Truth (Princeton Foundations of Contemporary Philosophy)
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7.5* REVISION
Now for the rival revision theory. The definition (16) of the jump makes perfect sense even for a total valuation T. But no total valuation is a fixed point. Rather the jump T* will disagree with the original T over the status of, for instance, any sentence B of type (8), any B equivalent to ~T (“B”). For such a B, if T(#B) = 1, then T[T (“B”)] = true, so T[~T (“B”)] = false, so T[B] = false, so T*(#B) = 0; and similarly if T(#B) = 0, then T*(#B) = 1. If we start with some total valuation U0 and consider U1 = U0* and U2 = U1* and so on, the truth value of such a paradoxical sentence will flip-flop as we go through the sequence of valuations. The jump represents a revision, not an extension.
If we want to consider a stage ω, by that stage there will be some numbers that from some point on have always received the same value. For instance, for a sentence A of L that is true (false), Um(#A) will be one (zero) for all m ≥ 1, while Um(#T (“A”)) will be one (zero) for all m ≥ 2, and so on. If we want to introduce a Uω, we will want it to give the value one (zero) to those numbers that have persistently been valued one (zero) in this way. But what of other numbers, whose value has been flip-flopping? Different revision theorists, Hans Herzberger and Anil Gupta and Nuel Belnap, take different approaches. The first would assign all such numbers zero, the second would assign them all whatever value they were assigned back at the beginning, while the third would allow a new arbitrary assignment of ones and zeros to them.
On any of the three approaches, the process may be iterated indefinitely. Any sentence that eventually reaches a point after which it is always true (false) may be called stably true (false). No fixed point will ever be reached, though there will be an ordinal by which the process is essentially complete, after which no stably true or stably false sentence ever changes truth value. Even for the simplest of the several versions of revision theory, this ordinal is much larger than the ordinal connected with Kripke's theory—a reflection of the greater complexity of the revision-theoretic approach. Still more elaborate constructions, combining elements of Kripkean and elements of the revision-theoretic approaches, have been undertaken by Hartry Field in order to incorporate a conditional-like operator into the language.
7.6* AXIOMATICS
Logicians have ways of classifying the complexity of sets of numbers. For instance, the set of code numbers of Tarski-true sentences of L is more complex than any set that is arithmetically definable, any for which membership is expressible by a formula of L. But the set of Tarski-truths is in some sense just beyond arithmetically definable sets in complexity. Perhaps unsurprisingly, the sets associated with Kripke's theory turn out to be more complex than those associated with Tarski's theory, and the sets associated with revision theory to be more complex still.
Logicians also have ways of classifying the logical strength of different theories. For instance, there is a natural set of axioms for arithmetic, formulated in the language L, called the first-order Peano axioms P. By Gödel's incompleteness theorems, neither it nor any other system can prove all truths expressible in L, and among the truths P cannot prove is (a coded version of) the statement that P is consistent. There is also a natural set of axioms, which we may call P', for arithmetic plus Tarski-truth for arithmetic sentences, which can be formulated in the language L', adding to the apparatus of P, as additional axioms, formalized versions of (13), and of (14) restricted to sentences not involving T. In P' one can prove the consistency of P (essentially by a formalized version of the argument that each axiom is true, and that deduction preserves truth). Because P' can prove the consistency of P and not vice versa, logicians count P' as being of greater “consistency strength.”
One would expect the natural system of axioms for arithmetic plus Kripke-truth to be higher up on the scale of consistency strength than arithmetic plus Tarski-truth, and the natural system of axioms for arithmetic plus revisionist-truth to be still higher. But actually, it is not entirely clear what “the natural system of axioms” for Kripke-truth is, and it is entirely unclear what “the natural system of axioms” for revisionist-truth is.
One branch of axiomatic truth theory begins with work of Solomon Feferman that was specifically directed towards axiomatizing Kripke's theory, and determining its place on the logicians' scale of strength. Workers in this area have still not achieved agreement on which candidate is “the” natural axiomatization, but have succeeded in finding the places of the various candidates on the scale.
Another branch of axiomatic truth theory began with joint work of Harvey Friedman and Michael Sheard that was directed towards a complete determination of which combinations of items from a list of candidate truth principles are consistent, whether or not the combination corresponded to any philosopher's conception. The principles considered include composition laws, introduction or elimination rules, and the like. Friedman also began work on determining the strengths of the consistent combinations. So far the branch of the theory concerned with consistent combinations has worked mainly against a background of classical logic. Though from several points of view partial logic might be a more natural choice, it is in many ways technically less tractable. In sum, despite a fair amount of work, many questions remain open.
Before leaving the topic of axiomatic truth theories, mention should be made of an objection sometimes raised against deflationism, based on the fact that a theory like P' is stronger than one like P. How “thin” or “anemic” can a property be, the critic asks, if the natural axioms for it enable us to prove things about other matters (such as consistency) that we couldn't prove before? While such a rhetorical question may be an understandable reaction to some deflationist sloganeering, if the basic idea of deflationism is that there is no more to the meaning of the truth predicate than a few simple rules for its use, then issues about the strength of formal theories are ultimately irrelevant. For it is well known that simple rules can be of great logical strength. If there is a threat to deflationism, it is perhaps in the complexity, rather than the strength, of the rules of axiomatic theories of truth. For if one had to think of the meaning of the truth predicate as somehow constituted by the principles of one of the axiomatic truth theories in the literature, one certainly could not say that the meaning is constituted by a few, simple rules.
CHAPTER EIGHT
Insolubility?
THOSE ENGAGED IN MAINLY TECHNICAL WORK of the kind considered in the preceding chapter do not generally discuss at any length whether their constructions are to be regarded as models describing our intuitive notion of truth, showing it to be consistent, or as prescriptions for modification of an intuitive notion of truth that is inconsistent. Those writing on the more purely philosophical side of the question of the paradoxes of truth do have to take a stand, and among them consistency theorists far outnumber inconsistency theorists. In surveying the more purely philosophical side of the subject in this chapter we will accordingly take consistency theories first, before turning to inconsistency theories, and then the relation between the consistency vs inconsistency debate and the realist vs antirealist vs deflationist debate.
8.1 PARADOXICAL REASONING
It will be well to have before us a particular paradoxical derivation. The one we wish to consider uses the equivalence principle not in the form of the T-biconditionals (which immediately raise questions about what kind of conditional is involved), but of rules of T-introduction and T-elimination. Writing T for “it is true that,” these and their negative counterparts read as follows:
(1a) from A to infer TA (1b) from TA to infer A
(1c) from ~A to infer ~TA (1d) from ~TA to infer ~A
Paradoxes result on applying such rules to such examples as
(2) (2) is false.
If we write p for (2) and F for “it is false that,” with falsehood as usual identified with truth of the negation, then passage back and forth among p and Fp and T~p is in effect just application of the rule of reiteration:
(3) from A to infer A
The only
other classical logical principle needed to arrive at the contradictory conclusions p and ~p is refutation by reductio:
(4) having seen B and ~B each to follow from A, to infer ~A
The further elaboration of the argument to get from negation consistency, or the deducibility of a pair of contradictory conclusions p and ~p, to absolute inconsistency, or the deducibility of any arbitrary conclusion q whatsoever, even “Santa Claus exists,” two more principles are needed, disjunction introduction and disjunctive syllogism:
(5) from A to infer A B
(6) from A B and ~A to infer B
The elaborated liar reasoning may be represented symbolically as follows, indenting steps that are merely an hypothesis and what follows under it, and leaving unindented those asserted categorically:
Two remarks will be in order. First, note that T-elimination was used at step (7d) hypothetically, to deduce a consequence from something merely assumed, while T-introduction was used at step (7f) only categorically, to deduce a consequence from something outright asserted. The reasoning could be rearranged to use T-introduction hypothetically and T-elimination categorically, but one or the other of the T-rules must be used hypothetically at some point.
Second, though negation is essentially involved in the derivation (7), paradox arises also without negation, using the conditional. The introduction and elimination rules for the conditional (stated in §6.5) and the rules of T-introduction and T-elimination together lead to absolute inconsistency: Curry's paradox derives the conclusion that Santa Claus exists by considering not “This sentence is false” but “If this sentence is true, then Santa Claus exists.” (Since it would take us too far afield to go further into the matter here, we must leave it as an exercise to the reader to figure out how.)
The same half-dozen strategies we saw used in §§4.3 and 4.4 to address problems of indeterminacy have been used to address the paradoxes. Considerations of space limit us to considering various strategies only in their simplest, most naive forms. Actual proposals in the literature—for citations see the “Further Reading” section following this chapter—are generally more sophisticated, with complications intended to forestall various objections.
8.2 “REVENGE”
Denial is represented by the claim that the equivalence principle is not after all a part or consequence of the intuitive notion of truth. Thus some claim that the principle “It is true that things are some way iff they are that way” is only a “habitual” generalization, on the order of “Birds fly,” not a genuinely universal generalization. This is usually maintained without offering any systematic account of the scope and limits and nature of the exceptions, except the indication that they are supposed to include liar-type examples. Assurances are made that we can really understand how it could be that either “(2) is not true” is not true, though (2) is not true, or that (2) is true, though “(2) is not true” is true. But those who find Moore's paradox (“It's raining outside, but I don't believe it is”) troubling will find this suggestion at least as much so.
Disqualification is represented by the claim that (2) does not express a proposition, perhaps the commonest initial reaction. This line is open to the same kind of objection that disqualificationism faced in connection with presupposition, namely, that supposed non-proposition-expressing sentences are found cited in explanation of action. As an eighteenth-century chemist, unaware of the failure of the presupposition that phlogiston exists, may explain an action by saying “The phlogiston has been removed from the air under the bell jar,” so a tourist ignorant of the paradoxical features of Epimenides' warning may cite that warning in explanation of why he skipped Crete on his Mediterranean tour, or a voter ignorant of the paradoxical features of Dean's accusation may cite that accusation in explanation of why she voted against Nixon.
But there is a far more serious problem, without parallel for presupposition, often referred to in the literature by a title like that of a B-movie in the horror genre: “The Revenge of the Liar.” The problem is that, given a purported solution to the liar paradox, it is generally possible to formulate, using the jargon of the solution, a new strengthened liar intuitively just as paradoxical as the original liar. If the solution to the ordinary liar is to claim that “This sentence is false” is in some sense defective, then the strengthened liar can be taken to be just “This sentence is false or defective.”
Specifically, if the “solution” to the falsehood-teller, “This sentence is false,” or
(8a) (8a) expresses a false proposition.
is to say that (8a) does not express a proposition, then the strengthened liar is just “This sentence either expresses a false proposition or no proposition at all,” which amounts to the untruth-teller
(8b) (8b) does not express a true proposition.
or “This sentence is not true.” (We leave it to the reader to think through how (8b) leads to paradox by a slight variant of the reasoning by which (8a) led to paradox.)
The would-be vindicator of the intuitive notion of truth who says of (8b) what was said of (8a), that it does not express a proposition, faces the reply, “So a fortiori (8b) does not express a true proposition, and since that is just what it says, it is true.” The would-be vindicator may then insist, “No, (8b) does not say that; (8b) does not ‘say' anything; (8b) does not express a proposition. So a fortiori (8b) does not express a true proposition, and contrary to what you say, (8b) is not true.” But then the would-be vindicator will be in the position of having said something (namely, (8b), which is to say, “(8b) does not express a true proposition”), and also having said that it is not true. The problem, sometimes called that of ineffability, is that the would-be vindicator's theory cannot be enunciated, at least not without saying something that, according to that theory itself, is untrue.
8.3 LOGICAL “SOLUTIONS”
Deviance is represented by proposals to adopt a logic of truth-value gaps, such as the Kleene strong trivalent logic, also called partial logic. (Adopting the intuitionistic logic advocated by antirealists would be no help with the paradoxes, since the principles of classical logic that lead to trouble in the liar and Curry cases are all accepted in intuitionistic logic, too.) According to partial logic, the truth values of compounds are determined from those of their components according to the adjoining tables, wherein 1 represents truth, 0 falsehood, and ? truth-valuelessness or gappiness. (The tables restate in symbols what those who have read §2.4 have seen stated there in words.)
In this logic, as the reader can work out from the tables, if B is gappy, so is A = B ~B. But B and ~B each follow by a truth-preserving rule (to infer a conjunct from a conjunction) from A, even though ~A is not true, and in this sense reductio (4) fails, and the paradoxical reasoning is blocked at step (7e). But the equivalence principle in the form of the rules (1abcd) is saved, if one takes TA to have always the same truth value (1 or 0 or ?) as A. That is, in effect, what the Kripke construction of the preceding chapter shows.
This “solution,” however, faces the problem of ineffability. For the would-be vindicator wants to say that (2) is neither true nor false, whereas ~Tp ~T~p comes out not true but gappy if p is gappy, so the enunciation of the view involves saying something that according to the view is not true.
Kripke is perfectly aware of the problem. For him, the fixed-point construction may be a model of natural language, but if so it is a model of natural language only at a “prereflective” stage. His own description of that model, in which he says that liar examples are neither true nor false, is from the perspective of a later and higher “reflective” stage of development. One conspicuous difference between the two stages is in their logics. The internal logic, the logic involved in the model, is a nonclassical one, the Kleene strong trivalent logic. The external logic, the logic of Kripke's proofs about the model, is as Kripke emphasizes classical, as in any other contribution to orthodox mathematics.
Kripke writes: “If we think of the minimal fixed point…as giving a model of natu
ral language, then the sense in which we can say, in natural language, that a Liar sentence is not true must be thought of as associated with some later stage in the development of natural language, one in which speakers reflect on the generation process leading to the minimal fixed point. It is not itself a part of that process…. The ghost of the Tarski hierarchy is still with us.”
An analogy offered by Soames can help to clarify this aspect of Kripke's view. Imagine someone pointing to a number of persons, none taller than four-foot-six, and saying that these paradigms, along with anyone shorter than any of them, are all smidgets, and then pointing to a number of other persons, none shorter than five-foot-six, and saying that these foils, along with anyone taller than any of them, are nonsmidgets. These specifications leave Mr. Smallman, who is exactly five feet tall, unclassified. For Soames, the (partial) specifications Kripke gives for what is to count as “true” (which those who have read §7.4 have seen set down as (13) and (14) of that section) are comparable to the partial specifications for the term “smidget.” Liar and other pathological sentences are analogues of Mr. Smallman.