How many of these interpretations are models of KB? Give a brief justification for your answer.
There are the same number of possible phi's (the KB doesn't constrain this).
For each of the phi's, there are two values for the pi(q). (Depending on whether q of the other individual (whichever of x and y isn't denoted by a) is true or false).
For the case where q is true of both individuals, p must also be true of both individuals. There are thus 8 interpretations where q is true of both individuals.
For the case where q is true of just one individual (it must be the individual denoted by a), p must be true of that individual and can either be true or false of the other individual. There are thus 16 interpretations where q is true of one individual.
Thus there are 8+16=24 models of KB.