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.In two important articles from the 1980s George Boolos challenges this traditional view (Boolos 19a).
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 \(xx\).
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.
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).
More controversially, it has been argued that the resulting formal system with plural as well as singular quantification qualifies as “pure logic”; in particular, that it is universally applicable, ontologically innocent, and perfectly well understood.
But the existence of two or more objects may not be semantically required; for instance, “The students who register for this class will learn a lot” seems capable of being true even if only one student registers.
It is therefore both reasonable and convenient to demand only that there be at least one object satisfying \(\phi(x)\).
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).