# Arguments Against Plural Quantification

What are some good arguments against taking plural logic, a slight extension of first order logic with plural quantification as a foundational logic?

Alternatively, what are some resources that discuss plural logic and take a position against it?

I've been reading about plural quantification recently, but have mostly been treating it as a simple alternative set theory or type theory with interesting technical details such as the non-existence of empty collections.

The SEP article in question and the Wikipedia article, which has similar content, spends a lot of time talking about the benefits of plural quantification mostly by appealing to plural morphology in languages such as English and overcoming potential objections to baking in set theory by showing a semantics that is similar to the semantics of first-order logic, but has an additional relation R that is the extension of the elementhood relation. The SEP article also mentions some results like the equi-interpretability of monadic second order logic and plural logic as justification for the ontological innocence of plural logic.

I can think of some arguments off the top of my head for why plural logic is a bad framework to commit to, or at least an unintuitive one. One of them comes from the example sentence given in the very start of the SEP article has unclear truth conditions.

(2) There are some apples on the table.

(2) might be true or might be false when there's exactly one apple on the table. English speakers will probably consider things like how much knowledge the speaker is likely to have about the number of apples on the table when interpreting this sentence and considering which situations are permitted or ruled out.

We can resolve this problem by requiring collections to be nonempty or requiring them to be nonempty and non-singular, but making the decision the same way for every plural locution in all contexts seems to sever the link with natural language.

This is an example of a counterargument against plural logic that isn't very original. I'm interested in more arguments like it or sources containing such arguments.

Apart from its expressiveness, shared deductive power with monadic SOL and ontic innocence, a main issue of plural logic is in some complicated cases it bears commitment to the existence of sets as referenced in this paper here:

One of the standard views on plural quantification is that its use commits one to the existence of abstract objects–sets. On this view claims like ‘some logicians admire only each other’ involve ineliminable quantification over subsets of a salient domain. The main motivation for this view is that plural quantification has to be given some sort of semantics, and among the two main candidates—substitutional and set-theoretic—only the latter can provide the language of plurals with the desired expressive power...

So once you have to commit to (arbitrary) sets for some involved cases, then you'll be considered problematic by many. Ironically, the paper offers a way out of set by expanding the other substitutional candidate to ground plural quantification.

• That's certainly NOT the standard view, but a very controversial claim. Proponents of plural logic have formulated various plural based model theories for plural logics, whose metalanguages make use of primitive plurals instead of sets. Boolos already showed in the eighties that a plural based standard semantics for plural logic has the same expressive strength as monadic second order logic. The crucial question rather is whether one should confine ontic commitment to first-order variables or extend it to variables of any category. Plural logic is innocent only if the first way is okay. Sep 2, 2021 at 23:36
• @sequitur thx for your critique! Your assertion is certainly perfect for everyday use and is much more expressive than standard FOL. However, if applied as a foundation of math then this nominalistic countably quantified logic is inadequate immediately. From my referenced paper you can see in section 5, "because definitions are finite inscriptions over a finite alphabet. This leads to a problem when we want to emulate set-theoretic second-order quantification over an infinite domain..it will be very hard to mimic the quantification over real numbers using this sort of substitutional reading. " Sep 3, 2021 at 0:49
• @sequitur so if you stay in FOL then using plural quantification has no issue with the added benefit of essentially reducing countably many function and relation non-logical symbols, but with increased logical symbols and syntax (simplify first order language signature while complicates underlying first order logic). But I may argue once you commit to collection, then you open the Pandora box which we learned collection contains sets and proper classes (2 monsters you have to differentiate). If you carry plural quantifiers to SOL as shown above, then more serious philosophical issue comes. Sep 3, 2021 at 2:22
• @sequitur Most human minds care the difference between singleton and plural within a finite upper bound like personal wealth and properties counting, but FOL cannot express finiteness due to compactness theorem of model theory even after adding plural quantifier. So plural quantifier cannot help this FOL "defect" at all. If we have to live in ℵ0 infinity by default, then FOL only keeps us within countable from uncountable (large cardinal) infinites such as function variables in SOL where there's no more "strange" non-standard numbers relative to us. Sep 3, 2021 at 3:04

I'm not sure why you would consider the understanding of the sentence "There are some apples on the table" as an argument against plural quantification. It is just as much a problem for ordinary first order logic, where you would represent it formally as meaning: there exists at least one apple on the table, or perhaps as: there exist at least two apples on the table. Whether you represent it using standard quantifiers or plural quantifiers does not matter. In either case, the exact understanding of the sentence, and whether it is appropriate given the number of apples that actually are on the table, depends on the context and the conversational implicatures.

What plural quantification allows us to do is to use first order logic for things that would otherwise require set theory or second order logic. A simple example is the ordinary distinction between a statement that is collectively true as opposed to distributively true. If ten people are asleep then those ten people are collectively and distributely asleep, but if ten people are surrounding Nelson's column, then they are doing so collectively but not distributively. We could say of the set of ten people that it has the property of surrounding Nelson's column, but it's nice to keep things simple and avoid set theory if we can. With plural quantification, we can say that there exists a plurality of people that surround Nelson's column. Speaking of a plurality here does not imply the existence of a thing like a set; what it does is to move the expression of plurality into the mechanics of quantification itself.

One can, naturally, ask about the semantics of plurality. But, it we steer by Quine's criterion, what exists is given by what we quantify over, not by the number or type of the quantifiers we use. Plural quantification allows us to neatly sidestep the issue of the existence of collections.