McG/McL, Ch 5: Indeterminate Truth

In this chapter, we prescind from higher-order vagueness and its grand unification with the Tarski hierarchy. Instead, we confront simple truth-value gaps, the kind that arise for words like ‘dommal’ and Carnap's version of ‘soluble’:


(Dommal) Being a dog is sufficient for being a dommal, and being a mammal is necessary.


(Soluble 1) Ax At, if x is placed in water at time t -> (x is soluble <-> x dissolves at t)
(Soluble 2) Ax Ay, if x and y are the same chemical substance -> (x is soluble <-> y is soluble)


We should note that on Carnap's intended reading of these postulates, there is no answer to the question, "is x soluble?" if x hasn't ever been placed into water. The rule for the use of "soluble" appears to presuppose both (i) that x has been placed into water, and (ii) that solubility does not cut across chemical-substance-kind lines.


McG/McL contrast these terms, with their artificially well-defined gaps, with more "penumbral" vague terms like "heavy":


"We know exactly when and why the term [Carnap's "soluble"] was introduced into the language, and we know exactly how it is used. More important, we can say exactly which facts are relevant to the term's application...relevant considerations are neatly localized." (1)


It seems that what is important about these terms is that epistemicism's case is weak for them. I can stipulate into existence a word like "soluble_c" [_c for 'Carnap'] or "dommal", and simply ‘refuse ‘to give conditions for their application which cover all cases. What shall we say about such terms? It is unlikely that other factors, like contextual ones, will fill in the gap that I have left open; the usage patterns of other speakers cannot do so either, since I have just introduced the term. Yet such a term *could* be so introduced and adopted into the language. (Indeed, it seems like many of our terms ‘do ‘come with very substantive presuppositions---what is less clear to me is whether it's right to understand e.g. "dommal" this way.) The authors conclude that "’soluble_c’ gives us [an] unmistakable example of [a] truth-value gap" (2). This is important to the dialectic because the existence of a (nonempty) gap is the supervaluationist's entering wedge.


The authors then consider what we might call "Williamson's gambit" in response to the Dommal problem. This is that "x is a dommal" is false of any non-dog, since the rules have not done enough to make it ‘true ‘of a non-dog. For Williamson, the allocation of truth is stingy; this is how we may adjudicate the truly gap-happy cases. An analogous ruling in the soluble_c case would make "x is soluble_c" false for anything that had never been placed in water. McG/McL respond that such a ruling would be seriously out of tune with the use speakers would make of the term once they had adopted it---since "dommal" is not ‘used’ like "dog", it seems artificial to give it the same truth-conditions as "dog" as a result of applying the arbitrary "truth is stingy" rule.


We should be cautious in reflecting on the "dommal" and "soluble_c" rules. Are they rules for ‘truth’, or are they rules for ‘usage’? Rules for usage--especially unembedded usage---will systematically underdetermine truth-conditions. While I might be able to stipulate usage rules (in affect I am doing this all the time, simply by using my words as I do) it is not so clear that I can stipulate truth-conditions, except in artificial, short-lived contexts; I have wide discretion over my ‘use ‘of sentences, and far narrower control over whether what I say with those sentences is ‘true’. Meditating on this distinction does seem to tell in Williamson's favor; after all, usage patterns to not rule out the discovery of informative identities, and it is unclear whether we ever stipulate gappy truth-conditions, even if gappy usage is common.


McG/McL go on to argue that the ‘(T)-for-utterances’ schema:

If u says that p, u is true iff p.


...is falsified by gappy usage, e.g. by the pattern of usage that would arise of "dommal" were adopted by the linguistic community. The authors reintroduce the idea of a tension between two different kinds of truth here, when they consider the question of whether we should embrace the schema.


Horn 1: "We embrace the schema as an a priori maxim that we intend to hold onto whether or not it reflects the facts of usage." We want "true" as a logical device and the schema's status is axiomatic.


Horn 2: We reject the schema for "dommal" and "soluble_c". In particular, we reject it because it entails bivalence via Williamson's argument from LEM. [?...not sure this is right; the text is a bit unclear here, since it actually seems to argue ‘from’ Bivalence ‘to ‘LEM!] (...And we accept LEM, because we accept classical logic.)


****


Then there is a bit of meditation on the epistemicist's take on things: the way he understands the terms "precise", "vague" and "determinately." The contrast here is between epistemicism and semanticism.



We rehearse the semanticists's diagnosis of the fallacious inference from "there is a red tile adjacent to a nonred tile" to "the word 'red' (or the concept ‘red’) has a sharp boundary" (16); again, the diagnosis rests on the intuitive pull of two competing, but distinct, notions of truth: one which supervenes on usage [where usage can be and is gappy] and another which is disquotational-classical and therefore leaves no gap.




***




We proceed now to the indictment that the supervaluationist cannot really respect classical logic because she cannot reason classically (even if he can get all the classical tautologies.) In a nutshell, the response will be this: she can reason classically all she likes with "plue". The restrictions on reductio, proof by cases, etc. that come with "determinately" should be seen as limitations on these proof schemas for an operator which ‘enriches’ the classical language:


"The valid inferences are the ones sanctioned by the classical predicate calculus, as described in any standard logic text. The semanticist isn't proposing a nonclassical definition of validity; she's proposing a nonclassical definition of truth. She regards the classically valid modes of inference as truth preserving, and she asks whether there are any other modes of reasoning, in addition to those identified by classical logic, that we can also count on to enable us to derive true conclusions from true premises. This quest should not be understood as the search for an expanded notion of validity, because the semanticist is perfectly content with the notion of validity as we have it already." (23-24)


So the semanticist McG/McL have in mind should be seen as distinguishing between ‘valid’ forms of inference on the one hand, and ‘truth-preserving ‘forms of inference on the other. Validity is classical validity. Truth preservation is ‘supertruth’-preservation. Not every truth-preserving argument is valid.

A helpful diagram would be this: Williamson's picture is that supervaluationist logic, which rejects reductio, DS, etc., imposes a severe *restriction* classical logic: that is because the supertruth-preserving inference forms are a *subset* of the valid ones. McG/McL respond by denying that; for them, the valid arguments are a subset of the supertruth-preserving ones. That's because they are keeping all of classical logic--with its reductios, DS's, and all the rest--and augmenting it with an operator, 'D'---whose associated forms of inference, it must be granted, do not include reductio and DS. It is possible for McG/McL to hold that they have included all of classical logic, contra Williamson, and that their picture is in harmony with the inferential practices of e.g. mathematicians who reason by reductio, because their system does countenance unrestricted reasoning by reductio in any context which does not include vague terms. Mathematical languages are precise*, so the mathematicians have committed no oversight. Moreover, we account for classically valid inferences in the general population by postulating that the popular standard for validity is the preservation of supertruth. This has no effect on our domain of classical logic, by Stone's theorem [see below]. And it prevents the "collapse" of the interderivability of "Tr 'p' " and "p".

...I think I understand this picture, but it still seems to me that the populace is faulted by it. After all, it means that when the populace reasons classically by cases, they leave a third case out. It seems like what might be required here is an argument that the populace *does* reason by three cases rather than two (the populace "includes the excluded middle") when the premises involve vague terms, which is, if we take Russell's argument, always; but I am not sure that the populace does this. I believe Keefe thinks the folk do this...


***

Finally, we move on to some many-valued metalogic to flesh out the contrasting picture. The distinction is made between (i) having many truth-values [real numbers between 0 and 1], on the one hand, and (ii) having many ‘designated ‘truth-values, on the other. If there is a boolean algebra defined on these many values, we have Stone's theorem:


[Stone's Theorem] For any Boolean algebra B and any proposed inference, the following are equivalent:

(1) The inference is valid in classical sentential calculus.

(2) The inference is strongly B-valid (preserves a truth-value of 1 from premises to conclusion)

(2) The inference is weakly B-valid (preserves truth value < b from premises to conclusion, where b is the threshold ‘designated’ value)


...as far as I can tell, the Boolean algebra bit boils down to this: the many-valued logic is truth-functional. So Stone's theorem simply tells us that if we keep classical consequence, the choice between strong or weak B-consequence is a non-choice. To wit: the expansion of classical logic undertaken by McG/McL's supervaluationist system is free to choose between the preservation of truth at a point (preservation of an intermediate degree of truth), and the preservation of global truth (preservation the highest degree of truth--supertruth.)


It could be the case that good inferences, the kind that appear in math journals, are cases of strongly B-valid inferences. Strongly B-valid inferences preserve supertruth. So we shouldn't think that, just because we are introducing a logic which restricts e.g. ‘reductio’, that we are contravening the math journals; it's just that ‘they’ were assuming the premises were supertrue, while we are giving a logic in which assumptions may have a lower degree of truth than that. McG/McL's suggestion is that this take on the data of ordinary (and extraordinary) inferential practices isn't ruled out by what we know about how people reason. What I have suggested is that, so far, we seem farther along the road to vindicating the mathematicians than the folk.


Of course, the key here is that the boolean operators are truth-functional! It is quite striking that McG/McL's form of supervaluationism is truth-functional, since of course this gives up the analysis of conditionals that seemed to be such a strength of the supervaluationist account...or perhaps it does not. The concern here is first and foremost with classical logic, which makes use of the material conditional. Nothing yet said prevents McG/McL from endorsing an e.g. metalinguistic analysis of ordinary language conditionals that could capture their non-truth-functional behavior in a supervaluationist framework.


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*...are they? It seems rather common for there to be gaps in mathematical terminology of the "dommal" kind, even if not of the more regular "heavy" kind.

Refl-Heck-tions II: Demonstrata and nonconceptual content

Heck (2000)'s delicate point about demonstrative phenomenal concepts, put two ways:

by Siegel in the SEP:

"Another point of debate raised by McDowell's concerns whether it is possible to form demonstrative concepts of the shade represented in experience in cases of illusion, when the shade represented in experience differs from the shade of the thing seen. If demonstrative concepts of color shades can pick out only shades actually had by the thing demonstrated (as Heck 2000 contends), then again McDowell's argument fails. However, it is again a matter of controversy whether demonstrative concepts are limited in this way. Yet another point of debate in this area is whether experience itself would be needed to anchor demonstrative concepts in the first place — in which case, it is said, they could not already be constituted by them (Heck defends this view)."

...and by Tye in his (2005):

"The conceptualist might respond that, whatever may be the case for the demonstrative expression`that shade', the demonstrative concept exercised in the experience is a concept of the shade the given surface appears to have. But, now, in the case of misperception, there is no sample of the color in the world. So, how is the referent of the concept fixed? The obvious reply is that it is fixed by the content of the subject's experience: the concept refers to the shade the given experience represents the surface as having. However, this reply is not available to the conceptualist about the content of visual experience; for the content of the demonstrative concept is supposed to be part of the content of the experience and so the concept cannot have its referent fixed by that content (Heck 2000, 496)."

The putative tension here is between "anchoring" and "constituting" ["being part of"]. I will take "anchoring" to mean "serving as the referent of" [as opposed to, say, having some epistemological meaning a la "serving as grounds for"].

The nonconceptualist claims that the fineness of grain of experience shows that there are nonconceptual contents, viz. contents of our experience which are not denizens of our volutary, thinking-and-imagining conceptual repertoire. The conceptualist reply is that we do have a concept for each shade Red(n), where n ranges across the many many (infinite?) values corresponding to lines on the spectrum. We don't have, say,individual proper names for them all, but instead, we can represent each Red(n) as ''that shade (of red)", where "that" is a demonstrative. Hence our concepts of the different Red(n)'s are demonstrative concepts.

Now consider the case of an illusion of Red(29). The content of the hallucinatory experience is obviously "o is Red(29)." Do we have a concept of Red(29)? We should ask: what does the "that" in "that shade"---which must be the concept we are deploying if we are deploying one at all---refer to? It cannot refer to anything in the real world, since by hypothesis there is nothing in the real world that we are seeing. Hence it must be (some constituent of) the experience itself which serves as the referent of "that". But then we do need the experience to supply the referent of the demonstrative---we do not already have a concept for each color we experience.

Clearly, something would be missing if the content of experience were something like

"object o has this color"

...if there is, ahem, no accompanying demonstrandum. Yet the sentence above is precisely what the accompanying conceptual counterpart of the experience is taken to be, containing only 'deployed concepts' of the agent's conceptual repertoire. Upshot: if we take the relevant conceptual state (probably belief) to rely on experience to supply referents for its demonstratives, then it cannot be the that very concept [the demonstrative one] which is part of the content of the experience. This would make experience self-referential; moreover, there just wouldn't be anything (else) for the demonstrative to refer to.

Tappenden on Unifying The Liar and the Sorites Paradoxes

Tappenden gives what might be called a pragmatic analysis of the assertability of the conditional premise of the sorites paradox:

(P2) If a man with c cents is poor, a man with c+1 cents is poor.

Analysis: semantically, P2 is gap. Why? For borderline cases of richness, we espouse strong Kleene tables. Therefore P2 has some instances which are neither true nor false: to wit, any instance of c in which having c cents makes one a borderline case of "poor".

Pragmatic upshot: you can't assert P2, but you can articulate it, where articulation is understood as a speech act distinct from assertion, with a different norm. It is a necessary condition for successful assertion of p that p express a true proposition [Tappenden footnotes Dummett here]; but truth is not a necessary condition for successful articulations. To articulate S is to claim that ~S is not assertable; articulation is for correcting (or preempting?) improper use by others. The relationship to semantic values of associated propositions is this: to correctly articulate S, it need not be the case that S is true; it need only be the case that S is not false. Hence P2's "articulability", and the strong "semantically positive" intuition we have towards such utterances, is explained without needing to postulate that P2 is true. [Note: I am not sure why we must say that S is not false--that it has this weaker semantic status--at all. Perhaps it could be false? He does note that articulation's perlocutionary effect does not, like irony's, depend on recognition of its falsity.]

Along the way, Tappenden makes some interesting, but not heavily supported, claims about the differences between different syntactic forms for (classically) logically equivalent sentences, when considered as the LFs of speech acts. For example, LEM sentences ("All the tiles are either red or orange") are claimed to function as "`sharp boundary' conditions" (565) and hence to be assertable only in the absence of hard (ie borderline) cases: to say that all the tiles are either red or orange is to say that, in our context, we should be able to sort them into two piles with nothing left unclassified. Likewise, to assert an existential ("some man is tall while his neighbor is short") is to implicate that a truthmaking last tall man can be identified.

On the other hand, logically equivalent sentences of the form
(Ax)~(Rx & ~Rx)
enforce weaker "no overlap" conditions: they function as claims that complementaries are exclusive, but not necessarily exhaustive. All this struck me as odd--particularly the claims about or-LEM sentences--because it simply didn't gel with my intuitions. Perhaps that is all that can be said about that.

One quite odd thing about Tappenden's discussion is that he categorizes Fine-ian penubral sentences with the corresponding tolerance sentences, where tolerance sentences are the ones that have the form of P2:

(Penumbral) if a man with c cents is poor, a man with c-1 cents is poor.
(Tolerance--P2) if a man with c cents is poor, a man with c+1 cents is poor.

Both of these sentences are, in Tappenden's taxonomy, "pre-analytic," and they both have the status that they are articulable without being assertable. This lumping-together is partially explained by noting that on 3-valued tables, both types of conditionals are gaps. But given that there is extensive discussion of the assertability and psychological import of these sentences, it is surely worth noting that (Tolerance) is a good deal less acceptable than (Penumbral), and that it is the former only which leads by sorites reasoning to a contradiction.

A final note on the vagueness portion: in the course of the paper, Tappenden makes an intriguing distinction between "essentially" and "inessentially" vague predicates, where it appears that the only essentially vague predicates are observational (e.g. "looks red.") [although he doesn't use enough examples to really confirm this hypothesis].

...and the Liar?

I wasn't sure I understood how the analysis was supposed to apply to the Liar--thereby unifying the two paradoxes--since I'm simply unsure of a very prelimiary point: how do you assert the Liar? The Liar is a sentence that refers to itself. I can refer to myself (with "I"), but I am not a sentence. An utterance can refer to itself (with "this utterance"), but an utterance is not a sentence either. I am not sure whether we can assert a sentence that refers to itself. (Do we thereby have to assert that it refers to itself? If we don't, how will we get the point across?)

If someone said to me:

(*) "This utterance is false"

I feel that my first reaction would be to say "...which utterance"? That is, I wouldn't at all feel confident that I knew which sentence had been asserted, because my instinct would be that the utterance contained an empty (or unidentified) demonstrative term.

I don't mean to be pedantic--but there is a need for some preliminary discussion of asserting the Liar here. Kripke, for example, thinks that you haven't asserted a proposition with (*), even though you did make an utterance.

In the absence of this discussion, what can be said? Tappenden is surely absolutely right that the analogue of bivalence for assertability does not hold:

(Bivalence-Assertability) For all sentences s, either s is assertable or *~s* is assertable. [*'s for Quine-corners].

But we did not need the Liar to show us this! Our lack of omniscience is sufficient. (Perhaps this is not discussed because Tappenden is using Dummett's truth-norm rather than a knowledge norm, a justified belief norm, or a Brandomian, reason-offering norm.) We wanted to know whether the Liar was true, and all we wound up with was the weaker observation that the Liar is not assertable. Tappenden does, however, make a bid to turn this into a solution the Liar by offering the following observation: since we can explain the non-assertability of the Liar without recourse to its truth-value, we are free to hold that both it and its negation are gap. That's good because holding that it has either non-gap truth-value leads to a contradiction.

Finally, we get an explanation of the 'semantically positive' status of these two:

(3*) The liar is true iff the liar is not true.
(C) (As)(True(s) v ~True(s))

...in terms of articulation. I am a bit confused about this, since I don't know exactly how Kripke's semantics for Truth (which Tappenden is taking on board) generates the truth-value Gap for sentences. However, the picture must be that they are gap, and their illusion of truth is explained by conversational norms.

***
Tappenden, Jamie. "The Liar and Sorites Paradoxes: Toward a Unified Treatment." J. Philosophy XC, no. 11, 1993.

with references to:

Kripke, S. "Outline of a Theory of Truth" J. Phil, LXXI, no. 19, 1975.