The discussion of the preceding section was intended to give an impression
of how we code up sentences as formulas of predicate logic. In this section,
we will be more precise about it, giving syntactic rules for the formation
of predicate logic formulas. Because of the power of predicate logic, the
language is much more complex than that of propositional logic.
The first thing to note is that there are two sorts of things involved in
a predicate logic formula. The first sort denotes the objects that we are talking about: individuals such as a and p (referring to Andy and Paul) are
examples, as are variables such as x and v. Function symbols also allow us
to refer to objects: thus, m(a) and g(x, y) are also objects. Expressions in
predicate logic which denote objects are called terms.
The other sort of things in predicate logic denotes truth values; expressions
of this kind are formulas: Y (x,m(x)) is a formula, though x and m(x)
are terms.
A predicate vocabulary consists of three sets: a set of predicate symbols
P, a set of function symbols F and a set of constant symbols C. Each predicate
symbol and each function symbol comes with an arity, the number of
arguments it expects. In fact, constants can be thought of as functions which
don’t take any arguments (and we even drop the argument brackets) – therefore,
constants live in the set F together with the ‘true’ functions which do
take arguments. From now on, we will drop the set C, since it is convenient to
do so, and stipulate that constants are 0-arity, so-called nullary, functions.
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