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We see this by distinguishing between two kinds of plural predication.
However, not only are such paraphrases often unnatural, but they may not even be available.
One of the most interesting examples of plural locutions which resist singular paraphrase is the so-called Geach-Kaplan sentence: How are we to formalize such sentences?
However, for present purposes it is simpler not to allow such predicates.
We will anyway soon allow pluralities that consist of just one thing.
In two important articles from the 1980s George Boolos challenges this traditional view (Boolos 19a).
He argues that it is simply a prejudice to insist that the plural locutions of natural language be paraphrased away.This translation allows us to interpret all sentences of \(L_\) and \(L_\), relying on our intuitive understanding of English. Applying \(\Tr\) to (\ref), say, yields: of plural first-order quantification based on the language \(L_\).Let’s begin with an axiomatization of ordinary first-order logic with identity.A predicate \(P\) that isn’t distributive is said to be For instance, the predicate “form a circle” is non-distributive, since it is not analytic that whenever some things \(xx\) form a circle, each of \(xx\) forms a circle.Another example of non-distributive plural predication is the second argument-place of the logical predicate \(\prec\): for it is not true (let alone analytic) that whenever \(u\) is one of \(xx, u\) is one of each of \(xx\).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).In introductory logic courses students are therefore typically taught to paraphrase away plural locutions.For instance, they may be taught to render “Alice and Bob are hungry” as “Alice is hungry & Bob is hungry”, and “There are some apples on the table”, as “\(\Exists \Exists (x\) is an apple on the table & \(y\) is an apple on the table & \(x \ne y)\)”.\] (That is, for any things, there is something that is one of them.) Let be the theory based on the language \(L_\) which arises in an analogous way, but which in addition has the following axiom schema of extensionality: \[ \tag \Forall\Forall [\Forall(u \prec xx \leftrightarrow u \prec yy) \rightarrow(\phi(xx) \leftrightarrow \phi(yy))] \] (That is, for any things\(_1\) and any things\(_2\) (if something is one of them\(_1\) if and only it is one of them\(_2\), then they\(_1\) are \(\phi\) if and only if they\(_2\) are \(\phi)\).) This axiom schema ensures that all coextensive pluralities are indiscernible. For ease of communication we will use the word “plurality” without taking a stand on whether there really exist such entities as pluralities.Statements involving the word “plurality” can always be rewritten more longwindedly without use of that word.