Abstract
Davidson conjectured that suitably formulated Tarski-style theories of truth can “do duty” as theories of meaning for the spoken languages that humans naturally acquire. But this conjecture faces a pair of old objections that are, in my view, fatal when combined. Foster noted that given any theory of the sort Davidson envisioned, for a language L, there will be many equally true theories whose theorems pair endlessly many sentences of L with very different specifications of whether or not those sentences are true. And if L includes words ‘true’, then for reasons stressed by Tarski, it’s hard to see how any truth theory for L could be correct. Moreover, each of these concerns amplifies the other. Appealing to possible worlds will not help with Foster’s Problem, for reasons that Chomsky discussed in the 1950s, and appealing to trivalent models of truth will not avoid concerns illustrated with Liar Sentences.
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Notes
This is the written version of an intentionally polemical talk presented at the Topoi conference in Torino. My thanks to the participants, and especially the organizers, for helpful comments and patience. I am also grateful to the anonymous referees, who provided valuable constructive remarks on the penultimate draft. Chapter four of Pietroski (2018) sets these issues in a somewhat different context, with still inadequate discussion of Lepore and Ludwig (2007). A fuller discussion would need to engage with Glanzberg (2004, 2015), along with other relevant literature on truth and the possibility of psychologized truth theories.
Drawing on many authors, Pietroski (2018) also stresses (i) the sundry ways that truth can depend on context, (ii) some implausible implications of combining empirically justified event analyses, inspired by Davidson (1967b), with truth-theoretic conceptions of meaning, and (iii) the relative ease of recasting, in internalist terms, many proposals framed in truth-theoretic terms.
The word ‘claim’ is polysemous, along with ‘assertion’, ‘thought’, etc. And even if we allow for the logical possibility of contradictory propositions—see Priest (1979, 2006)—this doesn’t warrant the radical hypothesis that (1) is true and not true, relative to the imagined context, as opposed to simply not true (because it fails to be truth-evaluable). If a theory of meaning implies that (1) is true given that it isn’t true, that tells against the theory. Moreover, if Davidson’s conjecture is combined with the idea that a Slang sentence can be true and not true relative to the same context, this heightens Foster’s concern. It’s hard enough to see how a theory can specify what a sentence Σ means by implying that Σ has a certain truth-theoretic property, even without adding the assumption that Σ can have this property if and only if it doesn’t.
Other utterances of (3) may have been true. But if a theory predicts that the chosen utterance of (3) was true, that tells against the theory, whatever logic one adopts; see note 3.
Even if the speaker is not “acquainted” with Neptune in Russell’s sense; see Evans (1982).
Correlatively, being explicit about notions of derivation—as Tarski and Chomsky were—reveals difficulties for Davidsonians. I read Foster in this light. Chomsky (1957, 1977, 1995) urged a conception of meaning like Strawson’s, in an overtly mentalistic idiom, while granting that internalistic meanings can provide “truth indications,” see Pietroski (2005, 2017b, 2017c).
I can’t prove that my favorite hammer doesn’t have a truth condition. But if sentences are tools that can be used to make claims, it isn’t surprising that a claim made by using a sentence Σ can be true in part because Σ has no truth condition. Suppose I put a hammer in a box after stating that if I ever put a hammer in a box, I will thereby be claiming that once the hammer is in there, three things in the box fail to be true. Then my odd claim, oddly made, is true if the box also contains a wrench and a screwdriver—or a wrench and a false sentence, or a wrench and (1). But the hammer isn’t true or false, not even if putting it in the box is a way of making a true claim.
See note 3. An anonymous referee suggested that perhaps relative to some contexts, (1) has a truth condition that cannot be realized. But if we allow for unrealized truth conditions, which would presumably differ from the realized truth conditions of boringly contradictory sentences like ‘A dog barked, and no dog barked’, the question is whether a theory that specifies (perhaps unrealized) truth conditions for Slang sentences would be a plausible theory of meaning for a Slang. To answer, we would need to know under what conditions distinct sentences are unrealizedly truth-conditionally equivalent.
It’s probably better to replace ‘Identical(x, Ernie)’ with something like ‘IsAnErnie(x) & c:Ernie(x)’, in which a predicate that applies to the many Ernies is contextually restricted by (uses of) an index associated with the noun; cp. Chomsky (1973). But this detail won’t matter.
The more interesting cases include [S [NP a [N cat]][VP [V ate [NP a [N fish]]]]] and [S [NP sm [N cats]][VP [V ate [NP sm [N fish]]]]]; where ‘sm’ indicates the (phonologically reduced) form of ‘some’ that is like the indefinite article for nouns that are not singular count nouns. The verb phrase [VP [V ate [NP a [N fish]]]] invites axioms like (C4) and (C5).
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(C4)
TrueOf([VP [V …][NP …]], x, c) ≡ ∃x'{TrueOf([V …], < x, x' > , c) & TrueOf([NP …], x', c)}
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(C5)
TrueOf([NP a [N …]], x', c) ≡ TrueOf([N …], x', c)
But plural nouns and “mass nouns” reveal familiar complexities.
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(C4)
One can adopt a special rule for instances of ‘True([S …], c) ≡ P’ or rewrite them, perhaps as instances of ‘∀x[TrueOf([S …], x, c)] ≡ P’, treating sentences as predicates true of everything or nothing. One can also adopt Lewis-Montague style theories with “function application” as the main combinatorial principle and correspondingly categorized lexical axioms that pair expressions with entities or functions that exhibit types from the now familiar Frege-Church hierarchy: basic things of type < e>, truth values of type < t>, and functions of type < X, Y > —i.e., functions from things of type < X > to things of type < Y>, where < X > and < Y > are themselves types of the hierarchy. Then axioms like those below
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(L1') ||Ernie<e>||c) = Ernie
(L3') ||snores<e, t>||c) ≡ λx.Snores(x)
(C1') ||[<Y> […<X>][…<X, Y>]]||c = ||[…<X, Y>]||c(||[…<X>]||c)
(C2') ||[<X> [<X> …]]||c = ||[<X> …]||c
yield parallel derivations like the one below.
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1.
||[<t> [<e> [Ernie<e>]][<e, t> [snores<e, t>]]]||c = ||[<e, t> [snores<e, t>]]||c(||[<e> [Ernie<e>]]||c) [C1']
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2.
||[<e> [Ernie<e>]]||c = ||[Ernie<e>]||c [C2']
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3.
||Ernie||c = Ernie [L1']
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4.
||[<e> [Ernie<e>]]||c = Ernie [2, 3]
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5.
||[<e, t> [snores<e, t>]]||c = ||[snores<e, t>]||c [C3]
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6.
||[snores<e, t>]||c = λx.Snores(x) [L3]
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7.
||[<e, t> [snores<e, t>]]||c = λx.Snores(x) [5, 6]
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8.
||[<t> [<e> [Ernie<e>]][<e, t> [snores<e, t>]]]||c = ||[<e, t> [snores<e, t>]]||c(Ernie) [1, 4]
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9.
||[<t> [<e> [Ernie<e>]][<e, t> [snores<e, t>]]]||c = λx.Snores(x)(Ernie) [8, 6]
But one still needs a rule that licenses replacement of ‘α’ with ‘γ’ given instances of ‘α = β’ and ‘β = γ’. One can also add a rule of lambda-calculation that licenses an additional step.
10. ||[<t> [<e> [Ernie<e>]][<e, t> [snores<e, t>]]]||c = Snores(Ernie) [9, λ]
But this rule of inference, which has nothing to do with the object language, is different in kind.
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See Chomsky (1964). Here, ‘#’ indicates that the string cannot be understood as having the indicated meaning. Note that ‘eager’ and ‘easy’ can be replaced with ‘reluctant’ and ‘hard’. The implausibility of (18b) is irrelevant.
Theorists can profess to be unconcerned about overdetermination. But if the task is merely to provide an algorithm that pairs each pronunciation with its attested meanings, without concern about whether the algorithm also pairs pronunciations with unattested meanings, then the task is easily accomplished. Specify some class of possible interpretations that include all attested meanings, and then offer the “universal” procedure that pairs each pronunciation with each of the possible interpretations—or each of the possible interpretations that could be constructed from the relevant lexical items; cp. Chomsky (1957).
A complication: deriving an instance of ‘True([S…], c) ≡ ∃x{Φx}’ will involve deriving intermediate theorems in which ‘TrueOf’, along with a description of some constituent of [S…], appear in ‘Φ’. But the intermediate theorems are not specifications of subtly different meanings. So let’s grant that such theorems—still partly metalinguistic on the right, and in this sense, not “fully discharged” specifications of truth conditions—don’t count for purposes of evaluating how many meanings the theory assigns to an expression of the object language.
See note 13. Davidson (1967a) thought it wouldn’t be a problem if a theory of the sort he imagined had theorems like (14T*) in addition to (14T); cp. the discussion of Tarski in section three below. In later work, Davidson proposed additional restrictions on using truth theories as theories of meaning, sometimes in the form of stipulations about the kinds of theories and evidence that “radical interpreters” would consider; see note 18. Along with much of the literature, I don’t think these further restrictions help; see Pietroski (2005), and for extended discussion, Lepore and Ludwig (2007). So I focus here on Davidson’s idea that given the right notion of derivation, derivable T-theorems will be suitably translational.
Tarski’s idea was not that (24T) is more “semantically legitimate” than (24T*). If being true is a matter of being satisfied by all sequences, then (24T) and (24T*) are truth-theoretically alike. Imagine a supplemented theory whose theorems include ‘(c = b + a) ≡ (0 = ei + 1)’ and ‘True(‘c = b + a’) ≡ Identical[3, Plus(2, 1)]’ and ‘True(‘c = b + a’) ≡ Identical[0, Plus(ei, 1)]’. Someone who saw only the third theorem, after it was derived from the first two, might think that ‘b’ has a meaning akin to that of ‘the result of raising the number whose natural logarithm is one to the power of the square root of negative one times the ratio of a circle’s circumference to its diameter’. But this would be a mistake. In my view, it’s bizarre to segregate theories of truth from theories of the entities that verify sentences that have truth conditions. But in any case, one can’t do this and then say—as Davidson (1986, p. 446) does—that “[T]here is no boundary between knowing a language and knowing our way around in the world.” If uninterpretive T-sentences can be distinguished from theorems of meaning theories, then pace Quine (1951), there is a theoretically interesting analytic/synthetic distinction.
Likewise for any proposed truth+ theory for L.
According to Davidson (1973, p. 134, his quote marks)
an acceptable theory of truth must entail, for every sentence s of the object language, a sentence of the form: s is true if and only if p, where “p” is replaced by any sentence that is true if and only if s is. Given this formulation, the theory is tested by the evidence that T-sentences are simply true; we have given up the idea that we must also tell whether what replaces ‘p’ translates s.
But even if we can get evidence that T-sentences are “simply true,” without ignoring the possibility that Slang sentences don’t have truth conditions, the proposed necessary condition doesn’t exclude Fosterized theories. So the issues about translation remain.
The derivation below parallels, line by line, the earlier one for [S [NP [N Ernie]][VP [V snores]]].
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1.
True([S [NP [N Kermit]][VP isn’t [A blue]]], c) ≡ ∃x{TrueOf([NP [N Kermit]], x, c) & TrueOf([VP isn’t [A blue]], x, c)} [C1]
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2.
TrueOf([NP [N Kermit]], x, c) ≡ TrueOf([N Kermit], x, c) [C2]
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3.
TrueOf([N Kermit], x, c) ≡ Identical(x, Kermit) [L6]
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4.
TrueOf([NP [N Kermit]], x, c) ≡ Identical(x, Kermit) [2, 3]
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5.
TrueOf([VP isn’t [A blue]], x, c) ≡ ~ TrueOf([A blue], x, c) [C4]
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6.
TrueOf([A blue], x, c) ≡ Blue(x) [L7]
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7.
TrueOf([VP isn’t [A blue]], x, c) ≡ ~ Blue(x) [5, 6]
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8.
True([S [NP [N Kermit]][VP isn’t [A blue]]], c) ≡ ∃x{Identical(x, Kermit) & TrueOf([VP isn’t [A blue]], x, c)} [1, 4]
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9.
True([S [NP [N Kermit]][VP isn’t [A blue]]], c) ≡ ∃x{Identical(x, Kermit) & ~ Blue(x)} [8, 7]
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1.
If we represent the three model-values for sentences with the numbers 1, ½, and 0, then given a Strong Kleene model: the values of ‘P & Q’ and ‘P v Q’ are, respectively, the minimum and maximum values of the conjuncts or disjuncts; ‘∀xΦx’ and ‘∃xΦx’ are treated similarly as conjunctions/disjunctions of their instances; and the value of ‘~P’ is 1 minus the value of ‘~P’; so if ‘P’ has the value ½, so does ‘~P’. Classical bivalent models can be viewed as special cases, with no sentences having the value ½.
See Beall et al. (2018) for a helpful accessible discussion and guide to the more technical literature.
If there is a truth-evaluable proposition that Linus isn’t true, maybe this proposition is klee, and the principle of excluded middle (for propositions) is false. That still leaves difficulties, including Curry’s paradox regarding conditionals and “revenge” puzzles concerning interpretations; see note 21. But these difficulties for certain proposals about truth do not justify implausible claims about Slangs.
Indeed, I suspect that pronunciation is evolutionarily more recent than the core system that generates meaningful combinations of lexical items that correspond to families of concepts; see Chomsky and Berwick (2017).
And even if one stipulates that (37) has the meaning of (36), this doesn’t guarantee that the Slang meaning determines the Tarskian satisfaction condition, much less that this meaning is compositionally determined by the components and syntax of (37).
For purposes of a class in elementary logic, I can evaluate some regimentations of a Slang sentence as better than others. Though for these purposes, I find myself assuming that words like ‘some’ and ‘all’ have meanings that are not mere extensions, while forgetting that Tarskian squiggles like ‘∃’ and ‘∀’ are syncategorematic.
Indeed, if Slang expressions have their meanings because of how their (alleged) truth-theoretic properties are mentally specified, then it seems unlikely that the mental specifications also have meanings; though perhaps they have contents that go beyond Tarskian satisfaction conditions.
Larson and Segal (1995) assume that such biconditionals are true, and that psychologized theories of meaning should help explain this.
Chomsky (1957) noted that everyone in the room speaks two languages can be understood in two ways: (a) everyone in the room is bilingual; or (b) two languages are such that everyone in the room speaks them. This invites the hypothesis, developed and defended by May (1985), that the string is structurally homophonous. Given a room, there may be worlds in which (a) or (b) obtains, but not both. Though both sentence meanings can be used to make correct reports, even if as a matter of necessity, every in the relevant room speaks the same two languages.
This was one reason for hoping that Slangs could be characterized in terms of a “base” component (analogous to a context-free grammar) and transformations, at least some of which preserve meaning, with recursion restricted to the base component—as opposed to more powerful (context-sensitive) grammars whose rewrite rules can license inversions, say from ‘BA’ to ‘AB’, and hence from ‘ABABABCCC’ to ‘AAABBBCCC’.
That seems right to me. The study of Slangs can lead one to posit various linguistic entities. But if it leads one to posit universes that physicists can do without—or abstracta that only certain semanticists want—something has gone wrong.
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