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:
- 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.
- 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 …’.
- 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’.
- 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’.
- 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!