Seminar III

The following exercises are concerned with the notions of consistency and inconsistency amongst sets of sentences, and the relation of semantic entailment.

I. Work out which of the following sets of sentences are semantically consistent, and which are semantically inconsistent.

1. $\{p \rightarrow q, \neg q \}$
2. $\{p \rightarrow q, \neg q \vee r, p \wedge \neg r \}$
3. $\{(p \vee q) \rightarrow r, \neg ( ( \neg p \wedge \neg q) \vee r)\}$

II. Work out which of the following semantic entailments are correct and which are not:

1. $\{p \rightarrow q\} \models (\neg q \rightarrow \neg p)$
2. $\{p \rightarrow q\} \models q$
3. $\{(p \vee q) \rightarrow r, \neg r \} \models \neg p$
4. $\{\neg p, p \wedge q \} \models r \vee q$

III. Is the following statement about the entailment relation correct or not?

For any set of sentences $G$, and sentence $A$,

$$G \models A \mbox{ if and only if } G \cup \{\neg A \} \mbox{ is inconsistent}$$