What is the difference between function and predicate?

Question

I am currently watching videos on formal semantics in Youtube. I find that the terms function and predicate are used a lot and that what they mean is similar. Functions take one or more arguments, and so do predicates. Typical predicates are content verbs, and sometimes these content verbs are called functions. Is a predicate always a function? Is a function always a predicate? Which of the two is the broader notion? Expressed in another way, are there functions that are not predicates, and are there predicates that are not functions? Thanks in advance for any guidance provided.

Answer 1

There is two kinds of "functions" that could be meant in your context:

1. Thinking of the content expressed by a predicate as a function which takes individuals as an input and gives a truth value as an output.

2. Special words, so-called functional verbs, nouns or adjectives, which are natural language expressions thought of as associating each possible argument with exactly one individual that the expression refers to, e.g. "father" is a functional noun where by "the father of Mary" we can refer directly to the person who is the father of Mary.

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An n-place predicate is typically thought of as denoting an n-place relation, i.e. a set consisting of tuples of n individuals. For example, "man" is a 1-place predicate that could denote the set {Peter, Bob} (meaning that Peter and Bob are men and noone else is in the situation), and "loves" is a 2-place predicate that could denote the relation {<Peter, Mary>, <Mary, Peter>, <Bob, Peter>} (meaning that Peter and Mary love each other, and additionally Bob loves Peter).

An n-place relation denoted by an n-place predicate can be expressed as an n-place function which returns a truth value. This is the so-called characteristic function of the set. For instance, the set denoted by the predicate "man" is associated with the function f such that f(Peter) = true, f(Mary) = false, f(Bob) = true, f(Susan) = false, and the "loves"-relation has the characteristic function f with f(Peter, Peter) = false, f(Peter, Mary) = true, f(Peter, Bob) = false, and so on.

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As a separate notion from the functions above, which evaluate to truth values, it is also possible to conceive of functions which evaluate to individuals.

An n-place individual function is something where you put in n individuals and get out an individual: [[f(x1, ..., xn)]] = y.

An n-place individual predicate is something where you put in n individuals and get out yes or no: [[P(x1, ..., xn)]] = true/false.

For example, "the father of __" is a 1-place function, since you can combine it with an individual and directly get out another individual, namely their father. "__ is a man" is a 1-place predicate, since you can combine it with an individual and get out true or false, depending on whether or not they is a man.

All n-place functions can be expressed as n+1-place predicates: [[f(x1, ..., xn)]] = y --> [[P(x1, ..., xn, y)]] = true, [[P(x1, ..., xn, z)]] = false for all z ≠ y. For example "the father of __" (has 1 slot and evaluates directly to the person who is the father) --> "__ is the father of __" (has two slots and evaluates to yes or no).