By Jon Cogburn
In AN EARLIER POST I generalized Evans' original argument against ontic vagueness to suggest a counterargument to those onticists who would respond to Evans by defending vague objects without vague identity. Here I want to do something similar, but aimed at semanticists who argue that vagueness lies in our representations of objects, not objects themselves.
(1) Evans' Argument and Semanticism to the Rescue
In natural language Evans' argument goes like this. Assume that it is indeterminate whether a and b are identical. Then a has the property of being such that it is indeterminate whether b is identical with a. But a does not have the property of being such that it is indeterminate whether it is identical with itself. So, b and a have distinct properties and are hence not identical with one another. But now on the assumption that it is indeterminate whether b is identical with a we have proven that b is not identical with a. But then any case of indeterminate identity will also be a case of flat out not being identical, which seems contradictory.*
Per contra, a represenation is vague when the relation of the representing system to the represented system is not one-one, but one-many. For example, a photograph which is so smudged that it might equally represent Brown or Jones or Robinson is vague. A small-scale map is usually vaguer than a large-scale map, because it does not show all the turns and twists of the roads, rivers, etc. so that various slightly different courses are compatible with the representation that it gives. Vagueness, clearly, is a matter of degree, depending upon the extent of the possible differences between different systems represented by the same representation. Accuracy, on the contrary, is an ideal limit (Keefe and Smith 2002, 66).
This easily becomes the view that reality is precise and that this is something more or less captured by more or less accurate representations. With a little bit of Quinean optimism you can even see the job of the philosopher as providing a "canonical notation" for science which is 100% accurate in this sense.
The semanticist's short answer to Evans' argument is thus just to note that in an ideal language appropriate for science it is never indeterminate whether b equals a, so we shouldn't worry about whether b's identity with a entails that it is not indeterminate whether b equals a.
If we regiment Evans' argument we can better appreciate the semanticist's long answer, which cant be freed from the Quinean mythology. Let "▽" stand for "it is indeterminate whether," "λx" stand for what it does in the lambda calculus (the property of x such that), and " " be the absurdity constant. Then, when fully expressed in a natural deduction system, the argument is:
- ▽(b = a) assumption
- λx[▽(x = a)]b 1, lambda abstraction
- ¬▽(a = a) truism
- | b = a assumption for ¬ introduction
- | | λx[▽(x = a)]a assumption for ¬ introduction
- | | ▽(a = a) 5, lambda cancellation
- | | 3,6 ¬ elimination
- | ¬λx[▽(x = a)]a 5-7 ¬ introduction
- | λx[▽(x = a)]a 2.4 = elimination
- | 8,9 ¬ elimnation
- ¬(b = a) 4-10 ¬ introduction
The key inference here is the first one. For the semanticist it is not indeterminate whether b equals a but rather whether a is referred to by "b". But then the first two lines of the proof would look like this:
- ▽("b" refers to a) assumption
- λx[▽("x" refers to a)]b 1, lambda abstraction
But if one handles the indeterminacy of the reference of "b" in the way suggested by Russell, and handled by supervaluationist versions of semanticism, then the lambda abstraction involves a de-dicto/de-re fallacy.** That the word "b" indeterminately refers to distinct objects, including a, does not entail that there is some individual b such that it has the property attributed to it in line 2 of the proof. The virtue of recognizing this is that one can adopt a theory of vagueness of the sort suggested by Russell without holding that we need to replace ordinary language and run of the mill properties with something Quine would have found kosher. Instead, we just offer a supervaluationist semantics for ordinary, vagueness tolerant, predicates.
One upshot to this is that semanticism is a little closer to Ungerian nihilism that we might have thought. Keefe and Smith write,
For example, the indeterminacy of the identity statement "Barney = P" (where P is an associated p-cat) could be taken to show that it is indeterminate to which of hte precise p-cats "Barney" refers. There is, then, no vague object that is Barney--indeed there is no unique object which is determinately the cat of that name (Keefe and Smith 2002, 52-3).
Pretty cool. In addition, both Kamp and Fine showed how degree theoretic semantics fit nicely with supervaluationism, which allows the supervaluationist to avail herself to degree theoretic blocking of the sorites paradox.
(2) Evans' Argument Against the Semanticist
So we seem to have that Evans' argument presents a special problem for the defender of ontic vagueness, but not one for the semanticist. I don't think this is right though, and suspect that people who find it plausible have been underestimating the power of the lambada calculus. In THIS POST I constructed an Evans type argument in terms of properties of type <e,t> (those canonically expressed by monadic predicates). The upshot of that is that Evans' argument threatens more than just vague objects, but vague properties as well.
From this, there should be no problem with constructing an argument for higher types. One that lambda abstracts over expressions type <t> should cause problems for vague states of affairs and one that lambda abstracts over expressions of type <e<e,t>> should cause problems for vague relations. Let's try it out with respect to the reference relation!
In what f0llows let R"x"y denote that x refers to y. Also, assume that "◻" is hyperintensional in the sense that "◻P" means that P is true at all worlds, possible or impossible. Then, two relations R and R' will be hyperintensionally the same if ◻∀xy(Rxy ↔ R'xy). This says that in every possible and impossible world they relate the same things. Unless one is a rabid Quinean (and some of my best friends are) this does not commit one to the existence of impossible worlds. In any case, one could run the same argument just using different notation for sameness of content. Or one could make sense of a hyperintensional box without talking about possible worlds. I use this notation because it tracks Montague's definition of identity at higher types, and thus shows how Evans' argument is generalizable. Of course Montague couldn't handle hyperintensionality, but impossible worlds fixes the problem easily.
So, again where R"b"a says that "b" refers to a, let us run Evans' argument on R.
- ▽◻∀xy(Q"x"y ↔ R"x"y) assumption (stating that it is indeterminate whether R and Q are the same reference relation)
- λX[▽◻∀xy(Q"x"y ↔ X"x"y)]R 1, lambda abstraction
- ¬▽◻∀xy(Q"x"y ↔ Q"x"y) premise (stating that it is not indeterminate whether Q is the same reference relation as itself)
- | ◻∀xy(Q"x"y ↔ R"x"y) assumption for ¬ introduction
- | | λX[▽◻∀xy(Q"x"y ↔ X"x"y)]Q assumption for ¬ introduction
- | | ▽◻∀xy(Q"x"y ↔ Q"x"y) 5, lambda cancellation
- | | 3,6 ¬ elimination
- | ¬λX[▽◻∀xy(Q"x"y ↔ X"x"y)]Q 5-7 ¬ introduction
- | λX[▽◻∀xy(Q"x"y ↔ X"x"y)]Q 2.4 semantics for λ calculus (by substitution from 2, since 4 says that R and Q are hyper-cointensional)
- | 8,9 ¬ elimination
- ◻∀xy(Q"x"y ↔ R"x"y) 4-10 ¬ introduction
The punchline here is that semanticism has no advantage over ontic accounts of vagueness, at least as far as Evans' argument is concerned. It was supposed to be reasonable to block the move from premise 1 to 2 in the original Evans argument because of the vagueness of the reference relation. But what analogous move would block the lambda abstraction with respect to the reference relation itself? I'm not seeing it. Meta-semanticism? What would that be?
(3) Further Questions
(3A) Merricks: (2001) argument:
Akiba and Abasnezhad attribute the following thought to Trenton Merricks:
To recall, supervaluationism postulates partial references to various precisifications: it holds that a vague expression partially refers to one precisification, partially refers to another precisificaiton, etc. But what is a partial reference? It's an indeterminate reference. So if epistemicism is incorrect and there are indeed partial references in reality, there ought to be ontic indeterminacy in reference relations. But simply, language is part of the world (Akiba and Abasnezhad 2014, 6).
I haven't read Merricks' paper yet, but I hope that the above argument can be seen as strengthening it.
(3B) Ontic Supervaluationism:
Ontic supervaluationists take vagueness to be in the world, but still find the basic framework of supervaluationism to be helpful in making sense of it. For example in Elizabeth Barnes' (2010) version vague objects are such that it is ontically indeterminate which of several possible determinate objects are the actual object in question. I think that ontic supervaluationists typically try to find other reasons to deny the lambda abstraction, in effect following David Lewis and defending de dicto (in premise 1) indeterminacy but not de re indeterminacy (which would license the lambda abstraction). I have no idea if or how the above argument intersects with such attempts. Barnes (2009) should provide guidance here, since Evans is in the title. It will be fun to read it and think about higher-order Evans arguments.
In THIS POST, I proposed a friendly amendment to Barnes' theory to save it from criticisms from Jessica Wilson and myself. I quite like the amended neo-Barnesian theory.***
All of the literature I've read thus far says that epistemicism, the view that both the world and our reference relations are in fact precise, doesn't face the kind of tu quoque arguments Merricks and I are raising for semanticist views. My intuition is that this is not the case, that in fact the epistemicist trades on sorites susceptible entities (such as how reasonable it is to believe something) when trying to make epistemicism sound plausible. But I haven't read Williamson's book on knowledge yet and fear that whatever weird stuff he says there is actually working to pre-emptively deflect such an argument.
*You can strengthen the argument to a clear contradiction if you add some modal principles. Contraposition gets you something very much like Kripke's necessity of identity. If two objects are identical, then they are determinately so. I'll consider some of this stuff in one or more future blog posts.
**See the introduction to Keefe and Smith as well as Akiba and Abasnezhad for a further discussion of this point.
***Though I'm alternatively drawn to denying premise 3. Huge swaths of the history of philosophy take it to be the case that self-identity is something earned, not given. I would love to see if Rescher or Seibt's analytic versions of process metaphysics elides or preservers this. In contrast to this Hegelian take on Evans argument, I earlier (and not very clearly) HERE suggested a more Kantian one, where the argument itself is a case of Moorean paradoxicality. The basic idea is that Evans showed that we're forced to think of distinct entities as determinately distinct, but this is of a piece with the idea that we believe anything we sincerely assert. But the universal assertibility of P therefore I believe that P, does not mean that you believe all P! There's an analogue with Berkeley's master argument about conceiving the inconceivable as well. I need to do a clearer post on this with respect to the proof theory of all three arguments.]
Akiba, Ken and Ali Abasnezhad, ed. 2014. Vague Objects and Vague Identity: New Essays on Ontic Vagueness. Dordrecht Springer.
Barnes, Elizabeth. 2009. "Indeterminacy, Identity and Counterparts: Evans reconsidered." Synthese 168, 81-96.
Barnes, Elizabeth. 2010. "Ontic Vagueness: A guide for the perplexed." Nous 44, 601-27.
Keefe, Rosanna and Peter Smith, ed. 2002. Vagueness: A Reader. Cambridge: The MIT Press.
Merricks, Trenton. 2001. "Varieties of Vagueness." Philosophy and Phenomenological Research, 62, 145-167.
- 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?)
- Vagueness notes 6 - A Proposed Generalization of the Evans/Salmon Argument (Not Involving Identity)
- Vagueness Notes 7 - Did Peter Simons discover a shorter, lambda free, version of the Evans/Salmon argument?