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?