Vagueness survey paper: II (vague metalanguages)

Ok, so here’s part II of the paper. One thing that struck me having taught this stuff over the last few years is how much the “vague metalanguages” thing strikes people as ad hoc if brought in at a late stage—it feels that we were promised something—-effectively, something that’d relate vague terms to a precise world—and we’ve failed. And so what I wanted to do is put that debate up front, so we can be aware right from the start that there’ll be an issue here.

When writing this, it seemed to me that effectively the Evans argument arises naturally when discussing this stuff. So it seemed quite nice to put in those references.

Sorry for the missing references, by the way—that’s going to be fixed once I look up what the style guide says about them. (And I’ll add some extras).


One textbook style of semantic theory assigns extensions (sets of objects) as the semantic values of vague terms (Heim & Kratzer xxxx). This might seem dubious. Sets of objects as traditionally conceived are definite totalities—each object is either definitely a member, or definitely not a member. Wouldn’t associating one of such totalities with a vague predicate force us, unjustifiably, to “draw sharp boundaries”?

On the other hand, it seems that we easily say which set should be the extension of “is red”:

[[“red”]]={x: x is red}

There’s no need for this to be directly disquotational:

[[“rouge”]]={x: x is red}

The trick here is to use vague language (“in the theorist’s metalanguage”) to say what the extension should be (“of the object-language predicate”). If this is legitimate, there’s no obstacle to taking textbook semantics over wholesale.

Perhaps the above seems unsatisfactory: one is looking for illumination from one’s semantic clauses. So, for example, one might want to hear something in the semantics about dispositions to judge things red, to reflect the response-dependent character of redness. It’s highly controversial whether this is a reasonable demand. One might suspect that it mixes up the jobs of the semanticist (to characterize the representational properties of words) and the metaphysician (to say what redness is, in more fundamental terms). But even if one wants illumination wherever possible in one’s semantic theory, there’s not even a prima facie problem here, so long one is still able to work within a vague metalanguage. Thus Lewis, in discussing his semantic treatment of counterfactuals in terms of the (admittedly vague) notion of similarity, says “I … seek to rest an unfixed distinction upon a swaying foundation, claiming that the two sway together rather than independently.” (Lewis, 1973, p.92). While Lewis recognizes that “similarity” is vague, he thinks this is exactly what we need to faithfully capture the vagueness of counterfactuals in an illuminating way. One might see trouble for the task of constructing a semantics, if one imposed the requirement that the metalanguage should (at least ideally?) be perfectly “precise”. But one would have to be very clear about why such a strong requirement was being imposed.

Let us leave this worry to those bold enough to impose such constraints on the theorist of language. Are there further problems with textbook semantics?

One worry might be that the resources it appeals to are ill-understood. Let us go back to the thought that sets (as traditionally conceived) are definite totalities. Then borderline-bald Harry (say) is either definitely in or definitely out of any given set. But Harry better be a borderline member of {x: x is bald}. Don’t we now have to go and provide a theory of these new and peculiar entities (“vague sets”) — who knows where that will lead us?

The implicit argument that we’re dealing with “new” entities here can be formulated as follows:

(1) Harry is definitely a member of S

(2) It is not the case that Harry is definitely a member of {x: x is bald}.

(3) So: S is not identical to {x: x is bald}

(Parallel considerations can be given for the non-identity of {x: x is bald} with sets Harry is definitely not a member of). The argument seems appealing, seeming to appeal only to the indiscernability of identicals. If we suppose S and {x: x is bald} to be identical, then we should be able to swap one for the other in the following context without change of truth value:

Harry is definitely a member of ….

But (1) and (2) show us this doesn’t happen. (3) follows by reductio.

The issues this raises are discussed extensively in the literature on the “Evans-Salmon” argument (and in parallel debates on the indiscernability of identicals in connection to modality and tense). One moral from that discussion is that the argument given above is probably not valid as it stands. Very roughly, “{x: x is bald}” can denote some precisely bounded set of entities, consistent with everything we’ve said, so long as it is indefinite which such totality it denotes. Interested readers are directed to (cite cite cite) for further discussion.

Vague metalanguages seem legitimate; and there’s no reason as yet to think that appeal to vaguely specified sets commits one to a novel “vague set theory”. But we still face the issue of how the distinctive puzzles of vagueness are to be explained.

2 responses to “Vagueness survey paper: II (vague metalanguages)

  1. I would say that similarity is not vague to one who finds two things similar –he has some definite sense of similarity –but, assuming he knows the specific aspects that two objects share jto make them similar, he can only speak in general terms to point out these aspects. In other words, he is limited to general categories when trying to describe specifics. He can say, for instance, that the two objects share exactly the same color blue–a blue that is warmer than aqua but cooler than an indigo blue–let’s say. These relevant terms all designate general categories. Thus, in a real way, the specifics are not conveyable. The person must go to the objects to determine whether he agrees or does not with the statement.
    Similarity is not like an address. It seems absurd to say that I went to the address I was given –but I disagree that the number or the street match the address. If the address system is correct–the ambiguity is more or less eliminated because the specific cases are all covered.
    But certainly, if the dwelling sought was behind the house and had a different number—one could say that one disagrees with the claim that the address given was correct—but you could say that the system did not cover all specific cases.
    The specificity is due to the elaborate convention of specific city and street names and numbers—-an agreement to limit ambiguity by imposing a narrowly circumscribed
    system that lists specific cases–streets and numbers and so on.
    A similar system could be created to determine similarity—-such that all could agree ahead of time to designate a large number of specific cases as similar —–but what a huge and cumbersome and inflexible system that would have to be!
    In the absence of such a system, we are stuck with ambiguity—and with the inability to convey in language the specific aspects that actually determine the similarity for the person claiming similarity (or difference or uniqueness for that matter).
    Unless I have agreed ahead of time with the claimant about this particular case–essentially by fiat–then of course I may disasgree with his claim when I see the colors, that is a result of the ambiguity of language.

  2. Hi, Henry again, it just occured to me…..

    All one can do is claim that two things are similar
    in some ways— claim that there is a sense or feeling of , or state of affairs, of similarity that has arisen –and hope that a similar notion or state of affairs will arise to others.
    I assert that when one points out similarity, one is pointing to the arising to be present, the existence of , this sense of similarity or state of affairs concerning the two objects.
    What makes two things similar is the arising of that notion, sense or feeling or sentiment or state of affairs. Without such—there are just two objects.
    I am assuming in this that the sense of similarity
    between two things is the same as the things actually being similar.
    It may be denied that the sense of similarity is objective –that is, if we assume an inner/ outer (subjective /objective)split, a sense of similarity does not indicate necessarily that there is similarity between wo things. But my reply to this is that any claim to objective similarity or difference is similarly due to the arising to the claimant of a sense of similarity, or similar state of affairs. Indeed, I assert
    that to claim that there is an inner/outer split is
    itself the result of the arising of the sense or state of affairs that there is a valid split. In other words, the claim of subjective validity of a sense of similarity comes from the same source as a claim for objective validity of a similarity.
    So, similarity is present whether conceived as subjective or objective presence.
    When one claims that two objects are similar or the same or different, one is claiming that there is a sense or a state of affairs, or both, that there has arisen a similarity.
    In some real way it is that arising that is the referent in the claim—-and the specific aspects of the objects which can be pointed to as determining similarity are irrelevant;
    for those same aspects may give rise in another to a claim of dissimilarity.
    Rather it is the arising of the sentiment itself –that there is a similarity–
    that is the real , the actual referent and in some fundamental way rather and not the objects or aspects.

    Formal logic does not recognize forms of logic that confound the effort to translate them into conventional logical notation.\–schemata.
    that feeling, that sense of sim ilarity—the assumption being that formal logic can capture all that is relevant to logic and deliberation.

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