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.Tags: English Homework AssignmentsData Paper Research ScrubbingAn Example Of An EssayEducation Social Inequality EssayA Small Place By Jamaica Kincaid EssayEssay Communication Barriers RelationshipExamples Of Methodology In Research PaperGood Thesis Statements For PtsdEditing Creative WritingEssay On Memory And Learning
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.
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 \(\phi(x)\).
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.It is therefore both natural and useful to consider a slightly richer language: arguments?Lots of English predicates work this way, for instance “… So if our primary interest was to analyze natural language, we would probably have to allow such predicates.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.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.\] (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.(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).