Reflections on the Liar

Bradley Armour-Garb's new collection, Reflections on the Liar, brings together a number of essays by accomplished authors who have not previously brought their talents sustainedly to bear on the Liar paradox. The volume thus presents an opportunity for scholars of the paradox to view the Liar from the perspectives of other related areas of philosophy. Among the many illuminating contributions, I will limit my focus to the four about which I have the most to say.Ian Rumfitt's contribution centers on a version of the Liar paradox that he formulates without invoking truth, from which he concludes that the Liar is “centrally about the limits of what can be said, and only derivatively about truth” (198). To obtain the Liar without truth, Rumfitt admits an infinite class of objectual variables P, Q, R, . . . that can take only the sentence position and that range over ways that things may be said to be (henceforth propositions). Also, Rumfitt posits a relation of saying which a person can bear to a proposition relative to a context, by uttering a sentence therein. Let ‘δP’ abbreviate ‘At noon on day d, [Professor] Brainstorm says that P’, and assume that at noon on day d, Professor Brainstorm utters the following sentence:(L) ¬ ∃ P (δP&P).Relative to that context, L is a Liar sentence. Rumfitt demonstrates this by deriving a contradiction from L, plus the assumption that Brainstorm utters L at noon on day d. Significantly, Rumfitt's derivation does not invoke truth. Call the paradox thus illustrated Rumfitt's Paradox.1On one reading of Rumfitt's conclusion that the Liar paradox is “centrally about the limits on what can be said,” there are no limits on what propositions there are, just limits on which sentences (or which utterances thereof) can express which propositions (Skyrms 1984; Gaifman 1988, 1992, 2000; Whittle [also in Reflections on the Liar]; Epstein 2017). But Rumfitt's view is that there are limits on what propositions exist. He denies that there is any such proposition as P0:(P0) There is nothing P such that (a) when Brainstorm utters ‘¬ ∃ P(δP&P)’ at noon on day d, Brainstorm says P, and (b) P.On first blush, in denying the existence of P0, Rumfitt simply embroils himself in contradictions. For any sentence that Rumfitt might use to communicate this denial would appear to express none other than P0. The simplest such sentence is Brainstorm's own ‘¬ ∃ P (δP&P)’, which, if it expresses anything at all, arguably expresses P0. At least initially, then, it is unclear how one could express the denial that P0 exists. This brings us to Rumfitt's solution.Rumfitt centrally distinguishes between rejecting ‘ ∃ P (δP&P)’ as untrue and asserting its negation ‘¬ ∃ P (δP&P)’ as true. Thus, Rumfitt posits signed formulae, which express these attitudes of acceptance or rejection toward formulae; and he (2000) presents “a system of logical rules which regulate inferential transitions between . . . signed formulae.” One enters ‘+A’ “to signify that A is accepted as true,” and “by writing down –A” one effects a rejection of A as untrue (207).The point is that rather than accepting the negation ‘¬ ∃ P (δP&P)’ and accordingly entering ‘+¬ ∃ P (δP&P)’, one can (and can only) simply reject ‘ ∃ P (δP&P)’, entering the strictly weaker ‘– ∃ P (δP&P)’. Importantly, rejecting ‘ ∃ P (δP&P)’ need not involve expressing, and therefore countenancing, any such proposition as P0 (which, again, would itself be such a P). Thus, Rumfitt avoids revenge.Two questions linger. First: is it really possible to reject a formula as untrue without countenancing the proposition that this formula is untrue? (This is a centrally motivating concern for Rumfitt's competitors Brian Skyrms, Haim Gaifman, and Bruno Whittle [and myself].) Second: how well does refusal to countenance P0 sit with the leading theories about propositions? The obvious merit of Rumfitt's solution makes these questions worthy of further research.James Shaw's contribution lays out the motivation for a compositional semantics of truth-talk that can handle semantic circularity, and identifies a desideratum facing any such program. The motivation comes from the fact that any fully general semantic theory must apply to its own terminology (214). If the sentences of this theory are to be given a compositional semantics then this semantics must be able to handle circularity (215).One particular challenge brought by circularity is that of defaulters. Contrast (4) and (T):(4) Everything Sid says today will be true.(T) (T) is true.Whereas English speakers tend to be uncertain about (T), they have a default practice of taking (4) to be true (provided that no other sentence Sid utters that day is untrue). Yet, both (4) and (T) involve semantic circularity. Shaw argues that speakers' judgments about (4) and (T) need to be taken seriously. If that is correct, then different varieties of semantic circularity must be handled in different ways. Unlike for (T), there must be something in the semantics of (4) that resolves any uncertainty in favor of the claim that (4) is true. This is Shaw's desideratum on compositional semantics.For Shaw the best way to meet this desideratum is to revise our received views about the semantic values of semantic expressions. In particular, we should reject Truth Extensionalism:(Truth Extensionalism) The compositional contribution of ‘true’ [to sentences that contain it] is[, like that of a normal predicate,] an extension (relative to a context/index pair, perhaps along with an anti-extension, etc.). (220)Instead, we should embrace a view called Truth-Proceduralism, which he attributes to Gaifman (1992):The meaning of ‘true’ is not just an extension, or intension, but some special kind of method. This is a method in which conventionally determined partial information about the set of truth values is used in a sequential, compositional assignment of truth values to utterances. We've always known that speakers proceed in assigning truth values to utterances in roughly specifiable patterns. What is unique about Truth-Proceduralism is that it constitutively links these patterns to the meaning of ‘true’. (243–44)To evaluate Shaw's assertion that the semantic value of ‘true’ includes a method as well as an extension, we must ask: what is the notion of semantic value for? Addressing this question, Seth Yalcin (2014: 30) identifies several different roles, one of which is that of explaining speakers' judgments about the truth or falsehood of various sentences. At least by that criterion, this procedure deserves to count as part of the semantic value of ‘true’: as we saw, one way to explain speakers' differential judgments about (4) and (T) is to posit an evaluation procedure that rules (4) to be true and (T) indeterminate.Bruno Whittle's contribution motivates the development of noncompositional approaches to the Liar paradox along the lines of Gaifman 1988, 1992, and 2000 (see also Epstein 2017), and sketches one new such approach. Most alternative views, he argues, face unsatisfactory expressive limitations of one or another sort. Here, Whittle considers Kripke 1975, Maudlin 2004, Gupta and Belnap 1993, and Field 2008. On all of these views, a language can either contain its own truth-predicate or contain an operator ‘∼’ that expresses exclusion negation (where if S is indeterminate then ∼ T(‘S’) is true), but not both. Either way, there are serious limitations on what can be said in the language.Whittle proposes to escape this situation “by accepting that [each Liar sentence] is an exception to the standard compositional rules” (311). To see the idea, let ‘b’ and ‘c’ both be names for the Liar sentence ‘¬Tc’. Then whereas the sentence ‘¬Tc’ says something about itself, the sentence ‘¬Tb’ says something about the sentence ‘¬Tc’ (that is, b), which is distinct from itself, ‘¬Tb’. For Whittle and other noncompositional theorists, this difference in referential patterns generates a difference in truth values: ‘¬Tc’ is neither true nor false; but this then renders ‘¬Tb’ true. What a sentence cannot say about itself can be said by another, distinct sentence of the same language.Initially, one might doubt that one can give a systematic semantics for a language that violates standard compositional principles. But, following Gaifman 2000, Whittle does exactly this. What is more, whereas Gaifman focused predominantly on tokens, Whittle's adaptation of Gaifman's semantics explicitly handles Liar sentence-types such as c above.Like its competitors, Whittle's solution faces revenge. Let sentences A and B be similar if and only if each is obtainable from the other by substituting coreferential expressions. Given this, if ‘d’ refers to E below, then E is a Liar sentence:(E) ∀ x (x is similar to d ⊃ ¬Tx).In Whittle's semantics, E receives the value n. Moreover, any sentence A similar to E also receives n. But this fact cannot be expressed (by a true sentence) in the language. For let ‘g’ be a new name of E, and then try to express what E cannot, using ‘g’ in place of ‘d’:(E′) ∀ x (x is similar to g ⊃ ¬Tx).E′ “will simply be one of the sentences ‘similar’ to E,” and so will receive the alethic status n (318).This objection shows that languages such as those Whittle describes have significant expressive limitations, violating the motivation for Whittle's view. But Whittle has a response: “This expressive restriction [is] fundamentally different” from those that plague the other competing views (319). Those views either “cannot express the general notion of truth” or cannot express exclusion negation. Either way, these expressive restrictions “prohibit natural claims that have nothing in particular to do with paradoxes,” such as the claim that every sentence is either true or not, where ‘not’ is understood in the exclusion sense (319). By contrast, for Whittle, “all we have are certain ‘strong’ paradoxes that we cannot say certain things about without ourselves saying something . . . that is neither true nor false” (319). This is “far less objectionable than not being able to express important general claims that do not . . . have anything . . . to do with paradoxes” (319).Turn now to Timothy Williamson's contribution. Historically, many approaches to the Liar paradox have involved replacing classical logic by another logic. But Williamson argues that this is a mistake: rather, classical logic ought to be scrupulously maintained. Williamson's argument relies on two premises: first, that philosophers should use abductive methodology when deciding between rival logics, and second, that this methodology strongly favors classical logic over any of its competitors, because classical logic exhibits an optimal combination of simplicity and strength. Williamson takes these considerations both to provide a strong prima facie case for classical logic and to support the retention of classical logic in the face of the semantic paradoxes in particular.Loosely speaking, a consequence relation is a relation that captures a notion of what follows from what.2 Since many such relations exist, we face a question of which one to reason in accordance with. To answer this question, Williamson makes the following proposal. For a consequence relation ├ and a set Γ of sentences, let Cn├ (Γ) be {φ: Γ├ φ}, the set of consequences of Γ under ├. Then we are to decide between rival consequence relations ├ and ├* by considering various sets Γ of independently well-confirmed sentences “such as well-established principles of physics,” and comparing the consequences Cn├(Γ) and Cn├*(Γ) of Γ under ├ and ├*, respectively. We choose the consequence relation under which Γ has the consequences that it properly should have (334).An immediate question about this method is, how do we figure out which consequences a given Γ “properly should” have? This is where abductive methodology comes in. For each consequence relation ├, Cn├(Γ) is to be thought of as a theory. Accordingly, the choice between distinct ├ and ├* amounts to a choice between theories, the sort of choice that scientists face in many other disciplines. “Scientific theories are compared with respect to how well they fit the evidence . . . but also with respect to virtues such as strength, simplicity, elegance, and unifying power” (334).Williamson then argues that there is a strong prima facie case for classical logic, since it is so far unmatched in “its combination of simplicity and strength” (337). Classical propositional logic is particularly strong, since it is Post complete—“the only consequence relation properly [containing] the classical one is trivial (everything follows from everything)” (337). In other words, concerning what follows from what, classical propositional logic is as permissive as one can get without saying that everything follows from everything. First-order classical logic is not Post-complete, but is, Williamson insists, stronger than its rivals “at least in the looser scientific sense [of being more informative], as well as being simpler than they are” (338). Here Williamson refers readers to Williamson 2007, 2013, and 2014.Now turn to the paradoxes. The semantic paradoxes are standardly taken to show that classical logic and disquotation are in tension: we must restrict one or the other. This makes for a case against classical logic that “is better than most cases . . . such as that from the sorites paradoxes,” because the principles of disquotation help to “compensate for the lost simplicity and strength of classical logic” (339). But nonetheless, Williamson insists that classical logic should not be revised. Whereas “the constants of classical logic seem to express absolutely fundamental structure,” the truth predicate and the quotation marks involved in disquotation “seem to express much less fundamental matters, specific to the phenomenon of language” (339). Thus, “the comparison between classical logic and disquotation looks analogous to the contrast between a successful theory in fundamental physics and a successful theory in one of the special sciences, such as economics” (339). In general, in such cases the better response is to revise the less fundamental theory.One response is to reject the analogy, and insist thatthe concept of truth is . . . fundamental to our usual understanding of classical logic, through both the standard truth-conditional account of the meanings of the classical logical constants and the standard Tarskian account of logical consequence as generalized truth-preservation. (340)However, any scientific theory makes heavy use of the classical logical constants, but not of the truth-predicate. Thus “restricting classical logic will tend to impose widespread restrictions on [the scientific theory's] explanatory power,” while “restricting disquotation makes no difference to their explanatory power” (340). Fans of disquotation could respond to this by restoring classical principles, such as P ∨ ¬P, for nonparadoxical sentences—invoking particular instances of these principles as auxiliary assumptions. “But then,” Williamson complains, scientists' explanations will be “ad hoc . . . in a way they are not for the classical logician, who derives them all from the simple, elegant, general principles of classical logic” (341). “Similar considerations,” moreover, “apply to other nonclassical treatments of the semantic paradoxes,” such as dialetheism. Williamson concludes that “the piecemeal reintroduction of instances of missing classical principles involves heavy abductive costs through loss of simplicity and elegance” (342)

Analytic philosophy · Contemporary philosophy · Epistemology · General interest · Philosophy · Philosophy and Theoretical Science

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