First Order Predicate Calculus

The predicate calculus includes a wider range of entities. It permits the description of relations and the use of variables. It also requires an understanding of quantification.

The language of predicate calculus requires:

Variables

Constants

---these include the logical constants

The last two logical constants are additions to the logical connectives of propositional calculus ---they are known as quantifiers. The non-logical constants include both the `names' of entities that are related and the `names' of the relations. For example, the constant dog might be a relation and the constant fido an entity.

Predicate

---these relate a number of entities. This number is usually greater than one. A predicate with one argument is often used to express a property e.g. sun(hot) may represent the statement that ``the sun has the property of being hot''.

If there are no arguments then we can regard the `predicate' as standing for a statement à la the propositional calculus.

Formulæ

---these are constructed from predicates and formulæ .   The logical constants are used to create new formulæ/ from old ones. Here are some examples:

Note that the word ``and'' used in the left hand column is used to suggest that we have more than one formula for combination ---and not necessarily a conjunction.

In the last two examples, ``dog(X)'' contains a variable which is said to be free while the ``X'' in ``X.dog(X)'' is bound.

Sentence

---a formula with no free variables is a sentence.

Two informal examples to illustrate quantification follow:

The former is an example of universal quantification and the latter of existential quantification.

We can now represent the problem we initially raised:
X.(dog(X)smelly(X))dog(fido)smelly(fido)

To verify that this is correct requires that we have some additional machinery which we will not discuss here.