Plural and Pleonetetic Quantification

Abstract

This essay has three parts. In the first, I make some general remarks about relational quantifiers, and, using ‘there are more ϕs than ξ’ as an example, I provide an application to syntax. A relational quantifier is one that binds one variable in each of an ordered n-tuple of open sentences, where n ≥ 2, but all my examples will concern cases where n = 2. The plural quantifiers ‘nearly every ϕ is ξ’ and ‘many ϕs are ξ’ are also best treated as relational. What is said in this part will be familiar to some readers, but knowledge of it is perhaps sufficiently patchy to justify including it. In the second part I make some remarks about domains of discourse appropriate for systems formalising reasoning with relational quantifiers. The matter is less straightforward than in the case of the standard, one-place quantifiers. This discussion involves mention of principles of inference that impose requirements of proportionality on plural quantifiers. The third and last part is devoted to the quantifier ‘most ϕs are ξ’, which is relational and evidently proportional. It is, to use Geach’s word, a pleonetetic quantifier. The close relationship between numerical principles and principles governing plural and pleonetetic quantifiers—a relationship already illustrated in the first two parts— raises the question how far the basic principles of plural and pleonetetic logic should ideally be from obviously numerical principles. I give some principles for monadic pleonotetic logic, which fall into two groups. In one group all the principles look straightforwardly logical—at least to my eye. The principles in the other group are formulated using only logical notions, but they are most naturally grasped via their relationship with numerical principles. I conclude, after providing examples of how to prove things with these principles, by leaving open the question whether a line can satisfactorily be drawn between logical and arithmetical reasoning.