Logic & truth-table drills

What the exam actually asks

Two logic tasks recur on every paper. Q1 hands you a definition such as "$P\leftrightarrow Q$ means $(P\to Q)\wedge (Q\to P)$" and asks you to prove the equivalence with a truth table. Q8(a) hands you English sentences and asks you to translate them into a knowledge base of implications. Both are mechanical once you have the drill in muscle memory — so we rebuild them cold here.

The truth-table method (the only method you need)

1. List every combination of the variables. Two variables ⇒ $2^2=4$ rows ($n$ variables ⇒ $2^n$ rows).
2. Add one column per sub-expression, evaluating inside-out.
3. Two formulas are logically equivalent iff their final columns are identical on every row.

The one fact that makes implication painless: $P\to Q$ is false on exactly one row — when $P=T$ and $Q=F$ — and true on the other three. Equivalently $P\to Q\equiv \neg P\vee Q$.

Worked example — the past-paper pair $(\neg P\to \neg Q)\equiv (P\vee \neg Q)$

This exact pair has appeared as a cold-retrieval question. Build both columns:

For $\neg P\to \neg Q$: it is false only when its antecedent $\neg P=T$ (i.e. $P=F$) and its consequent $\neg Q=F$ (i.e. $Q=T$) — that is the single row $P=F,Q=T$. Every other row is true. The right column $P\vee \neg Q$ is likewise false only at $P=F,Q=T$. The columns match on all four rows ⇒ EQUIVALENT. (Both reduce to $\neg (\neg P\wedge Q)$.)

The Q8 encoding step (logic half)

Forward chaining (L10.3) needs the sentences as implications. Translate each English statement into premise → conclusion:

• "When the cat is absent, the mice are present" ⇒ $cat\_absent\to mouse\_present$
• "When the dog is present, the cat is absent" ⇒ $dog\_present\to cat\_absent$
• "When the dog is absent and the cat is absent, the mice are present" ⇒ $dog\_absent\wedge cat\_absent\to mouse\_present$

Given facts $\{dog\_absent,cat\_not\_sleeping\}$. You fire these in L10.3 — here, just get the encoding exact.