Seminar VIII

The following exercises are concerned with the predicate calculus and interpretations.

I. For each formula of the predicate calculus given below, specify (i) an interpretation which makes the formula true; and (ii) an interpretation which makes the formula false.

1. $\forall x. (A(x) \rightarrow B(x))$
2. $\neg \exists x. (C(x) \wedge D(x))$
3. $\forall x. \forall y. ((R(x,y) \rightarrow R(y,x))$

II. What can you observe about the following formulas? (HINT: does it matter what interpretation you choose to evaluate them in?)

1. $\forall x. (A(x) \vee \neg A(x))$
2. $\exists x. (B(x) \wedge \neg B(x))$