*The traditional view, defended for instance by Quine, is that all paraphrases must be given in classical first-order logic, if necessary supplemented with set theory.*In particular, Quine suggests that (3) should be formalized as \[\tag\label \kern-5pt\Exists(\Exists\mstop u \in S \amp \Forall(u\in S \rightarrow Cu) \amp \Forall\Forall(u\in S \amp \textit \rightarrow v\in S \amp u\ne v)) \] (1973: 1: 293).

*The traditional view, defended for instance by Quine, is that all paraphrases must be given in classical first-order logic, if necessary supplemented with set theory.In particular, Quine suggests that (3) should be formalized as \[\tag\label \kern-5pt\Exists(\Exists\mstop u \in S \amp \Forall(u\in S \rightarrow Cu) \amp \Forall\Forall(u\in S \amp \textit \rightarrow v\in S \amp u\ne v)) \] (1973: 1: 293). But in recent decades it has been argued that we have good reason to admit among our primitive logical notions also the plural quantifiers \(\forall\) and \(\exists\) (Boolos 19a).*

For instance, (2) can be formalized as \[ \tag\label \Exists \Forall (u\prec xx \rightarrow Au \amp Tu) \] And the Geach-Kaplan sentence (3) can be formalized as \[ \tag\label \Exists [\Forall(u\prec xx \rightarrow Cu) \amp \Forall\Forall(u\prec xx \amp \textit \rightarrow v\prec xx \amp u\ne v)].

\] However, the language \(L_\) has one severe limitation.

Instead he suggests that just as the singular quantifiers \(\Forall\) and \(\Exists\) get their legitimacy from the fact that they represent certain quantificational devices in natural language, so do their plural counterparts \(\Forall\) and \(\Exists\).

For there can be no doubt that in natural language we use and understand the expressions “for any things” and “there are some In \(L_\) we can formalize a number of English claims involving plurals.

Next we need some axioms which for suitable formulas \(\phi(x)\) allow us to talk about the \(\phi\)s.

In ordinary English the use of plural locutions generally signals a concern with two or more objects.

However, for present purposes it is simpler not to allow such predicates.

We will anyway soon allow pluralities that consist of just one thing.

For our current purposes, it is convenient to axiomatize this logic as a natural deduction system, taking all tautologies as axioms and the familiar natural deduction rules governing the singular quantifiers and the identity sign as rules of inference.

We then extend in the obvious way the natural deduction rules for the singular quantifiers to the plural ones.

## Comments Plural To Thesis