[**Note: All vagueness posts are archived HERE.] **

In this post, I proposed a solution to the notorious Evans/Salmon argument against vague identity, one rooted in my work on Moore's Paradox. Here I want to generalize the argument in a way that I haven't seen it generalized in the literature yet. The literature is pretty big and I've barely dented it, so apologies if this has been done before.

From what I've read thusfar, there seems to be an assumption that if one could have vague predication without vague identity, then one wouldn't need to worry about the Evans/Salmon argument. This is my understanding of why there is a literature on whether one can have ontic vagueness without vague identity. But I've long nursed the suspicion that the assumption might be an artifact of participants in the debate restricting themselves to the first order lambda calculus. If one allows lambda abstraction over predicates in modal logic one can get a version of the argument that does not involve identity.

First, to be clear about the parallels, let's present the original Evans/Salmon argument using proper notation. Let "▽" stand for "it is indeterminate whether" upside down triangle, "λ" stand for what it does in the lambda calculus, and " " be the absurdity constant. Then, when fully expressed in a natural deduction system, the argument is:

- ▽(a = b) assumption
- λ
*x*[▽(*x*= b)]a 1, lambda abstraction - ¬▽(b = b) truism
- | a = b assumption for ¬ introduction
- | | λ
*x*[▽(*x*= b)]b assumption for ¬ introduction - | | ▽(b = b) 5, lambda cancellation
- | | 3,6 ¬ elimination
- | ¬λ
*x*[▽(*x*= b)]b 5-7 ¬ introduction - | λ
*x*[▽(*x*= b)]b 2.4 = elimination - | 8,9 ¬ elimnation
- ¬(a = b) 4-10 ¬ introduction

Again, I explain this more and analyze why I think this does not work as an argument against the ontic indeterminacy of identity claims here. But let's assume that it does. If we let our supposedly indeterminate claim be that it is necessarily the case that properties P and Q are coextensive, we can come up with a homologous argument. Consider:

- ▽◻∀
*x*(P*x*↔ Q*x*) assumption - λ
*X*[▽◻∀*x*(*X**x*↔ Q*x*)]P 1, lambda abstraction - ¬▽◻∀
*x*(Q*x*↔ Q*x*) premise - | ◻∀
*x*(P*x*↔ Q*x*) assumption for ¬ introduction - | | λ
*X*[▽◻∀*x*(*X**x*↔ Q*x*)]Q assumption for ¬ introduction - | | ▽◻∀
*x*(*Q**x*↔ Q*x*) 5, lambda cancellation - | | 3,6 ¬ elimination
- | ¬λ
*X*[▽◻∀*x*(*X**x*↔ Q*x*)]Q 5-7 ¬ introduction - | λ
*X*[▽◻∀*x*(*X**x*↔ Q*x*)]Q 2.4 semantics for λ calculus - | 8,9 ¬ elimnation
- ¬◻∀
*x*(P*x*↔ Q*x*) 4-10 ¬ introduction

From a proof-theoretic standpoint, the only fishy inference is from 2 and 4 to line 9. But if two predicates P and Q are necessarily coextensive, then as far as I can tell it should follow in any reasonable semantics from any Φ(P) that Φ(Q) (though one should check this with Montague's treatment).

One could try to run the argument with respect to higher order extensional lambda calculus on ▽∀*x*(P*x* ↔ Q*x*). In this language (P*x* ↔ Q*x*) should be sufficient to go from Φ(P) that Φ(Q). And it would be *much * easier to find examples where we intuitively want the indeterminacy to hold. But the problem is that ▽ itself behaves enough like a modal operator* so that the inference to step 9 looks *very* fishy. I don't know though. I also have the intuition that that the original argument is fishy in the same place (I'll have a post on this in the next few days), but that's probably because I don't find Kripke's arguments about the necessity of identity compelling. . .

In any case, I hope I've said enough to make clear that prospects for generalizing the Evans/Salmon argument via more powerful logics is at the very least an interesting topic.

[Notes:

*I haven't read Williamson's piece yet where he argues that s5 is the appropriate logic for vagueness. This intuitively seems wrong to me, for reasons I gave here. But Williamson knows what he's talking about, so I need to read the piece.]

- Is the Evans/Salmon argument against metaphysical indeterminacy merely a case of Moorean paradoxicality?
- Vagueness versus (Wilsonian/Brandomian) Underdetermination
- some problems for Elizabeth Barnes' account of vagueness
- Vagueness Notes 4 - Saving Barnes from the Wilson and Cogburn criticisms
- Vagueness notes 5 - Relevant HTML symbols (also, can someone fix the logical symbols Wikipedia page?)