> \varepsilon$” — it is “$\ge \varepsilon$”.
- Checking only one case in an islander puzzle. Make sure the fitting case is the only one.