4: Predicate Calculus: Language

Predicates

So far the most basic constituents into which we have analysed sentences have been sentences. Now we shall go further.

We can analyse the sentence

            Buttercup is a cow

into the predicate

            x is a cow

and the designator

            Buttercup.

A designator is a word or phrase which can be used to refer to a single thing, where "thing" is taken very broadly.

So the following are designators: proper names, such as "Buttercup", "John", "London"; non-count nouns, such as "water", "beauty", "2"; singular personal pronouns, such a "I", "she", "her", "him", "it"; definite descriptions, such as "the Dean of Christ Church", "my mother", "Henry's car", "4²".

A predicate is a matrix with spaces marked by individual variables, which yields a declarative sentence when the variables are replaced by designators.

Lower case letters from the end of the alphabet are used as individual variables: "x", "y", "z", "u", "v", "w"…

So "x is a cow" is a predicate, because if we replace "x" by "Buttercup" we get "Buttercup is a cow".

"x likes y" is a predicate, because if we replace "x" by "Buttercup" and "y" by "Daisy", we get "Buttercup likes Daisy". (And, if we replace both "x" and "y" by "Buttercup" we get "Buttercup likes Buttercup".)

A 1-place predicate is a predicate with one (type of) individual variable: e.g. "x likes Buttercup", "y likes y".

A 2-place predicate is a predicate with two different variables: e.g. "x likes y", "x likes y better than y's brother". And so on. (For the sake of completeness a sentence itself can be regarded as a 0-place predicate.)

When replacing variables by designators the rule is that the same variable must be replaced throughout by the same designator, but different variables (that is, variable types) may be replaced by the same or different designators. So both "Buttercup likes Buttercup" and "Buttercup likes Daisy" can be analysed into the predicate "x likes y" and (in the first case) "Buttercup" twice, and (in the second case) "Buttercup" and "Daisy". "Buttercup likes Buttercup" can also be analysed into the predicate "x likes x" and "Buttercup"; but "Buttercup likes Daisy" can't be analysed into the predicate "x likes x" and "Buttercup" and "Daisy".

Predicates and truth-functors

Up till now we have used truth-functors only to build up complex sentences out of simpler sentences. Now we will use them also to build complex predicates out of simpler predicates, (or predicates and sentences: but remember that a sentence is a 0-place predicate).

The following are examples of complex predicates:

            [x is a cow Ù x likes y]

            [x is a cow Ù x is brown]

            [x likes Buttercup Ù Buttercup is a cow]