Predicate Logic: Syntax

The trouble with propositional logic is that it is not possible to write general statements in it, such as ``Alison eats everything that she likes''. We'd have to have lots of rules, for every different thing that Alison liked. Predicate logic makes such general statements possible.

Sentences in predicate calculus are built up from atomic sentences (not to be confused with Prolog atoms). Atomic sentences consist of a predicate name followed by a number of arguments. These arguments may be any term. Terms may be:

Constant symbols

such as ``alison''.

Variable symbols

such as ``X''. For consistency with Prolog we'll use capital letters to denote variables.

Function expressions

such as ``father(alison)''. Function expressions consist of a functor followed by a number of arguments, which can be arbitrary terms.

This should all seem familiar from our description of Prolog syntax. However, although Prolog is based on predicate logic the way we represent things is slightly different, so the two should not be confused.

So, atomic sentences in predicate logic include the following:

• friends(alison, richard)
• friends(father(fred), father(joe))
• likes(X, richard)

Sentences in predicate logic are constructed (much as in propositional logic) by combining atomic sentences with logical connectives, so the following are all sentences in predicate calculus:

• friends(alison, richard) likes(alison, richard)
• likes(alison, richard) likes(alison, waffles)
• ((likes(alison, richard) likes(alison, waffles)) likes(alison, waffles)) likes(alison, richard)

Sentences can also be formed using quantifiers to indicate how any variables in the sentence are to be treated. The two quantifiers in predicate logic are and , so the following are valid sentences:

• bird(X) flies(X)
  ie, there exists some bird that doesn't fly.
• X (person(X) Y loves(X,Y))
  ie, every person has something that they love.

A sentence should have all its variables quantified. So strictly, an expression like `` X loves(X, Y)'', though a well formed formula of predicate logic, is not a sentence. Formulae with all their variables quantified are also called closed formulae.