Are there empty predicates?

Empty names are names that fail to refer, like 'Santa,' 'Pegasus,' and 'Planet Vulcan.' 'Santa Claus' fails to refer because (on most semantics for empty names) there is no entity that is assigned to 'Santa' as its referent. This is clearly distinct from another view (e.g. Frege's) that 'Santa' should be assigned e.g. the empty set as its referent. That is, there is a difference from having no referent and referring to the empty set -- for my cat has no referent, but '∅' refers to the empty set.

So are there empty predicates? That is, are there predicates that do not signify properties (or extensions, kinds, intensions (= functions from possible worlds to extensions), or whatever your preferred semantic value for predicates is). There are of course predicates whose extension is the empty set (e.g. 'is not identical with itself') -- these predicates signify uninstantiated properties (assuming you think predicates signify properties). But they still signify a property.

There is a fairly massive literature on empty names. (I can recommend Ben Caplan's 2002 dissertation as a nice survey of the empty names landscape.) But there is no talk of empty predicates -- is this because somehow every predicate, unlike names, automatically refers?

Related issue: Philosophers of science often say things like 'phlogiston' and 'caloric' fail to refer. Often, in explaining their claim "The word 'phlogiston' does not refer", these philosophers will say things like "The extension of the predicate 'is phlogiston' (or 'contains phlogiston') is empty." But having the empty set for your extension is different from failing to refer. So when we say that 'contains phlogiston' fails to refer, it seems like we should be saying that it has no (determinate?) extension, not that its extension is empty.

So are there any empty predicates? Are such things even possible? And can the usage of the philosophers of science be defended?


Hecky said...

You insist that there is no difference between lacking a referent and referring to the empty set, but I'm not sure I see the reason for this. Doesn't lacking any referent mean that the extension is empty? Put another way, what is the difference between lacking an extension and having an empty extension (i.e. referring to the empty set as the extension of the term)?

Greg Frost-Arnold said...

Hi Hecky --

Very fair point. See if the following does anything for you. (p.s. -- I assume that you didn't mean "You insist there is no difference", but rather "... there is a difference.)

Predicate terms, when they are functioning 'nicely' (= classically), have as semantic values an extension (the set of objects that satisfy 'x is P'), and an anti-extension (the set of objects for which 'x is P' is false).

I think (though I'm very interested in any argument to the contrary) that there is a difference between (i) a predicate term which has an extension that is empty and an anti-extension that is the whole domain of discourse, and (ii) a predicate that has neither an extension nor an anti-extension -- i.e., its extension and anti-extension are undefined/ non-existent. In short, (i) has a (certain kind of) semantic value, while (ii) does not.

There are clearly cases of (i) (e.g. 'is a flying human'); what I was asking in the post is whether there are any cases of (ii).

Hecky said...

I think I see what you're getting at now. Apologies for that egregious typo!

I confess I've never heard of the "anti-extension" of a predicate but it sounds obvious enough.

Part of the problem with this kind of question seems to be how one could properly call any term a predicate when it has neither an extension nor an anti-extension as in (ii). Terms must at least fail to refer in order to be called predicates, it seems to me (but I'm not sure how to put this point in terms of truth conditions).

So examples of undefined extensions might include multi-valued accounts of a predicate? Can there be predicates with totally undefined extensions in bivalent logic?

But if the extension and anti-extension of a term are "non-existent," then what allows me to call that term a predicate? Doesn't it minimally need to fail to apply in order to be a predicate? Put another way, doesn't it need to offer up some semantic criteria to determine whether it is true/false of objects, or indeterminately true? If I ask about the extension of "blork" but can't say what it means because it's a sound I just made up, then it would fail to have either an extension or an anti-extension, it seems. We have no semantic criteria and thus no way to determine either the extension or anti-extension. But this is just a nonsense sound, not even a word and certainly not a predicate.

So is the difference between (i) and (ii) that in (i) you are talking about "proper" predicates and in (ii) you are not talking about predicates but rather some other kind of term, or perhaps no term at all?

Also, can you say whether you are posing this question with bivalence in mind or whether multiple values are in play here?

Greg Frost-Arnold said...

Just a quick couple of thoughts:

1. I do not suppose bivalence (for me: 'every sentence is true or false') is right. It might be, but I think that stands in need of argument--for there are (to my mind) at least plausible candidates for sentences that are neither true nor false.

2. I think the way I'm conceiving of the distinction you are drawing in your latest comment is as follows: an empty predicate is at least grammatically a predicate; however, it may not have the semantic traits of a predicate.
The analogy is supposed to be exactly like so-called 'empty names,' like 'Santa' and 'Pegasus'. These refer to no object (on most usual views), but still can appear in sentences where (genuine) names can appear, to make a grammatically acceptable sentence.

Anonymous said...

Sorry for interrupting. This is Seyed. I think the question, i.e. are there empty predicates?, have different answers depending on the language and logic we are talking about. For example, if we have an un-typed language equipped with lambda operators bounding predicate variables (given that the logic is two-valued), we may introduce the following predicate:
[(Lambda X) ~(X exemplify X)]
But if the logic is two-valued and the language un-typed, this predicate cannot refer to any property. Assume it does. Call the property ‘F’. Then either F exemplifies F or F does not exemplify F. If F exemplifies F, then by lambda equivalence it follows that F does not exemplify F. If F does not exemplify F, again by lambda equivalence and the rule of double negation, it follows that F exemplifies F. So, the above-mentioned predicate is only a finite string of symbols in our language that does not refer to any property, though it might be said that it is a predicate, “grammatically speaking”!

Greg Frost-Arnold said...

Hi Seyed --

Thanks for stopping by! I think you are right. When I posted this, I was talking about this with the guy in the office next to mine (James Woodbridge), and he suggested that there could be supposed properties that were in some way, shape, or form contradictory or inconsistent or whatever. And you've provided a very nice, clean example of such here -- thanks.

Note: James thinks that there is no property of truth (or any other semantic properties) because of the semantic paradoxes. He's written a lot on this idea; check his webpage linked above if you're interested.