Some recent discussions of the metaphysics of properties are weakened by neglecting the possibility that higher-order resources might be sui generis.
So this paper is not just about the implications of adding sui generis second-order quantification to our ideology, but of adding higher-order resources more generally. 3.2 I will also assume the intelligibility of another piece of higher-order apparatus, namely, predicates of predicates. Second, I think I know what ‘∃ X Xa’ means on the intended reading the fact that I cannot manifest that knowledge by providing a synonymous English sentence casts no doubt on that, since English might lack the resources to express some things that higher-order predicate calculus can (cf. First, the claim that all quantification is either objectual or substitutional is a Quinean dogma which has never been established by argument (Williams, 1976, p. What could ‘∃ X Xa’ mean if interpreted neither objectually nor substitutionally? It is not my purpose to deal with this challenge here, but I will make a couple of remarks to signal my attitude towards it. Some will hold that this project is pointless because the sort of quantification I am discussing is unintelligible. Then in the remainder of the paper I will argue that bringing in irreducible second-order quantification defeats three of the principal arguments for the existence of properties. 2 I will set out Jones’s arguments and present some objections to them. As I see it, introducing this sort of quantification weakens the case for belief in properties. Nothing essential is thereby lost.) I agree that it reshapes the debate, but I do not agree that it allows us to establish the existence of properties. (Jones’s paper in fact covers properties and relations, but for simplicity I will just consider here what he says about properties. Jones argues that introducing irreducible second-order quantification reshapes the debate, and indeed settles it in favour of the existence of properties. Nicholas K. Jones ( 2018) has already given an answer to that question.
In particular, I will ask what difference adding this piece of ideology makes to the debate within metaphysics over the existence of properties.
In this paper, I will explore the implications of taking second-order quantification to be sui generis, irreducible to first-order quantification. (I will call this reading of the quantifier ‘substitutional quantification’.) All these interpretations take second-order quantification to be reducible to first-order quantification (over properties, sets, or linguistic expressions respectively). Alternatively, we might read ‘∃ X Xa’ as ‘There is a linguistic expression such that substituting it into “_a” yields a truth’. Another objectual reading takes them to concern sets: on this reading, ‘∃ X Xa’ would mean ‘There is a set containing a’. For example, we could read these formulae as disguised talk of properties and instantiation: then ‘∃ X Xa’ would mean ‘There is a property had by a’. How should formulae such as ‘∃ X Xa’ be interpreted? We might read them objectually. For example, the formula ‘∃ X Xa’ involves quantification into predicate position ( second-order quantification) rather than nominal quantification. It appears in the English sentence ‘Something is red’ and the formula ‘∃ x F x’ of predicate calculus. Nominal quantification is quantification into the syntactic position held by a singular term.
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