The semantics of predicate logic is defined (as in propositional logic) in terms of the truth values of sentences. Like in propositional logic, we can determine the truth value of any sentence in predicate calculus if we know the truth values of the basic components of that sentence. An interpretation function defines the basic meanings/truth values of the basic components, given some domain of objects that we are concerned with.
In propositional logic we saw that this interpretation function was very simple, just assigning truth values to propositions. However, in predicate calculus we have to deal with predicates, variables and quantifiers, so things get much more complex.
Predicates are dealt with in the following way. If we have, say, a predicate P with 2 arguments, then the meaning of that predicate is defined in terms of a mapping from all possible pairs of objects in the domain to a truth value. So, suppose we have a domain with just three objects in: fred, jim and joe. We can define the meaning of the predicate father in terms of all the pairs of objects for which the father relationship is true - say fred and jim.
The meaning of and
is defined again in terms
of the set of objects in the domain.
X S means that for every object X in the domain, S is true.
X S means that for some object X in the domain, S is true.
So,
X father(fred, X), given our world (domain) of 3
objects (fred, jim, joe), would only be true if father(fred, X)
was true for each object. In our interpretation of the father
relation this only holds for X=jim, so the whole quantified
expression will be false in this interpretation.
This only gives a flavour of how we can give a semantics to expressions in predicate logic. The details are best left to logicians. The important thing is that everything is very precisely defined, so if use predicate logic we should know exactly where we are and what inferences are valid.