Eliminating singular quantification

I’ve been thinking idly about plural quantification over the last day or so (the things one does on ones holidays…).

The general idea is that we can go beyond standard first-order predicate logic, by adding distinctively plural quantification. So, in addition to quantifying by saying “there is something such that it is Mopsy”; we may also say “there are some things such that Mopsy is one of them”. Oystein Linnebo has a really nice summary of plural logics in the Stanford Encyclopedia.

The setting I was thinking of is less expansive than the systems that Oystein concentrates on (those he calls PFO and PFO+). The way it is less expansive is this. the languages of PFO and PFO+ includes both singular quantification/singular terms and plural quantification/plural terms in its primitives. I want a system that has only plural quantification/terms as primitive. This means that rather than taking the relational primitive “is one of”, holding between singular terms and plural terms, as primitive, I’ll take “are among”, which holds between pairs of plural terms. The payoff may be this: singular terms, variables and predication may turn out to be “dispensible”, in the same sense that Russell’s theory of descriptions showed that individual constants were dispensible. This may well be stuff that is already covered by the literature (or just obvious). If so, I’d be very happy to get references!

I will be taking it that the language of plurals contains predicates of plural terms. In this way, we follow what Linnebo calls L_PFO+ rather than L_PFO. Now generally we can distinguish between plural predicates that are distributive; and those that are non-distributive. Linnebo’s examples are: the distributive predicate “is on the table” (if some things are on the table, then each one of those things is individually on the table); and the non-distributive predicate “forms a circle” (Some things can form a circle, even though there is no sense in which each individually forms a circle). Linnebo says that he does this to allow for non-distributive predications; but part of my motivation is to allow also for distributive plural predications. Syntactically, we need not pay attention to this (though if the semantic treatment of distributive and non-distributive plural predicates is to differ, we might want to differentiate them syntactically: introducing two sets of predicates. I’m going to ignore such refinements for now.)

Here’s the language:

1. L_Plural has the following plural terms (where i is any natural number):

* plural variables xxi;
* plural constants aai.

2. L_Plural has the following predicates:

* a dyadic logical predicate <. (to be thought of as are among);
* non-logical predicates Rni (for every adicity n and every natural number i).

3. L_Plural has the following formulas:

* Rni(t1, …, tn) is a formula when Rni is an n-adic predicate and tj are plural terms;
* t < t’ is a formula when t and t’ are plural terms;
* ~φ and φ&ψ are formulas when φ and ψ are formulas;
* (Ev)v.φ is a formula when φ is a formula and vv a plural variable.
* the other connectives are regarded as abbreviations in the usual way.

What I’m interested in is whether we can develop a natural logic of plurals on the basis of this language: and if so, what its expressive power would be.

An immediate task would be to reintroduce singular quantification. The intuitive thought is that singular quantification can be thought of as a special case of plural quantification, where we somehow ensure that there is just one of them. The trick is to show how this can be done without circular appeal to singular quantification.

My thought (roughly) is to treat this as the following restricted quantifier [Exx : (yy)(if yy < xx then xx < yy].

Why will this play the role of singular quantification? Well, just because if you’ve got a plurality of things, which is such that every subplurality is also a superplurality, it’s got to be a plurality consisting of just one thing (I’m assuming that there are no “null” pluralities). Now, of course, L_plural doesn’t contain restricted quantifiers. But it’s easy enough to find things that play the role of restricted quantifiers (formally, we’ll define a paraphrase from L_PFO+ into L_plural that’ll play this role). In parallel fashion, we can get a paraphrase of sentences containing singular terms, and paraphrase them into something that only uses plural vocabulary.

E.g. “(Ex)Elephant(x)” may go to: “(Exx)((yy)(if yy < xx then xx < yy)& Elephant(xx))”. And “Runs(Susan)” may go to: “(yy)(if yy < Susan then Susan < yy)& Runs(Susan) )

Now, it seems to me that there are some interesting questions of detail about how best to formalize the “intuitive” logical theory for L_plural that I’ve been working with. But let me leave the this for now. Question is: does the above elimination of singular quantification and terms in favour of plural quantification and terms seem tenable? Does the paraphrase work on the “intuitive” reading of L_plural. Can people see any obstacles to formalizing this intuitive logic for L_plural?


6 responses to “Eliminating singular quantification

  1. hi Robbie – well this is a good start [commenting on the quantification issue / plural count nouns]. Have you thought about incorporating non-count nouns as well? I have something of a vested interest in that myself. best, Henry Laycock

  2. My memory is failing me at the moment, but I think a logic that takes plural predication as primitive has been worked out by Beyong-Uk Yi (see his recent [=last couple of years] Logic and Meaning of Plurals (I), (II)” in Journal of Philosophocal Logic) and I forget what Smiley and Oliver do in their recent article in the same journal, but it might be what you’re thinking of, too.

    Sorry if I’ve sent you on a wild goose chase, but perhaps this is what you are looking for.

  3. Thom McKay’s Plural Predication develops a language in which plural predication is primitive and singular quantification is had in the way you propose: i.e. [Exx : (yy)(if yy < xx then xx < yy]. He calls `<' `among'.

  4. Hello Robbie. I’ve been thinking about this problem for years, is it not the other way round? Can we not reduce plural quantification to the singular kind. Intuitively, singular terms are basic. Mill argues that sentences like ‘Peter and Paul preached in Galilee and Jerusalem’ reduce to four sentences (Peter preached in Galilee & Peter preached in Jerusalem &c). We connect plural reference to plural quantification by some appropriate substitution rule, e.g. just as we move from ‘any x is an apostle’ to ‘Peter is an apostle ‘, so we move from ‘any x’s are apostles’ to ‘Peter and Paul are apostles’. I.e. substitute for ‘any x’s’ any expression consisting of concatenated proper names (perhaps with the limiting case being a single proper name).

    That does not mean there exists some ‘Peter and Paul’ thingy, only that we can make that substitution. This implies ‘Peter is an apostle and Paul is an apostle’, as above. Thus, only singulars exist.

  5. Hi all,

    Thanks for this, and apologies for the delay in getting back to you. I managed to pick up a copy of McKay’s stuff, and so will be able to see what’s going on there soon. Yi’s stuff looks good too.

    On the non-count noun stuff: I really haven’t thought about it! I guess the way I’ve got things set up at the moment, the predicates are mostly natural construed as involving count-nouns. The idea then is to define the singular application (“is an elephant”) as a limiting case of plural application (“are elephants”).

    Suppose we’ve got a non-count predicate like “is water”. How will that interact with the machinery? Well, there does seem to my naive ear to be both plural and singular applications of mass nouns like “water”: we can make sense “those puddles are water” as well as “that puddle is water”. So you might hope that the same story would go through here. But I’m aware that this might be hopelessly naive (I just don’t know much about the syntax or semantics of mass nouns and the like). Henry: is there a feature of non-count nouns that makes things especially problematic? What’s the best place to go to find out about it?

    Ocham. Interesting. So the idea is that you reduce plurally quantified sentences (substitutionally) to plural named sentences, and then reduce those further to singular stuff. Be interesting to see how this would work out in detail for e.g. the Geach Kaplan sentence “some critics admire only one another”.

    I’ve got one worry of principle. Some plural predications don’t seem to “divide down” to singular predications in the way you sketch. E.g. on one reading, “John and Jim carried a piano” doesn’t entail “John carried a piano”; in the way that “John and Jim are human” does entail “John is human” and “Jim is human”. That threatens to block the final stage of your procedure. I guess the standard names are “collective” and “distributive” plural predication respectively.

  6. I was going to mention this but didn’t fit neatly into a single comment. Of course, non-distributive predicates seem to be a problem. But they are not. ‘John carried the piano with Jim’ is true when the singular predicate ‘- carried the piano with Jim’ applies to John, and also true when the singular predicate ‘John carried the piano with -‘ applies to Jim.

    The only difference is that with distributive predication you can split predication over a conjunction, tyically ‘and’.

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