(Most people who write on the subject make this concession.) This gives rise to the , which are the instances of the schema \[ \tag \Exists \phi(u) \rightarrow \Exists \Forall (u\prec xx \leftrightarrow \phi(u)) \] where \(\phi\) is a formula in \(L_\) that contains “\(u\)” and possibly other variables free but contains no occurrence of “\(xx\)”.
(That is, if something is \(\phi\) then there are some things such that everything is one of them if and only if it is \(\phi\).) In order fully to capture the idea that all pluralities are non-empty, we also adopt the axiom \[ \tag \Forall \Exists (u \prec xx).
Plural quantification has also been used in attempts to defend logicist ideas, to account for set theory, and to eliminate ontological commitments to mathematical objects and complex objects.
The logical formalisms that have dominated in the analytic tradition ever since Frege do not allow for plural quantification.
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.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.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.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?For now, the formal languages \(L_\) and \(L_\) will be interpreted only by means of a translation of them into ordinary English, augmented by indices to facilitate cross-reference (Boolos 1984: 443–5 [1998a: 67–9]; Rayo 2002: 458–9).(More serious semantic issues will be addressed in Section 4, where our main question will be whether our theories of plural quantification are ontologically committed to any sort of “set-like” entities.) The two clauses of this translation which are immediately concerned with plural terms are The other clauses are obvious, for instance: \(\Tr(\phi \amp \psi) = (\Tr(\phi)\) and \(\Tr(\psi))\).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 & \(x \ne y)\)”.