Liar Paradox, The
LIAR PARADOX, THE
Attributions of truth and falsehood under certain conditions generate the "liar paradox." The most famous illustration of this comes from the Epistle to Titus, in which St. Paul quotes approvingly a remark attributed to Epimenides: "One of themselves, even a prophet of their own, said, The Cretans are always liars, evil beasts, slow bellies. This witness is true" (King James version). Let us suppose that Epimenides, the Cretan prophet, did say that the Cretans are always liars, and let us consider the status of his utterance—call it E —under the following two conditions. (1) A Cretan utterance counts as a lie if and only if it is untrue. (2) All Cretan utterances, except perhaps E, are untrue. Now, if E is true, then, since E is a Cretan utterance, not all Cretan utterances are untrue. Hence, Cretans are not always liars (by (1)), and so E must be untrue. On the other hand, if E is untrue, then indeed all Cretan utterances are untrue (by (2)). Hence, Cretans are always liars (by (1)), and so E is true after all. Both the hypotheses, that E is true and that E is not true, yield, therefore, a contradiction. Yet the steps in the argument are all apparently valid, and the initial setup is not impossible. This is the liar paradox.
The paradox was discovered by Eubulides of Miletus (fourth century BCE) and has exercised logicians down the ages to the present time. (See Bocheński 1961, Spade 1988.) For principally two reasons, interest in the paradox was especially great in the twentieth century. First, arguments similar to that found in the liar wreaked havoc in several prominent logical systems (e.g., those of Gottlob Frege and Alonzo Church). This prompted a search for systems that were immune from paradox. Second, the rise of semantical studies created a need for a better understanding of the notions of truth, reference, and the like. The notions are fundamental to semantical investigations, but the paradoxes reveal a profound gap in our understanding of them. (The notion of reference, like other semantical notions, exhibits, under certain conditions, paradoxical behavior.)
The liar and related paradoxes raise a number of difficult conceptual problems. One is the normative problem of designing paradox-free notions of truth, reference, and the like. Another is the descriptive problem of understanding the workings of our ordinary, paradox-laden notions. The work on the paradoxes in the first half of the twentieth century is, perhaps, best viewed as addressing the normative problem. The work in the second half is best viewed as addressing the descriptive problem. Some of this work is outlined below.
Let us sharpen the descriptive problem a little. For simplicity, let us restrict our attention to a fragment, L, of our language that contains no problematic terms other than "true." All other terms in L have, let us suppose, a classical interpretation. How should "true" be interpreted? A natural demand is that the interpretation must validate the T-biconditionals, that is, all sentences of the form,
(T) "B " is true if and only if (iff) B,
where B is a sentence of L. The argument of the liar paradox shows, however, that every possible classical interpretation of "true" is bound to make some T-biconditionals false. (This is a version of Alfred Tarski's indefinability theorem.) How, then, should we interpret "true"? Should we abandon the natural demand? Or the classical framework? Or the naive reading of the T-biconditionals? Essentially, the first course is followed in the contextual approach, the second in the fixed-point approach, and the third in the revision approach.
The Contextual Approach
This approach takes "true" to be a context-sensitive term. Just as the interpretation of "fish this long" varies with contextually supplied information about length, similarly, on the contextual approach, with "true": Its interpretation also depends upon contextual information. There is no consensus, however, on the specific information needed for interpretation. In the levels theory due to Tyler Burge and Charles Parsons, the context supplies the level at which "true" is interpreted in a Tarskian hierarchy of truth predicates. In the Austinian theory of truth developed by Jon Barwise and John Etchemendy, the relevant contextual parameter is the "portion" of the world that a proposition is about. In the singularity theory of Keith Simmons, the relevant information includes certain of the speaker's intentions.
Contextual theories assign to each occurrence of "true" a classical interpretation, though not the same one to all occurrences. This has several characteristic consequences: (1) Occurrences of "true" do not express global truth for the entire language (by Tarski's indefinability theorem). They express instead restricted or "quasi" notions of truth; the former possibility is realized in the levels theory, the latter in the singularity theory. (2) Truth attributions, even paradoxical ones, have a classical truth-value. Paradox is explained as arising from a subtle, unnoticed, shift in some contextual parameter. (3) Classical forms of reasoning are preserved. But caution is in order here: Whether an argument exemplifies a classically valid form turns out to be nontrivial. For example, the argument "a is true, a = b ; therefore, b is true" exemplifies a classically valid form only if "true" is interpreted uniformly, but this is nontrivial on the contextual approach.
The Fixed-Point Approach
This approach interprets "true" nonclassically. It rests on an important observation of Saul Kripke, Robert Martin, and Peter Woodruff. Consider again the language L, and assign to "true" an arbitrary partial interpretation 〈U, V 〉, where U is the extension and V the antiextension (i.e., the objects of which the predicate is false). We can use one of the partial-valued schemes (say, Strong Kleene) to determine the sentences of L that are true (U ′), false (V ′), and neither-true-nor-false. This semantical reflection defines a function, κ, on partial interpretations; κ(〈U, V 〉) = 〈U ′, V ′〉. The important observation is that κ has a fixed point: There exist 〈U, V 〉 such that κ(〈U, V 〉) = 〈U, V 〉.
Certain partial-valued schemes have a least fixed point, which is a particularly attractive interpretation for "true." It is also the product of an appealing iterative construction: We begin by supposing that we are entirely ignorant of the extension and the antiextension of "true"; we set them both to be ∅ (the null set). Despite the ignorance, we can assert some sentences and deny others. The rule "Assert 'B is true' for all assertible B ; assert 'B is not true' for all deniable B " entitles us to a new, richer interpretation, κ(〈∅, ∅〉), for "true." But now we can assert (deny) more sentences. The rule entitles us to a yet richer interpretation κ(κ(〈∅, ∅〉)). The process, if repeated sufficiently many times, saturates at the least fixed point.
Under fixed-point interpretations, the extension of "true" consists precisely of the truths and the antiextension of falsehoods. The T-biconditionals are, therefore, validated. They are not, however, expressible in L itself: fixed points exist only when certain three-valued functions, including the relevant "iff," are inexpressible in L.
The Revision Approach
This approach holds truth to be a circular concept. It is motivated by the observation that truth behaves in a strikingly parallel way to concepts with circular definitions. Suppose we define G thus:
x is G = Dfx is a philosopher distinct from Plato or
x is Plato but not G.
The definition is circular, but it does impart some meaning to G. G has, like truth, unproblematic application on a large range of objects. It applies to all philosophers distinct from Plato and fails to apply to nonphilosophers. On one object, Plato, G behaves paradoxically. If we declare Plato is G, then the definition rules that he is not G ; if we declare he is not G, the definition rules that he is G. This parallels exactly the behavior of truth in the liar paradox.
The revision account of truth rests on general theories of definitions, theories that make semantic sense of circular (and mutually interdependent) definitions. Central to these theories are the following ideas. (1) A circular definition does not, in general, determine a classical extension for the definiendum (the term defined). (2) It determines instead a rule of revision. Given a hypothesis about the extension of the definiendum G, the definition yields a revised extension for G, one consisting of objects that satisfy the definiens (the right side of the definition). (3) Repeated applications of the revision rule to arbitrary hypotheses reveal both the unproblematic and the pathological behavior of the definiendum. On the unproblematic the revision rule yields a definite and stable verdict, irrespective of the initial hypothesis. On the pathological this ideal state does not obtain.
The ingredient needed to construct a theory of truth once we have a general theory of definitions is minimal: It is just the T-biconditionals, with "iff" read as "= Df." This reading was suggested by Tarski, but, as it results in a circular definition, it can be implemented only within a general theory of definitions. Under the reading, the T-biconditionals yield a rule of revision. Repeated applications of this rule generate patterns that explain the ordinary and the pathological behavior of truth. The revision approach thus sees the liar paradox as arising from a circularity in truth. The approach has been developed by, among others, Anil Gupta, Hans Herzberger, and Nuel Belnap.
The three approaches, it should be stressed, do not exhaust the rich array of responses to the paradoxes in the twentieth century.
Antonelli, A. "Non-Well-Founded Sets via Revision Rules." Journal of Philosophical Logic 23 (1994): 633–679.
Bocheński, I. M. A History of Formal Logic. Notre Dame, IN, 1961.
Chapuis, A. "Alternative Revision Theories of Truth." Journal of Philosophical Logic 24 (1996).
Epstein, R. L. "A Theory of Truth Based on a Medieval Solution to the Liar Paradox." History and Philosophy of Logic 13 (1992): 149–177.
Gaifman, H. "Pointers to Truth." Journal of Philosophy 89 (1992): 223–261.
Gupta, A., and N. Belnap. The Revision Theory of Truth. Cambridge, MA: MIT Press, 1993.
Koons, R. C. Paradoxes of Belief and Strategic Rationality. Cambridge, U.K.: Cambridge University Press, 1992.
Martin, R. L., ed. The Paradox of the Liar, 2nd ed. Reseda, CA, 1978. Contains a useful bibliography of material up to about 1975; for later material consult the bibliography in Gupta and Belnap, 1993.
Martin, R. L., ed. Recent Essays on Truth and the Liar Paradox. New York: Oxford University Press, 1984. Contains the classic papers of Parsons, Kripke, Herzberger, and others; a good place to begin the study of the three approaches.
McGee, V. Truth, Vagueness, and Paradox. Indianapolis: Hackett, 1991.
Priest, G. In Contradiction. Dordrecht: Nijhoff, 1987.
Russell, B. "Mathematical Logic as Based on the Theory of Types." In Logic and Knowledge. London: Allen and Unwin, 1956.
Sainsbury, R. M. Paradoxes. Cambridge, U.K.: Cambridge University Press, 1988.
Simmons, K. Universality and the Liar. New York: Cambridge University Press, 1993.
Spade, P. V. Lies, Language, and Logic in the Late Middle Ages. London: Variorum, 1988.
Tarski, A. "The Semantic Conception of Truth." Philosophy and Phenomenological Research 4 (1944): 341–376.
Visser, A. "Semantics and the Liar Paradox." In Handbook of Philosophical Logic, edited by D. Gabbay and F. Guenthner, Vol. 4. Dordrecht: Reidel, 1989.
Yablo, S. "Hop, Skip, and Jump: The Agonistic Conception of Truth." In Philosophical Perspectives, edited by J. Tomberlin, Vol. 7. Atascadero, CA: Ridgeview, 1993.
Yaqūb, A. M. The Liar Speaks the Truth. New York: Oxford University Press, 1993.
Anil Gupta (1996)
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