I'm learning FOL, and in the book I'm reading, I found an exercise containing a logical expression which should be evaluated to true/false, though I can't see how it would be possible.
The exercise contains these three logical sentences:
$$ a)\: \forall{x}\forall{y}\forall{z} \:[(P(x, y) \: \land \: P(y, z)) \: \supset \: P(x,z)] \\ b)\: \forall{x}\forall{y} \: [(P(x, y) \: \land \: P(y, x)) \: \supset \: (x = y)] \\ c)\: \forall{x}\forall{y} \: [P(a, y) \supset P(x, b)] $$
The exercise demands to find an interpretation for each sentence, where the sentence itself is false and the other two are true. I think I can do that, but I have a problem with sentence $c$: it contains free variables. How do I evaluate it to true/false, in order to find an interpretation where it is either true or false? It seems wrong to me to simply assume $\forall{a}\forall{b}$ anyway.
$a$ and $b$ are individual variables: in this case, we need a variable assignment function $\mu$ to give them a "meaning". The truth value of a formula is fixed for an interpretation and a variable assignment; changing the variable assignment may affect the truth value of the formula.
See: Ronald Brachman & Hector Levesque, Knowledge Representation and Reasoning (2004), page 21 and page 22: