Final version: Feb 17

Resources: HLW, Chap 5 “Truth”

A.

For each of the following sentences, please determine whether it is a tautology, a contingency, or a contradiction. Explain your reasoning in a way that is easily understandable, e.g. by pointing to a truth table or by giving an assignment of truth values to atomic sentences.

1. $((P\vee \neg P)\wedge (Q\vee R))\vee (\neg Q\wedge \neg R)$

2. $(P\to Q)\vee (Q\to R)$

3. $(P\vee Q)\wedge (\neg R\to S)$

4. $(P\wedge \neg Q)\to (R\vee (\neg P\wedge Q))$

B.

For each of the following arguments, please determine whether it is valid or invalid. Explain your reasoning in a way that is easily understandable, e.g. by pointing to a truth table or by giving an assignment of truth values to atomic sentences.

1. $P\vee Q,P\:\vdash \: \neg Q$

2. $\vdash (P\leftrightarrow Q)\vee (Q\leftrightarrow R)\vee (P\leftrightarrow R)$

3. $(P\to R)\vee (Q\to R)\:\vdash \: (P\vee Q)\to R$

C.

1. On pset1 you were asked to translate the following sentence.

   Dumbledore will be killed, and either McGonagall will become head of school and Hogwarts will flourish, or else it won’t flourish.

   Discuss (in a half page or less) whether this sentence implies that if Hogwarts doesn’t flourish, then McGonagall did not become head of school. (A good answer will discuss different ways that this sentence might be symbolized, and for each such symbolization, whether it implies that $\neg H\to\neg M$.)

2. Find a sentence that uses only the connectives ∧ and ¬ and that is logically equivalent to $P\to Q$. Explain (in no more than half a page) how you know that these sentences are equivalent.