Vagueness

This post was too long to post as a comment at Clark Goble's "Peirce and the Liar II" thread at his blog Mormon Metaphysics. But I don't feel sure enough about it to post it as a a full-blown "official" post at my own regular blog! For some some earlier discussion which might clarify it, skip down this page to a pair of my earlier comments at Clark's blog which I reproduce here. For the complete thread, go to "Peirce and the Liar II"

A: "Skippy, Lassie, and Flipper: a certain one of them is a mammal."

B: "All three of them are mammals."

A: "Yes, but I'm saying that a certain one of them is an mammal."

But did A say that a certain one of them is a mammal? Or did A "merely" say "a certain one of them is a mammal"?

C: "Skippy is a mammal!"

D, later: "C said that a certain one of them was a mammal, but I forget which one C said."

Although the OED says that "a certain..." tends to suggest an unwillingness to explicitly specify, actually it is often meant to indicate that a specification will be forthcoming, perhaps gradually. It often is a lot like a demonstrative "this" pointing to near-future items or events internal to a discussion.

E: "A certain man went to the store. Who was it?"

F: "It was I, I went to the store, with you."

E: "Yes, you did, but I mean, a certain man, the one of whom I'm thinking, went to the store."

"The one of whom I'm thinking" is an example of such specification, with its "the" signifying the specification's intent of sufficiency. Such uses of "a certain" are in order to acknowledge or warn of a temporary, transitory vagueness. Figuring out who E is thinking of may be a tall order (except for "cold readers" like Poe's Dupin), but logically the specification "the one of whom E is thinking" is not vague. However, it is usually so difficult to succeed on early guesses at what such a speaker is being so coy about, that it might as well be vague, so it's used in expository examples of vagueness, to get the idea across. (Coyness is, of course, vagueness intended with a "cute" attitude. "Don't you know who I'm talking about? C'mon, try and guess.")

Now, the term "this" is like a kind of "a certain" which (more sincerely) presumes that the listener knows (or will easily know after a moment) which thing is being designated.

Now, the kind of deductive logic which one studies in Quine's textbooks is not really dialogical but more monological. But there one can still make a distinction between the objectual (ranging over objects even when the objects are unspecified) and the substitutional. I will attempt now to argue that the substitutional constant corresponds, or is better regarded as corresponding, to the "this" or "that" or "yon," and that the objectual constant should correspond to the "a certain," and that the standard schematic predicate terms -- which are already considered substitutional -- can do double duty as the "of this kind," "of that kind," better than as the "of a certain kind". The problem is that, when we see schematic predicate terms in the textbook, we usually have no idea what explicit predicates they stand for. So shouldn't they be taken in the sense of "of a certain kind"? But it needs to be remembered -- we also usually don't care what they "really" stand for, and we're not supposed to care. We read them and suppose ourselves as being like the listeners in a dialogical situation, but really it is a monologue and we might as well be the speakers of those toy examples. And indeed we are encouraged to freely substitute explicit predicate terms as if we get to decide for what the schematic predicate terms really stand -- not because we are dealing with generals but because we stand as speakers getting to decide what predicate terms we will veil.

But an objectual schematic term would change that (and probably take us out of first-order logic) -- then we would per force stand as listeners, from whose substitutional "list" the necessary specification may be absent, while the facts discussed become, so to speak the speaker -- a speaker being specific but not clearly expressing itself. The objectuality brings an element of uncertainty. But why can't one be uncertain about which of three things already on the substitutional list is being referred to? "This sentence refers to a certain one among Skippy, Lassie, and Flipper." That sentence is false, since it performs no such reference. "I refer to a certain one among Skippy, Lassie, and Flipper." That sentence could be true in context in real life, and we can pretend it's true for the toy example. But, assuming that the basis of its uncertainty will not be the listener's unfamiliarity with those TV characters, then the basis will be the involvement of a singular event not on the listener's substitutional list -- the choice of reference by a distinct speaker. It is the pertinent relation of Skippy, Lassie, and Flipper to this event, this fourth "object" not on the listener's substitutional list, which heightens the vagueness. (It is on the speaker's list, but not on the listener's.)

In short, we should not treat the schematic term -- when it doubles as a constant pronoun -- as being vague unless we are forced to do so by its being objectual. My motivation is to conflate the non-explicit/explicit distinction with the objectual-substitutional distinction by treating the substitutional constant pronoun as being such that it might as well be explicit -- as a "this" or "that" rather than an "a certain". So far my justification for this move is not sitting with me too badly. I'm afraid to ask what you might think of it! I know you [Clark] want to get back to the Liar's Paradox.

Thus, the substitutional constant pronoun, like the substitutional variable pronoun, is "merely" a convenience which spares us from having actually to create or repeat a list or a name (in practical terms, it's often quite necessary, and the substitutional variable pronoun also accomplishes feats of distinction and cross-reference which the expanded alternation itself can't do unaided, at least not without difficulties that I don't see how to surmount, but a logician could probably give a pretty plain answer on this), while the objectual constant pronoun, like the objectual variable pronoun, introduces real uncertainty. The objectual constant pronoun is, furthermore, vague, referring invariantly to something specific in itself but not speaking itself clearly to us as listeners.

I know you [Clark] want to get back to the Liar's Paradox, and it's been hard for me to go along with it until I know what I think about vagueness. So now might be a good time for you to move that forward, or to start a new thread on it if you wish, or at peirce-l, since [the current thread at your blog] is so weighed down with the discussion of vagueness itself which had helped raise the subject of the Liar's Paradox again.

Update: Here's a chart.

	General Scope		Singular Scope
	Variable (ranging as a GENERAL over kinds)	Constant	Variable (ranging as a GENERAL over things)	Constant
Objectual	of one or another kind	of a certain kind (VAGUE in kind)	something or other	a certain thing (VAGUE)
Substitutional	of one or another kind (among the kinds)	of this kind, that kind (DETERMINATE in kind)	one or another (of the things)	this, that, it (DETERMINATE)

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Posted by BenðUde11 September 29, 2005 10:56 AM at Clark Goble's "Peirce and the Liar II" thread at his blog Mormon Metaphysics

Clearly a vague thing will be as specific as the specific thing which, qua veiled, is the vague thing. "A certain horse" is more specific than "some, any horse" and, notwithstanding its vagueness, as specific as "Secretariat." Clearly, too, at "supreme" generality, it's pretty hard to get vague. "A certain supreme genus" sounds like a comically intended vague-ification of "being," unless one also considers "non-being" a supreme genus, but it's probably best not to let such extremity questions unsettle this sort of discussion.

Now, if, instead of mixing various modes of generality as I have done, we adhere to singular forms for consistency's sake, then the question of generality and specificity which we've been discussing appears as a question of variability and constancy. The variable "some/anything x" is no less veiled than the constant "some/a certain thing j" but, since "some/anything x" ranges through all things, there's less uncertainty and vagueness about what is or could be behind the veil. If we lift the veil, we get an explicit alternation of all indexed things: "Alf or Beth or Cam or Delta or Ephraim or Zelda or Hettie or Thaddeus or Ida or Caspar or Mona or Newt or...[etc.]" The attempt to use this alternation as the variable leads to two problems: (1) the resultant variable still needs pronounlike elements in order make distinguishable and cross-referrible applications of it, as the standard variable allows easily (e.g., as in "ExEy Gxy" or "something and something else orbited each other"); and (2) the resultant variable is, per force, substitutional rather than objectual (unless we "break the rules" in order to include an objectual variable in the alternation in order to include things which we can't specify). If there are things which, for whatever reason, we can't specify, then they can't be in the explicit alternation, but they can still be regarded as included in the range of the standard objectual variable "x."

So it seems to me that the real parameters involved here are at least three: (1) variability / constancy, (2) veiledness (pronounlikeness) / explicitness, (3) general scope ("horse," "some kind of," "a certain kind of," etc.) / singular scope ("John," "something/anythiing," "a certain thing," etc.). The vague appears as the veiled or pronounlike constant. As a composite of the veiled and the constant, the vague will, as a constant, be more specific than the variable (I'm using the word "specific" as a kind of generic term for the "specificity" pole in each of these parameters), as long as the parameter of general scope/singular scope is "held constant" so to speak (i.e., as long as we adhere to singular forms for consistency's sake or to general forms for consistency's sake, unlike what I was doing in my earlier comments).

A few notes: (1) Between the extremes of universal and singular scopes, two very distinct intermediate kinds of scope can be defined, and since variation and constancy mirror the scopes in their own way, two very distinct intermediate kinds of constancy/variation can be defined as well. I don't, at least as yet, see a neat way to do that in the case of veiledness/explicitness. (2) The distinction between the veiled (pronounlike) and the explicit might turn out to be best annexed to or conflated with the objectual/substitutional distinction. If so, it may be possible to formally explore intermediate forms of veiledness/explicitness via the objectual/substitutional distinction.

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Posted by BenðUde11 September 30, 2005 11:50 AM at Clark Goble's "Peirce and the Liar II" thread at his blog Mormon Metaphysics

Ponderings over whether one can fairly conflate the veiled/explicit distinction with the objectual/substitutional distinction has led me to some pretty delicate (or so they seem to a ham-handed fellow like me) considerations about (1) the dialogical contexts of vagueness, (2) the extent to which the hardly traced-out bits of logical formalism in terms of which I've been considering vagueness may be fairly considered to represent a few elementary permutations of the dialogical setup such as to enhance or reduce the "sense" of vagueness, and (c) whether "veiled" is too strong a word for its side of that broader distinction which I've called the "veiled/explicit" (i.e., maybe I should just say "nonexplicit/explicit"). I realize I'm being vague, but I'm making these comments just to say that I'm still working on it, could go various ways on it, etc. - posted by The Tetrast, initially @ 10/01/2005 03:12:00 PM [top of post] [top of page] Copyright © BenjaminðA.ðUde11.