Let ‘Let n be such an x’ Be: A Reply to Meléndez Gutiérrez

I defend Haze’s argument against the Breckenridge-Magidor theory of instantial reasoning from an objection by Meléndez Gutiérrez. According to Breckenridge and Magidor, in reasoning like ‘Some x is mortal. Let n be such an x…’, the ‘n’ refers to a particular object but we cannot know which. This surprisingly defensible view poses an obvious threat to widespread notions in the philosophy of language. Haze argues that the theory leads to absurdity in cases like ‘Some x is unreferred-to by any expression. Let n be such an x…’ and should therefore be rejected. Meléndez Gutiérrez counters that Haze’s argument is just a case of Berry-like paradox and thus fails to refute the Breckenridge-Magidor theory. I argue that the analogy breaks down: unlike the intuitively compelling and widely believed well-ordering principle about positive integers, the principle drawn from Breckenridge and Magidor that plays a supposedly analogous role enjoys no such status, and is instead simply shown to be false by Haze’s reductio. The possibility of such a response is obscured when Meléndez Gutiérrez portrays Haze’s argument as involving a stipulation that ‘n’ is to refer to an unreferred-to object. On the contrary, Haze’s argument does not assume that expressions like ‘n’ work by means of referring at all, and simply lets stipulations like ‘Let n be such an x’ be themselves, without imposing a theory on them. Once this is clarified, we can see that Haze’s argument is unaffected by Meléndez Gutiérrez’s objection.