Predicate logic

We begin with the extension of propositional logic to predicate logic and then to modal predicate logic. We look first, from a logical perspective, at what happens when we translate from ordinary language to predicate logic without modal operators. We focus on a very small fragment of both ordinary language and predicate logic: a fragment large enough to show the impact of possible worlds semantics on a range of topics and problems. We then turn to what happens when possible worlds semantics is added to predicate logic.

The first thing about translating from ordinary language to predicate logic is that there is far more to consider than when translating to propositional logic. Predicate logic forces us to understand what is said in ordinary language in a way that does not always accord with our simplistic intuitions. Our intuitions have to be informed and changed. We have to look for features, expressions and underlying meanings in natural language that match the structures and meanings in predicate logic. Here we see clearly that ordinary language and predicate logic are quite different languages. Fortunately, we do not have to begin with complex sentences to see what is important.

Basic logical features

Consider the simple sentences:

Kermit is a frog.

Kermit is green.

These sentences contain a singular term and two general terms. The singular term is “Kermit”. It is a naming term that we take to refer to just one entity.