Kripke outline

By Mary Lee,2014-05-29 22:57
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Kripke outline



     Saul Kripke‘s ‗Outline of a Theory of Truth‘ [1975] has been the most influential publication on truth and paradox since Alfred Tarski‘s ‗The Concept of Truth in Formalized Languages‘ [1935]. It is thick with allusions to related unpublished work, and

    the present account will provide some information on this additional material, but the ubiquitously cited ‗Outline‘ must remain the main focus in the limited space available.


     Let us dispose at the outset of the potentially distracting issue of the bearers of

    truth. Suppose Y says ‗What X just said is true,‘ and Z asks, ‗But what did X say?‘ Then

    Y may answer with either a direct or an indirect quotation of X, perhaps saying ‗X said, ―Snow is white,‖ and that‘s true,‘ or perhaps saying ‗X said that snow is white, and that‘s true.‘ Since it seems that a direct quotation denotes a sentence while a that-clause denotes

    a proposition, it seems that Y is attributing truth in one case to the sentence X uttered and in the other to the proposition X thereby asserted.

     Kripke applies ‗true‘ and ‗false‘ to sentences, though he holds that some sentences are neither; but some philosophers apply ‗true‘ and ‗false‘ only to propositions, and some hold that every proposition is one or the other. Is there more than a merely verbal difference here? Well, nothing Kripke says is incompatible with the view that ‗true‘ and ‗false‘ apply to sentences only in a sense derivative from their application to


    propositions (a sentence counting as true or false only insofar as it expresses a true or a false proposition), or with the view that a sentence can fail to have a truth value only by failing to express a proposition (rather than by expressing a proposition that fails to have a truth value). Kripke focuses on sentences rather than propositions because he finds it

    1clear that sentences can refer to themselves and unclear whether propositions can do so.

     In the pre-Kripkean literature at least four kinds of apparently self-referential sentences in natural language can be distinguished, giving rise to four different kinds of liar paradox. On the one hand, it appears that a sentence may refer to itself directly. This may be through demonstratives, as with:

(1a) This very sentence is false.

Or it may be with proper names, as when we let ‗Pseudomenon‘ name the sentence:

(1b) Pseudomenon is false.

    On the other hand, it appears that a sentence may refer to itself indirectly, by describing a certain sentence while being itself the unique sentence fitting that description (or by making a generalization about sentences of some kind while being itself a crucial instance of that kind). This may be so for what may be called structural reasons, as with the Grelling-Quine example:


    (1c) ‗yields a falsehood when appended to its own quotation‘

     yields a falsehood false when appended to its own quotation.

    Or it may be so for what may be called historical reasons, as when the only sentence on the blackboard in room 101 happens to be:

(1d) The only sentence on the blackboard in room 101 is false.

    Another type common in the literature arises when reference numbers for displayed sentences are used in philosophy papers, as with (1a-d) above. We may then get something like this:

(1e) (1e) is false

    It is unclear whether this type is best assimilated to one of the other types or considered sui generis.

     In cases (1a,b,c) the self-reference is, so to speak, intrinsic, but in (1d) it is

    extrinsic. What may be the earliest form of the liar, in which Epimenides the Cretan says that everything said by Cretans is false, which is paradoxical only on the historical assumption that everything else said by Cretans is false, is also of the extrinsic kind. Kripke does not deny the possibility of the (1a) kind of self-reference, argues for the possibility of the (1b) kind, recalls the possibility of the (1c) kind, and emphasizes the


    possibility of the (1d) type, from which he draws the conclusion that it hopeless to look for any intrinsic test for paradoxicality. He also uses the (1e) type without comment.

     As for the possibility of self-reference in formal as contrasted with natural languages, Gödel proves that under fairly modest conditions or at any rate, under

     there will conditions fairly modest for a language in which to discuss mathematics

    always exist a sentence that in effect refers to itself and attributes to itself any desired property expressible by a formula of the language. (The phrase ‗in effect‘ here is by way of acknowledgment of the complication that in the Gödel situation as standardly presented one does not literally have reference by linguistic expressions to linguistic expressions, but only to code numbers of linguistic expressions.) Moreover, under the

    same fairly modest conditions, a wide range of syntactic properties (including provability) will be expressible by formulas of the language. Kripke in the ‗Outline‘ displays little interest in formal languages so artificially truncated as not to meet Gödel‘s fairly modest conditions.

     While it is self-reference and consequent dependence of the truth or falsehood of a sentence S on the truth or falsehood of S itself that is at the root of the liar examples, 00

    it has been recognized since mediæval times that paradox can also arise when S depends 0

    on S while S depends on S, as when Socrates says ‗What Plato is about to say will be 110

    false,‘ and Plato says ‗What Socrates just said was true.‘ Longer circles are also possible, and Kripke recognized also the possibility of problematic examples resulting from infinite sequences where S depends on S, which depends on S, which depends on S, 0123

    2and so on.


     In mathematics a dependence relation is called non-well-founded if there is a non-

    empty set having no element not dependent on an element of the set. (This covers all three cases of a depending on itself, circles with adepending on adepending on … a 00 1 n

    depending on a, and infinite sequences with a depending on adepending on a and so 001 2

    on. The ‗bad‘ sets in these different cases are {a} and {a, a, , a} and 001n

    {a, a, a, }.) Of course, the rigorous mathematical terminology cannot properly be 012

    applied until we have a rigorous mathematical definition of the dependence relation at issue. Kripke picks up from the literature the alternative term ‗ungroundedness,‘ which prior to his treatment had been used only in an intuitive way, and aims eventually to give it a rigorous definition.


     The liar paradox was introduced by Eubulides and much discussed by Chrysippus and others in ancient times, while it and related paradoxes, under the label insolubilia,

    3were much discussed by Bradwardine and others in the middle ages. Modern

    discussions began in the period around 1900, when the set-theoretic paradoxes were being discovered, from which period date paradoxes about the notion of definability (Berry‘s, Richard‘s, König‘s) that brought truth-related notions into disrepute among

    mathematicians. Tarski, envisioning significant mathematical applications of such notions, sought a partial restoration of their reputation in work that is the starting point for all later discussions, Kripke‘s included.


     Tarski‘s work has both a positive and a negative side. Both start from the assumption that a predicate may be deemed a truth predicate for a language just in case

    we have the following for all sentences of the language:

    ) if and only if A (2) T(a

where the term a denotes the sentence A as the quotation ‗‗Snow is white‘‘ denotes the

    sentence ‗Snow is white.‘ There is no need for a separate falsehood predicate if one

    assumes, as do Tarski and his successors, that the falsehood of an item is equivalent to the truth of its negation and the truth of an item to the falsehood of its negation.

    On the negative side, Tarski observes that if, as with natural language, the language for whose sentences the predicate expresses truth itself contains that very predicate, and has means of self-reference, then universal applicability of (2) leads to contradiction. Hence the slogan: ‗A language cannot contain its own truth predicate,‘ or in terminology whose felicity is open to question, ‗A language cannot be semantically closed.‘ Tarski concludes that the intuitive notion of truth, expressed by the predicate ‗is true‘ of natural language, is contradictory. He therefore seeks to rehabilitate not this

    intuitive notion, but only a restricted version, sufficient for his envisioned applications, predicated only of sentences of formal language not itself containing the truth predicate.

     Kripke emphasizes that it is a gross mistake to say that Tarski bans self-reference. He could not have done so even if he had wanted to, owing to Gödel‘s results. If he bans anything, it is the presence of semantic predicates (such as truth) as contrasted with

    syntactic predicates (such as provability). Even here ‗ban‘ is not the right word, since


    when Tarski says ‗A semantically closed language is impossible,‘ what he means is not an imperative, ‗Thou shalt not make unto thee any semantically closed language,‘ but rather a declarative, ‗You could not have a semantically closed language even if you

    wanted to.‘

     On the positive side, Tarski proposes for a formal languageof a certain kind to

    give a mathematically rigorous definition of a predicate for which all instances of (2) admit mathematically rigorous proof. The kind of formal language Tarski considers has predicates and terms, from which may be formed atomic sentences, from which may be formed other sentences using negation, conjunction, disjunction, and universal and existential quantification. There is a universe of discourse, and elements of that universe

    -tuples of elements are assigned as are assigned as denotations to the terms while sets of n

    extensions to the n-place predicates.

     On the simplifying assumption that every element of the universe of discourse is the denotation of some term, the extension of a one-place predicate F is completely

    determined by a total assignment of truth values, true or false, to all atomic sentences of form F(a), wherein a may be any term subject to the proviso that a and b have the

    same denotation, then F(a) and F(b) must be both true or both false. Similarly in the

    many-place case. From the assignment of truth values to atomic sentences the assignment of truth values to other sentences is determined by the rules of classical logic.

     The rule for negation has already been mentioned: A negation is true if and only if what it negates is false, and vice versa. A conjunction is true if all conjuncts are true, and false if all or some conjuncts are false and any others true. Disjunctions are treated dually (a disjunction is false if all disjuncts are false, and true if all or some disjuncts are


    true and any others false). On our simplifying assumption, quantifications are treated analogously (a universal quantification is true if all instances are true, and false if all or some instances are false and any others true, and existential quantifications dually). Without the assumption that every element of the universe of discourse is the denotation of some term, the definition is more complicated, and requires an auxiliary notion of satisfaction. The details need not be recalled here, since the definition given in any present-day logic textbook is close to Tarski‘s.

    The connection between the negative and positive sides of Tarski‘s work is that

    his truth definition just discussed is given not in the ‗object language‘ itself, but in a ‗metalanguage.‘ On Tarski‘s approach, if we wanted a notion of truth not just for the original formal language L, but for the language L containing of the original language 01

    plus a predicate T for ‗is a true sentence of L‘ we would need another truth predicate T; 001

    and if we wanted a notion of truth not just for L, but for the language L containing L 121

    plus a predicate T for ‗is a true sentence of L,‘ we would need yet another truth 11

    predicate T; and so on. But in practice the applications made by Tarski and by 2

    subsequent workers in model theory, the branch of logic originating with his paper,

    hardly ever involve such a hierarchy of languages and truth predicates indeed, they

    hardly ever require any truth predicate beyond the initial one.


     Some subsequent philosophical writers many of whom seem to have been,

    unlike Tarski, seeking to vindicate the intuitive notion of truth, seeking to show that it is subject only to apparent paradoxes, not real antinomies have suggested that, contrary


    to Tarski‘s conclusions, a language can contain its own truth predicate, if one gives up

    the classical assumption of bivalence (that every sentence is true or false) and allows truth-value gaps. (If there is anything that may properly be called a ‗ban‘ in Tarski, it is an unstated prohibition against considering such gaps.) Kripke‘s main aim in the ‗Outline‘ is to produce a rigorous version of the truth-value-gap proposal.

     Quite independently of considerations about paradoxes, several circumstances in which truth-value gaps arguably arise have been noted in the philosophical literature. These include failure of existential presupposition (‗The King of France is bald,‘ said

    after the abolition of the French monarchy), vagueness (‗The King of France is bald,‘

    said under the French monarchy at a time when the reigning monarch has lost much but not all of his hair), and nonsense (‗The King of France is a boojum,‘ said at any time).

     Suppose then we work with a language that is a modification of the kind considered by Tarski, in that each n-place predicate is not just assigned an extension, or set of n-tuples of elements of the universe of discourse of which it is true, but an extension and anti-extension, a set of n-tuples of which it is true and a set of n-tuples of

    which it is false, with the two sets non-overlapping, but with their union perhaps not the whole universe of discourse. On the simplifying assumption that every element of the universe of discourse is the denotation of some term, the extension of a one-place predicate F is completely determined by a partial assignment of truth values, true or false,

    to some atomic sentences of form F(a), wherein a may be any term subject to the

    proviso that a and b have the same denotation, then F(a) and F(b) must be both true or

    both false or both without truth value. Similarly in the many-place case.


     But by what rules is the partial assignment of truth values to other sentences to determined from the partial assignment of truth values to atomic sentences? Different schemes of rules have been proposed for evaluating logical compounds some or all of whose logical components may lack truth value, with some schemes looking more plausible for some types of truth-value gap and others for others, and with all schemes agreeing with the classical rules recalled earlier in cases where all logical components do have truth values.

     On the Frege weak three-valued scheme, any compound having a component

    without truth value is without truth value. On the Kleene strong three-valued scheme, a

    conjunction is false if at least one conjunct is false, and without truth value if least one conjunct is without truth value and no conjunct is false. Disjunction is treated dually (a disjunction is true if at least one disjunct is true, and without truth value if at least one disjunct is without truth value and no disjunct is true). Universal and existential quantification are treated analogously to conjunction and disjunction (a universal quantification is false if any instance is false and without truth value if at least one instance is without truth value and no instance is false, and dually for existential quantification).

     On the more elaborate van Fraassen supervaluational approach, a partial

    valuation of atomic sentences determines a set of total valuations, namely, the set of all those that agree with the partial valuation as far as it goes. A non-atomic sentence then counts as true under the original partial valuation if it is true under all these total valuations, and false if false under all, and otherwise without truth value. Thus in contrast to the Kleene scheme, where a conjunction of two sentences without truth value is

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