Vagueness, epistemicism and theism
March 1, 2011 — 12:08

Author: Alexander Pruss  Category: Divine Command Existence of God  Tags: ,   Comments: 15

Epistemicists say that our vague natural language is, in fact, fully sharp. If I place grains of sand onto a sheet of paper, there will eventually be a grain of sand such that prior to placing it, there was no heap, and after placing it, there was a heap. We don’t know which grain it is, but we know there is one on the basis of the following argument. Let Gn be the sand after the nth grain has been placed. Then, G1000000 is a heap, and G1 is not a heap. It is a logical consequence of this that there is a number n, between 1 and 1000000, such that Gn is not a heap and Gn+1 is. And it’s obvious that there is no number n which we know to be as above. So, epistemicism is true–there is a boundary, and plainly we don’t know where it lies.

The above is a very plausible argument. But it runs into two kinds of problems. First, the incredulous stare: it just doesn’t seem like there should be such an n. This has some force, but only if the alternative to epistemicism is something other than revising logic. Plus the epistemicist can give a good explanation of why we are mistaken here. We have a tendency, often exploited by anti-realists, especially in ethics and aesthetics, of confusing what we cannot know with what there is no fact about. Still, the incredulous stare does indeed have a pull on me here.

Second, there is this argument: Language is defined by our practices. Our practices underdetermine which number n is such that Gn fails to fall under the predicate “is a heap” but Gn+1 does fall under it. But something falls under the predicate “is a heap” if and only if it is a heap. Hence, there is no fact about which number n is such that Gn is not a heap but Gn+1 is. One might try to deny that language is defined by our practices or that our practices underdetermine the number n, but unless there is a theory of how language is defined in such a way as to determine the number n, this is intellectually unsatisfying.

But theism seems to make it possible to be an intellectually fulfilled epistemicist.

For the theist can accept the following theory. God thinks perfectly precise thoughts with no vagueness of any sort. Our language comes to us from God. Just as my use of the words “Beijing” and “quark” get their meanings from other people’s earlier use of it, so too our language ultimately gets its meaning from God’s decision as to what should mean what. God thinks perfectly sharply, and then set perfectly sharp boundaries for human language’s predicates. He didn’t, in general, inform us as to the perfectly precise independent specifications of the boundaries. But our language is, nonetheless, perfectly precise.

There are two paths to further development. One path has it that each time our practices have seemingly created a new predicate, God was behind the creation. Jones hears an idea and says it is “cool”. No one has used the word earlier in this sense, and the usage takes off. But, in fact, Jones didn’t introduce the word by herself in to the language. God cooperated with Jones and filled out the vagueness in Jones’ concept of the “cool”.

The second path is that God defined precise rules by which bits of language gain meaning. These rules are every bit as precise and deterministic as the laws of Newtonian physics. These rules specify what exactly falls under the predicate “is cool” when the predicate is introduced by Jones in such-and-such a way under such-and-such circumstances.

Both paths further divide into two variants: a constitutive and a causal variant. On the constitutive variant, God’s intentions as to what should mean what (specifically in each case, on the first path, and under more general descriptions, on the second) at least partly constitute what meaning-facts there are–that would be akin to divine command theory (in its divine-will variant). Perhaps God participated in giving us a rule of language formation or a bit of language, and we inherit this in the way in which on Kripke’s theory the meaning of “Socrates” is inherited by us from earlier uses by others.

On the causal versions, God causes meaning-facts. These meaning-facts may be embedded in the natures of speakers–that would be a Natural Law version–or they may be “out there” (wherever “there” is). (I like the Natural Law version most.)

This gives us a cool argument:

  1. (Premise) G1 does not fall under “is a heap”.
  2. (Premise) G1000000 does fall under “is a heap”.
  3. There is a number n such that Gn+1 falls under the predicate “is a heap” and Gn does not. (Follows by classical logic from 1 and 2)
  4. (Premise) The best explanation of (3) is that human language was created by an agent whose thoughts suffer from no sort of vagueness.
  5. (Premise) Every non-supernatural agent’s thoughts suffer from some sort of vagueness.
  6. Probably, there is a supernatural agent whose thoughts suffer from no sort of vagueness and who created human language. (Inductively from 3-5)

I am not endorsing epistemicism. I am still pulled to thinking the sentence-proposition relation is many-many and that classical logic governs propositions rather than sentences. But the above line of thought, and the comparison with Natural Law, makes epistemicism very attractive to me. If God, in creating human beings, can create them with a nature that grounds normative facts about them, he can create them with a nature that defines meanings as well. (Moreover, there may be a reduction of meaning to normativity. Here’s an example of the form such a reduction might take: a type A of action is an asserting that p if and only if A ought to be refrained from unless p.)

  • Ted Poston

    Hi Alex,
    Nice post! You do a good job motivating epistemicism via classical logic. One point at which your theistic story differs from Williamson’s account is over the asymmetry of truth and falsity. As I recall (it’s been quite a while since I’ve looked this up), Williamson holds that a statement with a vague predicate goes false at the first hint of less than perfect use. As an analogy think of supervaluationalist approaches: some sentences are super-true, other’s true on some admissible precisifications, etc. As the analogy re the asymmetry of truth and falsity goes, for Williamson a sentence is true iff it’s super-true. On your theistic account, you preserve an intuitive symmetry between truth and falsity. It’s interesting to note that often symmetry principles can be used to argue against epistemisticism.
    So here’s the question I had, but first a story. Suppose I invent (or, am the originator of) of language L. L contains a predicate ‘B’ that I give it perfectly precise rules for the use of ‘B’. Now, L migrates to another community. They don’t learn all the rules for L but do well enough to count as using L. But one thing I notice about there use of ‘B’. There’s systematic changes in how they use ‘B’ for penumbral cases. People refuse to use ‘B’ or ‘~B’ for those cases. People think there’s just no fact of the matter. Some theorist of L have develop alternative semantics for ‘B’. Etc. So the question: does ‘B’ mean the same thing in their mouths as in mine? [For an example of a phenomenon like this look at how the definition of ‘nonplussed’ has changed]

    March 2, 2011 — 7:57
  • Mike Almeida

    For the theist can accept the following theory. God thinks perfectly precise thoughts with no vagueness of any sort.
    That God thinks perfectly precise thoughts certainly does not entail anything like epistemicism, and I don’t think you’re suggesting that it does. Supervaluationists can agree that God has precise thoughts (but again I’m not sure why this is necessary). Supervaluationists can also agree that we can have such thoughts, too. The view that vagueness is in the language is consistent with the view that semantic decisions can be made that fully precisify the language. It would be hard to know what the motive might be to do so, nonetheless, we can. God might be a supervaluationist who has made such a decision.

    March 2, 2011 — 8:21
  • Ted:
    The point at which the statement goes false is surely vague (at least epistemically). Does the Williamson view support the common definition of falsity: p is false iff ~p is true?
    The drift question is interesting, and different versions of the theistic epistemicism may have different answers at some level. If God is directly involved in every new introduction of a term into the language (this is like occasionalism), then the question is going to be: Did God decide to make people use ‘B’ with a new meaning? If God puts into place perfectly precise rules for language introduction (and I take these rules to be linguistic in a general sense–they govern the significance of the gestures and acts by which we introduce bits of language, and these gestures an acts are linguistic in a general sense), but which we nonetheless do not know the precise content of, then there will also be an answer to the question, but we won’t know it with certainty, since we don’t know the rules with certainty.
    Nonetheless, on both views, we have good reason to think that the rules are such as to ensure that the meanings typically match usage pretty well (otherwise, scepticism about what we mean ensues), and this is compatible with both the occasionalist picture and the rule-based picture (occasionalist certainly can say that there are patterns in divine decisions). If so, then we will have good reason to think that we’re dealing with a new word.
    There will, of course, be epistemically vague cases of semantic shift, in which we won’t be able to tell whether we’re dealing with a new word or not, but on the view, there is an answer.
    Indeed, I didn’t think that the claim that God thinks perfectly precise thoughts entails epistemicism. The claim does entail that there is no “vagueness in the world”, but one can interpret supervaluationism as compatible with that.
    Currently for me one of the biggest problems with supervaluationism is the problem (I don’t know who was the first one to point it out; I got it from Sorensen’s SEP entry) that it comes out as super-true that there is an n such that (a) G_n is not a heap and (b) G_{n+1} is a heap. Since the point of supervaluationism was to protect the intuition that there is no such n, standard supervaluationism undercuts its own motivation. I am inclined to think this is a merely technical problem, to be overcome with technical resources, but I could be wrong about that.

    March 2, 2011 — 9:12
  • Ted Poston

    It’s been a long time since I’ve thought about Williamson’s asymmetry claims. The relevant section is pp. 205-209 in his Vagueness book. On p. 208 he writes, “the concepts of truth and falsity are not symmetrical. The asymmetry is visible in the fundamental principles governing them, for (F) is essentially more complex than (T), by its use of negation. The epistemic theorist can see things way: if everything is symmetrical at the level of use, then the utterance fails to be true, and is false in virtue of that failure. In that sense, truth is primary. At the level of truth and falsity, there is no symmetry to break.” (p. 208)
    Re your problem with supervaluationism: that’s the problem of higher-order vagueness. One of the major advantages of Williamson’s view is that it handles higher-order vagueness as well.

    March 2, 2011 — 9:38
  • Mike Almeida

    Currently for me one of the biggest problems with supervaluationism is the problem (I don’t know who was the first one to point it out; I got it from Sorensen’s SEP entry) that it comes out as super-true that there is an n such that (a) G_n is not a heap and (b) G_{n+1} is a heap.
    That’s not a problem, or shouldn’t be. Certainly there is an n such that Gn is not a heap and Gn+1 is a heap. There is always a sufficently large increment n to make that true. There might even be a smallest increment n such that that is true. But what would be a problem is if, for some increment n, Gn is not a heap and Gn+1 is a heap such that there is no smaller increment n-j such that Gn-j is not a heap and G(n-j)+1 is also not a heap. If there is no such smaller increment, then you’ve got a precise border at some level of vagueness. But even if there is such a border it’s infinitely high. I don’t think that presents too serious a problem, since it is nonetheless true that any *possible border* you might select between that G’s and non-G’s at any level is vague. There won’t be any closest border just as there is no earliest time after 8am. To put it another way, I’d worry about such an argument if I thought it also solved the cable guy paradox. I’m confident it doesn’t.

    March 2, 2011 — 10:18
  • Mike:
    The increment is one. We’re adding one grain of sand. G_n is a heap of sand with n grains and G_{n+1} is a heap of sand with n+1 grains.
    Well, I accept a complexity asymmetry between truth and falsity. Falsity is the truth of the negation, so it is more complex.
    I don’t see how the problem of the existence of such an n is the problem of higher order vagueness, but I may be missing something obvious. I took the problem of higher order vagueness to be the problem that “definitely bald” are “it’s supertrue that x is bald” also vague. But the problem Sorensen points out is a different one–the problem is that the sentence “There is an n such that G_n is not a heap but G_{n+1} is a heap” is true for every precisification of “heap”, and hence is super true.

    March 2, 2011 — 11:50
  • Ted:
    By the way, the motivation of epistemicism that I’m using is from Sorensen’s “Metaphysics of Words”.

    March 2, 2011 — 11:52
  • Ted Poston

    Thanks for the Sorensen paper! Re the drift issue: I’m not clear on how your response goes. You say that “we have good reason to think that the rules are such as to ensure that the meanings typically match usage pretty well.” But in the drift issue I took it to be false that meaning constituting rules match the usage. Suppose that A is a borderline case of B. 1/3 of the community asserts the sentence ‘A is B’; 1/3 asserts the sentence ‘A is not B’; and the other 1/3 refuse to assert or deny, thinking, perhaps there’s just no fact of the matter. On your theistic approach the last third are definitely wrong and one of the other is definitely right. One important thing to observe here is that the speakers of the community don’t defer to God for the use of ‘B’. It’s not like they point to the first introduction of the term and give its introductory use as the baptism (so to speak). It looks like ‘B’ has drifted away from its original precise use by God. (Note that as you set it up: the use of a term by the meaning-constituting community (=God) is perfectly precise; God uses predicates like ‘bald’ in a perfectly precise manner. But for Williamson, the meaning-consistuting community (=us) the usage is not perfectly precise. This is where Williamson tells the extra story about a sentence going false at the first sign of divergence in usage. I think there are some important differences re theistic epistemicism and Williamson’s epistemicism around this area)
    Re higher-order vagueness: The problem Sorenson points to is a problem that arises from precisifying languages (a higher order problem). Let ‘b’ be a constant in the penumbral region. It’s not supertrue that if b grains is not a heap then b+1 is a heap; we can draw the boundary different ways. But given that we are going to draw a precise boundary it’s supertrue there is an n such that G_n is not a heap but G_{n+1} is a heap. Another way to see it: the original vagueness problem is to give an account of vague predicates. Supervaluationalism does that via precisifications. But now it faces the problem that on any precisification there’s going to be a sharp cutoff.

    March 2, 2011 — 13:25
  • Ted:
    Thanks for the clarification on the sharp cutoffs existing on each precisification.
    “It looks like ‘B’ has drifted away from its original precise use by God.”
    I am sorry–I misunderstood your original example through not reading carefully enough. I now see the problem.
    I think there are two possibilities. One is that God has in fact shifted the boundary for ‘B’, so now we have a genuinely new term, but the way the new boundary is positioned (maybe it’s a messy and jagged frontier) makes the folk particularly apt to erroneously think there is no fact of the matter. The other is that the folk is just wrong here.
    But your point does connect up with one of my own reasons for accepting something like supervaluationism, namely the possibility of underspecified terms. Suppose I tell you this. I am going to use the term “xyzzy” from time to time and:
    – anything whose electric charge is more than exactly +10 Coulombs counts as xyzzy
    – anything whose electric charge is less than exactly -10 Coulombs counts as not xyzzy.
    I say no more. I simply don’t tell you anything about what happens from -10C to +10C, inclusive. And I haven’t bothered to figure it out for myself, either. I have no opinion on what happens between -10C and +10C (not even an opinion on whether there is a fact of the matter), because it doesn’t practically matter in the contexts in which I use “xyzzy”.
    On the theistic epistemicism, it seems that either (a) God decides where the boundary for “xyzzy” lies or (b) the introduction of “xyzzy” fails, and sentences using “xyzzy” are meaningless.
    But (a) just seems implausible to me. For one, my stipulation was perfectly symmetric between positive and negative charge. But a precisification would have to specify whether something with exactly zero electric charge is xyzzy or not xyzzy, and there just seems to be no reason for it to be one or the other. On what grounds would God decide this? (This is not insurmountable. After all, maybe God can just decide based on what is good for us. Or maybe God can initiate a coin-flip process that decides this.)
    On the other hand, if we accept (b), then we have the problem that the following arguments are valid:
    A1. Your hair is xyzzy.
    A2. So, your hair does not have a charge lower than -20C.
    B1. Your hair has a charge greater than 20C.
    B2. So, your hair is xyzzy.
    Yet how can they be valid if A1 and B2 are meaningless?
    This applies not just to predicates. One can also do it with connectives. I “introduce” the connective “ond”. I tell you that “p ond q” is true if p and q are both true and “p ond q” is false if p and q are both false, and I specify that “ond” uses the introduction rule for conjunction and the elimination rule for disjunction. The resulting system is consistent (albeit incomplete) and we can reason just fine with it. Does God really bother to decide, or does God really institute laws that decide, what the full truth table of “p ond q” is?
    I suppose the theistic epistemicist can bite the bullet on these cases in either way. The options–say that God does in fact determine the underspecified regions or say that the terms are meaningless–do not at least violate classical logic, which is more than one can say for many non-epistemicist solutions.

    March 2, 2011 — 14:06
  • Interestingly, it seems that this theistic epistemicism can solve the problem with the simple epistemicist view that Hawthorne finds in the case of “person”.
    I also think what I’m giving ends up being a species of the hyperinflationism Hawthorne is talking about.

    March 2, 2011 — 14:24
  • Mike Almeida

    Mike: The increment is one. We’re adding one grain of sand. G_n is a heap of sand with n grains and G_{n+1} is a heap of sand with n+1 grains.
    Alex, sure, of course, the increment is 1. But 1 what? If you go by grains then certainly there will be a sharp cut off. But that presents no problem at all. What could be the problem? The problem arises if there is no increment smaller than a grain (let that be G-, a slightly smaller grain) such that G- is not a heap and G- +1 is not a heap. But there is such a smaller grain. In order for there to be a sharp cut off, there must be some increment Gi in grains such that Gi is not a heap, Gi + 1 is a heap, and for all increments Gi- smaller than Gi, it is also true that Gi- is not a heap and Gi- + 1 is a heap. That is true iff. there is a sharp border between G’s and non-G’s. But as I mentioned, even if there is a sharp border at the limit, it is true of every possible border that it is non-sharp.

    March 2, 2011 — 16:43
  • Mike Almeida

    Maybe a word of clarification would help. If there is a discrete transition between heaps and non-Heaps then it must be true that for any increment Gi of any dimension it is true for some i added to some amount of sand that Gi is not a heap and Gi+1 is a heap. It’s not enough to show that there is some increment Gi of some dimension such that, for every precisification of ‘Heap’, Gi is not a heap and Gi+1 is a heap. To take an obvious example: let the increment be the entire heap of sand. We begin at Gi = 0, which is not a heap and Gi+1 (the entire heap) which is a heap, which is true on every way of precisifying ‘heap’. Presumably the same will be true for half-heaps of sand. Does Sorensen claim to have a sound proof that, for any increment i of any dimension, it is true for some i added to some amunt of sand that Gi is not a heap and Gi+1 is a heap?

    March 2, 2011 — 17:26
  • John:
    I agree completely. It depends on the surrounding environment. It also depends on the size of the grains and the shape of the pile. (If it’s too flat it’ll be more a hump than a heap, and if it’s too sharp, it’ll be a tower or column.)
    That’s why instead of asking about numbers n such that n grains are a heap, I suppose a single process of placing grains on a particular piece of paper, and ask at which point in this particular process a heap came into existence. (Interestingly, it could be that adding a grain of sand to the heap makes it cease to be a heap, too, by causing an avalanche that makes the shape be wrong for a heap. But I did not assume in the argument that if G_n is a heap, then G_{n+1} is. I guess I should have also assumed that the 1000000 grains aren’t spread out so much as not to be a heap.)
    Here’s a fun question: What is the smallest number n such that it is logically possible to have a context in which n grains of sand to be a heap of sand? I think n is no greater than four. There will be contexts in which four grains of sand, three on the bottom layer and one on the top layer in a pyramid shape, will be a heap. Imagine a bare environment, large grains of sand, and logically possible observers that are the size of small ants. Could one do it with three? I think so. It would be harder to balance them, but if they are sufficiently irregular, one could have two on the bottom and one on top, and they wouldn’t fall. Could one do it with two? I think “heap of Fs” requires at least three Fs (one could heap two Fs on top of each other, but they wouldn’t form a heap). So, if I am right, then there is a non-vague answer to this question–it’s two.
    Likewise, there is probably an answer to the question of what is the smallest number of terminal scalp hairs that one could have and be non-bald in some context (you can have lots of vellus hair and not be bald). I think the answer is one. No vagueness there. (I still sometimes wonder whether philosophers don’t misuse the term “bald” in talking of the Sorites paradox. I am not 100% sure it’s actually true that someone who has one terminal hair counts as bald. But enough people use the example, and they unlike me are native speakers of English, that I am pretty happy to defer.)
    The vagueness comes in in these cases when we fix a context.

    March 3, 2011 — 8:51
  • Brandon Reams

    Could someone refer me to some introductory and moderate works in the area of supervaluation/epistemicism?

    March 3, 2011 — 21:52
  • Mike:
    You can prove this in classical logic for any interval that has the property that a finite number of applications of the interval will span the distance between the extremes. (So it won’t work for an infinitesimal interval.) This is because in classical logic, given names a_1,…,a_n, we can prove from: ~P(a_1) and P(a_n) the disjunction:
    (~P(a_1) and P(a_2)) or (~P(a_2) and P(a_3)) or … or (~P(a_{n-1}) and P(a_n)).
    So if the disjunction is true, it must have a true disjunct, and hence there must be a k such that ~P(a_{k-1}) and P(a_k).
    Here’s a sketch of a proof of the big disjunction. Let D be the big disjunction. Then by applying De Morgan (once to the whole disjunction, and then once to each component), ~D holds if and only if:
    (P(a_1) or ~P(a_2)) and (P(a_2) or ~P(a_3)) and … and (P(a_{n-1}) or ~P(a_n)).
    For a reductio, suppose ~D. Then by conjunction-elimination we have:
    P(a_1) or ~P(a_2)
    P(a_2) or ~P(a_3)

    P(a_{n-1}) or ~P(a_n).
    We also have:
    Now, from ~P(a_1) and (P(a_1) or ~P(a_2)), you can prove ~P(a_2) by classical logic. From ~P(a_2) and (P(a_2) or ~P(a_3)) you can prove ~P(a_3) by classical logic. Continuing this procedure, we generate a proof of ~P(a_n). But that contradicts P(a_n), and hence by reductio ad absurdum, we need to reject our assumption ~D. Thus, D follows by classical logic from ~P(a_1) and P(a_n).

    March 4, 2011 — 8:39