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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.

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    Could you link to those youtube videos so we have more context? Commented Dec 1, 2022 at 14:52
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    Thank you for the contentful answer! An example of the videos I am watching is here: youtube.com/watch?v=cg9F0_4mJ_E In this video, he seems to use the terms function and predicate and verb almost interchangeably
    – Buffoon
    Commented Dec 2, 2022 at 3:38
  • I think the main source of the confusion is that the instructor is slightly sloppy in his language usage when he says "a transitive verb is a two-place function"; the TLDR to your question is: An n-predicate is a kind of expression (sequence of sounds/letters) that denotes (≈ means in a given situation) a function from n individuals to a truth value. Commented Dec 2, 2022 at 15:57

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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.


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.


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).

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    You give the example the father of __ as a one-place individual function. Would it be incorrect to say that it is a one-place predicate? Perhaps predicates always map to truth values, in which case the father of __ cannot be viewed as a predicate. If that's correct, then it seems that a predicate is always a function (one that maps to a truth value), but there are functions that are not predicates (e.g. the father of __). Is that right?
    – Buffoon
    Commented Dec 2, 2022 at 3:41
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    "Predicate" is a kind of linguistic expression, like e.g. "subject argument". "Function" is the kind of mathematical object used to model the meaning of this expression. The conversion between a set/relation and its characteristic function is trivial enough that those two can be used interchangeably without much problem, but one does typically take care to distinguish between the predicate (syntax) and the set/function it denotes (semantics). Commented Dec 2, 2022 at 15:30
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    "the father of __" is an individual function since when you fill the slot, you get something that refers to an individual; "__ is the father of __" is an individual predicate since when you fill the slots you get something that is true or false. One could decompose it by saying that "father (of)" is a predicate denoting a relation between two individuals, "the father of __" is a function extracting the only matching object into a function value, and "__ is the father of __" again turns it into a predicate comparing the subject for equality with the function value. Commented Dec 2, 2022 at 15:40
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    Yes, the denotation of an individual predicate always maps to a truth value, whereas the denotation of an individual function always maps to an individual. The confusing thing here is again the two kinds of "function"; both predicate expressions and functional expressions (like "father of") denote what we think of as a function in the mathematical sense, but the former maps individuals to truth values whereas the latter maps individuals to individuals. Commented Dec 2, 2022 at 15:47
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    The kind of predicates we've talked about so far take individual arguments and denote functions from individuals or truth values. For predicates that take propositions as arguments, we need something more complicated, called intensions, and the function the predicate denotes will map from such intensions of sentences and individuals to truth values. Commented Dec 3, 2022 at 17:17

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