What is a predicate?

Reworking some introductory chapters on QL languages, I thought I should make some  general introductory, pre-formal, remarks about names and predicates. But then, before talking about arities, about sense vs extension, etc., there’s a more basic question: what is a predicate? A simple question, but with a surprising range of answers to be found in the literature. Some candidates:

  1. A predicate is just an expression, as in ‘is wise’ in the ordinary language sentence ‘Socrates is wise’, or ‘loves’ in Romeo loves Juliet’ — an expression which can combine with names or other suitable expressions to form sentences.
  2. A predicate is an expression with gaps, as in ‘ is wise’ or ‘ loves ’. Or since gaps are hard to spot we may use gap-markers, which aren’t part of the gappy predicate expression but just signal where the gaps are, and represent the predicates as ‘… is wise’ or ‘… loves …’.
  3. A predicate is an expression with gaps and a rule as to whether the gaps have to be filled in the same way — so in a different notation, we might distinguish
    ‘① loves ②’ from  ‘① loves ①’.
  4. A predicate is a linguistic function — for example, the function that sends the name ‘Socrates’ to the sentence  ‘Socrates is wise’, sends the name ‘Plato’ to the sentence  ‘Plato is wise’, sends the name ‘Aristotle’ to the sentence  ‘Aristotle is wise’ and so on. Or for another example, the function which sends  ‘Seneca’ to  ‘Seneca killed Seneca’, sends ‘Mark Anthony’ to ‘Mark Anthony killed Mark Anthony’, sends ‘Lucretia’ to  ‘Lucretia killed Lucretia’.
  5. A predicate is a property of sentences — perhaps the property shared by ‘Seneca killed Seneca’, ‘Mark Anthony killed Mark Anthony’, ‘Lucretia killed Lucretia’ …

And there are other variant candidates. What to do? Wrangling about the right answer, engaging e.g. with the arguments of Frege, Geach, Dummett and others, is not what I would want to be doing in an introductory logic book! In fact, I’d like to be able to write mostly as if predicates are simple expressions which come with rules about how they combine, and not get any fancier. But still, if there were a compelling argument for going with another story about the nature of predicates and rejecting the rest, I wouldn’t want to be corrupting the youth by blithely giving the wrong account.

It’s mighty cheering then to (re)read Alex Oliver’s paper ‘What is a predicate?’ in The Cambridge Companion to Frege where he argues — I think pretty convincingly — that

Whereas other authors argue for different candidates, I propose that there is nothing to choose between them. Anything goes: each is equally serviceable. … The choice between them can be made arbitrarily, or, when the context allows, it can be left unmade.

Hooray! I can point in the Further Readings to Alex’s arguments as justification enough for the way I write about predicates, and leave things at that. And if you don’t know Alex’s paper, you should do:  take this as a warm recommendation!

A minor footnote, though. I was wondering whether to talk of one-place, unary, or monadic predicates. Surely a trivial decision! And yet …

Start from the sentence ‘If Socrates is a philosopher, Socrates is wise’ and remove the recurring name to get the expression ‘If ① is a philosopher, ① is wise’, with gaps to be filled the same way. This expression can be used to attribute the property (unary relation) of being-wise-if-a-philosopher to an individual object, or we can quantify in, e.g. to say that everyone has that property. It is natural to class this expression, then, as a unary predicate belonging with other simpler unary predicates that can be used to attribute unary relations. However, it would seem a bit unhappy to say that this is a one-place predicate, given that — as presented — it wears on its face the fact there are two places for a name to go. This gives us some reason perhaps to prefer the more technical sounding ‘unary’ (or ‘monadic’). Which surely has been said before somewhere!

12 thoughts on “What is a predicate?”

  1. Dear Peter, just give the students ONE predicate they understand, like: to be a prime number. Then say there are many others, some of them clear, others less clear, but that’s just as it is with most things… no need to solve eternal questions to learn something.

    1. I basically agree!

      I’m fond Quine’s maxim,”where it doesn’t itch, don’t scratch”. But equally, I didn’t/don’t want to write something that on reflection I think is false — hence my spending some time thinking again, for my own sake really, about what predicates might be. That’s changed some phraseology in the book in minor (probably unnoticeable!) ways: but I’m happier …

      1. I think that sounds like the right approach. Try to say something as simple and straightforward as you can that will also hold up if the reader returns to it once they know more.

        I also think JvP has a good point. One of the pitfalls than can entrap an author who knows a lot about a subject is that they will overcomplicate what they say because they are aware of various problems and controversies.

        1. Oh yes, a definite pitfall. I’ve lost count of the number of paragraphs at various places I’ve written and then deleted when the “does this overcomplicate things?” test is applied. Which isn’t to say that I’ve deleted all I should have!

  2. In many applications a predicate is a function from a universe of discourse $latex X$ to a binary value in $latex \mathbb{B} = \{0, 1\},$ in other words, a characteristic function or indicator function $latex f : X \to \mathbb{B},$ and $latex f^{-1}(1),$ the fiber of $latex 1,$ is the set of elements denoted or indicated by the predicate.  That is the semantics, anyway.  There are of course many formal languages whose syntactic expressions serve as names for those functions and nominally speaking we may call those names “predicates”.

    1. Among philosophers, at any rate, ‘predicate’ is used for a linguistic/syntactic item — your predicate (=function) is the candidate (pretty much Frege’s candidate!) for what the linguistic item picks out/denotes/refers to. I’d strongly deprecate calling that a predicate, which perhaps just shows my upbringing! Of course, Frege (and others) would also deprecate calling a predicate (in the syntactic sense) a name of a function, but that’s another story!

  3. I much prefer (in this context of teaching baby logic) option 3. And, about your last point: I found that students had no trouble thinking of the predicate mined from ‘Maynard loves himself’ as one-place. In class I would give something like:
    Maynard took the book from Edna who gave it back to him.
    and they would find the 6 gaps but still count it as 3-place.

    1. Right! It’s just that — until I was (re)writing a few sentences in the book — I hadn’t been quite so consciously struck before that we don’t just count by places by … erm … places, and so a sentence signalling that was missing but needed!

      1. Why don’t you count places by places? Why is it so important to go into this at such a level of detail about English syntax and semantics that you have to explain that there might be a rule that says some gaps have to be filled by the same thing?

        If you are going to do that, however, then I think you ought to say more than in IFL1 about what bits of English syntax can become gaps, and it ought to be consistent too, so that “apples” won’t be a gap-candidate in one place and a predicate in another.

        For “Maynard took the book from Edna who gave it back to him”, why not just have three gaps: “… took … from … who have it back to him”? Why go to the trouble of turning “who” and “it” and “him” into gaps only to say that have to be filled by the same things as other gaps?

        Or why not say there are 6 gaps without a same-filling rule? After all, Emma might give the book to someone else, rather than back to Maynard, or someone else might return the book.

        Also, why can’t there be gaps for the verbs and prepositions? Maynard might have given an apple to Emma, and Emma might have eaten it.

        All that ought to be explained, it seems to me, once you head down this road.

        1. Indeed, and I did that in those classes. The closest to what I did is I think Hodges treatment in his baby logic book. Certain words are designators (proper names, singular pronouns, non-count nouns and definite descriptions). Each gets a syntactically based “test”. (well, with the pronouns it’s a list). etc.

  4. I don’t like any of the candidates very much, though 4 and 5 are by far the worst. If I had to pick, I’d go with 1 if it would be enough for the things I needed to discuss, else 2. If I reached a point where it seemed I needed to resort to 3, I might try to change other things so that I could avoid it.

    Re the unary vs one-place, etc footnote, I can see the problem — if you go down the road of saying there can be more gaps than are allowed to have distinct gap-fillers. But why go that way?

    If we start with “John loves himself”, the more natural unary gapping is “… loves himself”, not “(1) loves (1)”. It’s true that someone might say “John loves John”, but then I’d argue that they are using the binary “… loves …” but happen to be filling both places with “John”.

    For something like “If Socrates is a philosopher, Socrates is wise”, I’d start with understanding what that is saying, rather than mechanically putting in gaps. (You have to understand it anyway in order to say which gaps have to be filled in the same way, so it’s not a step you can avoid.) Is it part of the meaning that the very same person who’s a philosopher is also wise? (That seems most likely, though it’s possible to imagine contexts in which that’s not what’s meant.) If so, the predicate when written as English with gaps can be “If … is a philosopher, that person is wise”.

    BTW, the idea that that’s a predicate seems quite strange to me, regardless of how it’s written. Are we supposed to believe that someone could have the property, not of being wise (perhaps inferred because they’re a philosopher) but of being-wise-if-a-philosopher? It’s not for nothing that our logics treat ‘if’ differently!

    There’s a similar issue on pages 20-21 of IFL1 when it talks of “… loves …” and says it “has two empty places waiting to be filled up by names like ‘Romeo’ and ‘Juliet’ (or by more complex terms like ‘Someone in this room’ and ‘every philosopher’).”

    In any case, what is the task? Is it saying what a predicate is and giving examples using English with gaps to express them? Or is it identifying the predicates in arbitrary, existing English sentences? If it’s the latter, then I think a host of difficulties flock in. For a start, you’ll have to have rules about what can be replaced by a gap (is it just noun phrases?) and what (if any) changes in the rest of the sentence are allowed. (For example, if the sentence is “I love Juliet”, does “love” have to stay “love” or can it be “loves”?)

    There’s also the question of what you say about syllogistic reasoning. For example, in IFL1, page 6, you discuss

    All F are G
    n is F
    So, n is G

    and say ‘F’ and ‘G’ hold the place for predicates which you say are “expressions which attribute properties or pick out kinds of thing”. So here “apples” would be a predicate.

    1. Thanks for all this!

      The last point I can get easily out of the way. In the lastest version of the early chapters of IFL2, any (loose) talk of ‘predicates’ has gone — because, as you point out, it doesn’t really square with what is said later, so easier just to rephrase. And that does free me up to use ‘predicate’ as a semi-technical logicians’ term later.

      I’ll say a bit more about the rest when I’ve reread (and re-thought about) Dummett who I recall has some interesting things to say about the use of reflexive pronouns (like ‘himself’) in natural language as a way of reducing places in what Dummett would call complex predicates. But I’m coming round to the idea that it might be better on more than one count — less misleading, for a start — to talk of open sentences rather than complex predicates.

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