Truth (Princeton Foundations of Contemporary Philosophy)

by Burgess, John P. ; Burgess, Alexis G.

  Presumably the pre- or unreflective speaker will simply go forward with what he has been told, classifying lots of children and a few adults as smidgets, and a few children and lots of adults as nonsmidgets, and saying nothing about Mr. Smallman, of whose existence he may be unaware. The reflective speaker, by contrast, will ponder what she has been told by way of explanation of the term “smidget,” notice that she has been given no basis to call Mr. Smallman a smidget or a nonsmidget, and perhaps be tempted to conclude that he is not a smidget, though not a nonsmidget, either. The sense in which she says that Mr. Smallman is not a smidget is associated with “a later stage in the development of the language, in which one reflects on the process.” The long and short of it is that Kripke, with his prereflective/reflective distinction, can make and does make no claim of achieving a total vindication of the intuitive notion of truth.

  Doublespeak is represented by the proposal to distinguish, in hopes of improving on Kripke and exorcizing Tarski's ghost, not Kripke's two stages in the development of the notion of truth, but two speech acts, strong affirmation and weak affirmation. The former is ordinary assertion, or putting forward as true, while the latter is putting forward as unfalse. The gappiness of liar examples, ~Tp ~T~p, can then be at least weakly affirmed, and to that extent the ineffability problem can be overcome, and perhaps a vindication claimed.

  Let us look a little closer at this proposal. If one distinguishes strong and weak affirmation, one must then double various other notions. One must distinguish strong rejection (ordinary denial or assertion of the negation), from weak rejection, the weak affirmation of the negation. (Weak rejection somewhat resembles the metalinguistic negation considered in §4.4.) Likewise one must distinguish strong entailment (ordinary implication or following in a truth-preserving manner) from weak entailment (or following in an unfalsehood-preserving manner). As strong entailment preserves strong affirmability, so weak entailment preserves weak affirmability. (The logics of strong and weak entailment differ; for instance, disjunctive syllogism preserves strong affirmability but does not preserve weak affirmability in the case where A is truth-valueless but B is false.) While we are at it, we may as well define strong veracity to be ordinary truth, and weak veracity to be unfalsehood. Then we can say that, weak or strong, affirmation is putting forward as veracious, rejection is affirmation of the negation, and entailment is veracity- and affirmability-preserving. In the jargon of the literature on many-valued logics, the strong and weak versions of trivalent logic differ in that, while they have the same truthtables, the former takes only 1 while the latter takes both 1 and ? as “designated values.”

  So far, so good, but there is a problem. With the intuitionistic logic advocated by antirealists, while not all classical forms of mathematical reasoning are acceptable, a great many still are, and a substantial body of intuitionistic mathematics has been built up. In particular, when intuitionists prove formal results about their logic, they are careful always to reason intuitionistically, so that their results will be theorems of this intuitionistic mathematics. By contrast, as Solomon Feferman has said, with gap logic “nothing like sustained ordinary reasoning can be carried on.” The lack of a useful conditional-type operator is one handicap, but only one. So it is no surprise to find that advocates of gap logic, in proving results about gap logic, never restrict themselves to gap-logically acceptable forms of inference. This is the real “Revenge of the Liar,” raising doubts whether gap theorists believe their own theory seriously enough to let it affect their practice.

  8.4 “PARACONSISTENCY”

  Another type of “solution” advocates recognizing truth-value gluts rather than gaps. Partisans of this approach reject disjunctive syllogism (6), and block the paradoxical reasoning at its last step (7j). They allow negation inconsistency but not absolute inconsistency, a combination known as paraconsistency, and speak of the rival gap approach under the contrasting label paracompleteness, a term not traditionally used by partisans of the gap approach themselves. But glut theorists, too, face an ineffability problem. It is not that they cannot enunciate their theory without saying something that, according to that theory itself, is untrue (as well as unfalse), but that they cannot enunciate it without saying something that is false (as well as true); and so they, too, are led to resort to a kind of doublespeak.

  The most important fact about the glut theory is one well known to specialists but rarely mentioned in the literature, namely, that it is identical to the gap theory except for terminology: One group pronounces the 1 and 0 and ? of the trivalent tables as “true” and false” and “gappy,” while the other pronounces them as “unfalse” and “untrue” and “glutted”; one group applies the ordinary terms truth and assertion and denial and implication to the strong versions of veracity and affirmation and rejection and entailment, while the other applies them to the weak versions. If glut theories are perceived as radical, exciting, and transgressive, while gap theories are perceived as moderate, dull, and conventional, that is a testimony to the power of advertising, for there is no more real difference between the two than between a used and a “pre-owned” car.

  This is all that really needs to be said here about “paraconsistent” vs “paracomplete” approaches. An option genuinely distinct from both would be presented by a quadrivalent or four-valued logic with both gaps and gluts (where it might be held, for instance, that the truth-teller is gappy, while the falsehood-teller is glutted). Such an option would still face, perhaps even more acutely, the problem of a split between the “external logic” and the “internal logic.”

  8.5 CONTEXTUALIST “SOLUTIONS”

  Dependency is represented by “solutions” that retain classical logic, but reject the formalization (7) as involving a fallacy of equivocation. For (7) represents all tokens of “(2) is false” by the same letter p, whereas (it is claimed) different tokens, because they occur in different contexts, express different propositions, and can have different truth values.

  One type of contextualist theory claims that the extension of the truth predicate can be context-dependent in this way. In its simplest version, it is just the view that each occurrence of the truth predicate bears an inaudible, invisible Tarski-style subscript. (The problem of practical feasibility mentioned in §7.1 for audible, visible subscripts does not arise, since “context” is supposed to supply the subscripts, whether or not the speaker or writer can.)

  How is the imputed subscript to be determined? The most simplistic proposal would be just this, that in any stretch of discourse, one starts with subscript zero, and raises the subscript only when the principle of charity requires one to do so in order to be able to construe some argumentative step as cogent. Of the various rules of inference we have considered, the only one that might require such subscript-raising is T-introduction (1a) (along with its negative counterpart (1c)), which on the kind of view we are contemplating is only appropriate with the proviso that the subscript on T must be higher than any in A. Thus in the deduction (7) the subscript would have to be raised at step (7f).

  Restating the deduction in words rather than symbols (again indenting hypothetical but not categorical steps), it then becomes the following:

  Read without the subscripts, each step considered in isolation is at least initially and superficially plausible, but according to the present analysis, the last is fallacious. It appears to be an instance of the valid (6), but owing to the difference in subscripts between (9i) and (9e), it is really an instance of the invalid

  (6') from A B and ~A' to infer B

  This seems very tidy, but it is hard to see how to apply the policy of giving lower subscripts to earlier-occurring tokens in a case where two tokens occur simultaneously, as in the old Socrates-Plato example, or the postcard paradox, where one is presented with a postcard with “What is written on the other side of this card is true” on one side, and “What is written on the other side of this card is false” on the other side.

  Worse, strengthened liars
threaten:

  (10a) What I am now saying is of a type that has no true tokens.

  (10b) What I am now saying is not true on any level.

  The proposal under consideration seems to require that it is impossible to quantify over all tokens or contexts or levels or subscripts. But why should this be impossible, and why should one believe that (10ab) don't accomplish it?

  A variant version of this strategy distinguishes levels of propositions rather than of truth. Since any hierarchy of propositions brings with it a hierarchy of truth predicates (with truthi being truth for propositionsi) this is not so much an alternative to as an elaboration of the proposal we have been considering. An example like (2) is construed in an elaborated fashion as involving an implicit quantification over propositions, somewhat as follows:

  (11) There is a proposition that is expressed by (11), and any such proposition is false.

  So the specific form of context-dependence claimed to be involved is variation from context to context of the domain over which some quantifier ranges.

  This specific phenomenon, quantifier-domain variance, is widely recognized by linguists and philosophers of language, usually illustrated by such examples as the following:

  (12) All the beer is in the refrigerator.

  where it is not reasonable to understand the quantifier as ranging over all the beer in the whole wide world, but rather over all the beer in the house, or all the beer for the party, or some such, depending on the context in which (12) is uttered. Note, however, that the speaker in (12) is not intending to make any absolutely universal statement about all beer, while the solution to the liar paradox being contemplated requires, in order to avoid strengthened liars, that anyone who speaks of “all propositions,” even if she is trying with all her might to say something about absolutely all propositions, must fail in the attempt. This is something quite different.

  Moreover, this proposed solution invites an obvious ineffability objection. The would-be vindicator of the intuitive notion of truth seems to need to maintain

  (13) All propositions that appear to be about all propositions are not really so, but only about some restricted range of propositions.

  And (13) is in an obvious way self-stultifying: If it is not really about all propositions, then it falls short of its purpose, but if it is really about all propositions, then it is a counterexample to itself. This is an embarrassment faced long ago by Russell, who in his own solution to the Russell paradox, his theory of types, wanted to say something very like (13). Russell invented a doctrine, adopted by proponents of the kind of contextualism under consideration, of a special kind of unnegatable systematically ambiguous or schematic utterance, through which (it is claimed) the thing the theorist wants to say, though it cannot be said, yet can be shown. Critics don't see it.

  8.6 INCONSISTENCY THEORIES

  Defeatism seems the natural inductive conclusion from the observation that one “solution” after another to the liar paradox fails to convince. Nonetheless, avowed inconsistency theorists are scarce, though the authors of this book are among them. The more usual reaction to the failure of old “solutions” is to propose some new one.

  While for consistency theorists the question is how to defend our intuitive notion of truth, for inconsistency theorists it is whether that notion should, in view of its inconsistency, be renounced or revised or retained. In a specialist area that puts a premium on rigor (think: pure mathematics), retention is not an option. The notion must be renounced at least until someone (think: Tarski) devises a consistent revision. If someone else (think: Kripke) later devises another, there is no reason not to adopt as many as are available and useful, so long as one distinguishes them, using different ones for different purposes.

  For the general public, by contrast, the liar paradox poses no threat. It is something discovered in historical times by a philosopher and passed on by books to later philosophers, but most people never encounter it. Hence for philosophers to urge the general public to renounce or revise their notion of truth would be futile. It will be retained no matter what philosophers say, and in speaking with their neighbors philosophers will be obliged to speak as their neighbors do, whatever they themselves think. But what should philosophers think? Even among the small number of inconsistency theorists at least three different attitudes can be discerned.

  A first answer is suggested by Matti Eklund's endorsement of the principle that when nothing can meet all the conditions built into the meaning of some term, then the distinction the term in fact marks (or should be understood as marking) is the one that comes closest to doing so. On such a view, though it was not intended that the equivalence principle should hold anything less than absolutely universally, and though each step of the liar paradox reasoning is indeed intuitive, the equivalence principle in fact holds only “habitually,” and contrary to intuition one of the steps of the liar paradox reasoning fails. Assuming classical logic is retained, it must be either the T-elimination step (7d) or the T-introduction step (7f). To determine which of the two it is, one would have to determine whether the true/untrue distinction that comes closest to meeting the conditions built into the meaning of the truth predicate is one that marks the liar paradox as true or one that marks it as untrue.

  To make this determination will be no easy task. If we understand “coming as close as possible” as “retaining as many T-biconditionals as possible,” we have to confront the fact that there is not just one but many a maximal consistent set of T-biconditionals (a consistent set that becomes inconsistent if even one more T-biconditional is added). Should we understand something to be marked true only if it is so marked whichever of these sets one chooses? Or should we understand the true/ untrue distinction as ambiguous or otherwise indeterminate? The issues are further complicated by Vann McGee's observation that any arbitrary conclusion q whatsoever is equivalent to some T-biconditional Tp ↔ p. (Just choose a p that says Tp ↔ q.)

  It might be thought that the solution here would to be to establish some priority among T-biconditionals, holding some more worthy of retention than others. But there can be no basis for doing so if the equivalence principle is the whole of the meaning or intension of “true,” and if extension is determined solely by goodness of fit with intension. For the equivalence principle is simply a universal generalization, and as such does not discriminate among its instances. Perhaps, then, we should take extension to be determined not by fit with intension as such, but by fit with intension and the use we make of it. This would justify giving higher priority to retaining useful over useless or harmful T-biconditionals. The inconsistency theorist would then be like a Tarskian trying to replace an inconsistent notion of truth with a useful substitute, but with this difference, that once the useful substitute is found, the inconsistency theorist will claim, not indeed that this is what we intended all along, but at least that this is the extension we had succeeded in capturing all along. A skeptic may wonder how the inconsistency theorist can be justified in making such a self-congratulatory claim—to call it an application of the “principle of charity” would be to misapply the maxim “charity begins at home”—but a prior question is why one would even want to.

  A second answer to the question what attitude the philosopher should take, involving no such claim, is explored by A. G. Burgess in his doctoral dissertation, the option of adopting the same “fictionalist” stance towards truth that so many philosophers have advocated adopting towards so many other things (from mathematics to modality to morals). Central to fictionalism is a distinction between what one says about the “fiction” (what philosophers say to each other in the philosophy seminar room) and what one says within it (what philosophers say to members of the general public on the street).

  About the “fiction of truth” one might say

  (14a) The liar paradox shows that the notion of truth is ultimately incoherent.

  as one might say

  (14b) Sherlock Holmes is not a real person but a character
invented by Arthur Conan Doyle.

  Within the fiction one might say

  (15a) To assert that some proposition is true is equivalent to just asserting that proposition.

  as one might say

  (15b) Sherlock Holmes shares rooms with Doctor Watson in Baker Street.

  One maintains an attitude of genuine belief towards what one says about the fiction, and to what one says within it if prefaced by the disclaimer “According to the pertinent fiction.” Towards what one says within the fiction, taken without the disclaimer, one only maintains a pretence of belief. Philosophical discussions of fictions in the literal sense of novels and short stories and the like—Kendall Walton's discussion of prop-oriented make-believe is especially often cited—can be drawn on for principles about when one can and can't fairly say “According to such-and-such a fiction, p.”