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Sunday, December 9, 2012

I Before E: Liar, Liar, Theory on Fire



Lets say that a Human Language is a spoken or signed language that human children can naturally acquire. Let a Tarskian Language be a language for which there is a finitely specifiable theory of truth. If L is a Human Language, then for each sentence S of L, a human child has the capacities required to understand S. If L is a Tarskian Language, then there is a (true) theory such that for each sentence S of L, there is a corresponding theorem of the following form: True(S) P; where ‘≡’ is the material biconditional. Let Kaplanian Languages be those that are fundamentally like Tarskian Languages, while allowing for some context sensitivity of the sort illustrated with (1).
(1)  I wrote this.
If L is Kaplanian, then for each sentence S of L, there is a corresponding theorem of the following form: for each assignment A of values to context-sensitive elements of S,
True(S, A) F(A); where A might, for example, be such that the thing it assigns to I wrote the thing that A assigns to this (or a corresponding deictic index). We can also speak of Davidsonian Languages, for which there are truth theories whose theorems concern certain spatiotemporally located utterances of sentences. But at least for now, lets not worry about any differences between Davidsonian and Kaplanian languages.
Here, I want to focus on examples like (2) and their bearing on thesis (D).
                        (2)  The second numbered example in this post is false.
                        (D)  for each Human Language L, there is a theory of truth
                             that can serve as the core of an adequate theory of meaning for L
(D) goes beyond the conjecture that Human Languages are Davidsonian/Kaplanian.
My concern is that even if one can (pace Tarski) specify theories of truth for Human Languages, a theory that consistently assigns truth conditions to examples like (2) will be too sophisticated to serve as the core of theory of meaning for a Human Language.
For simplicity, Let Lari be a name for the second numbered example in this post (leaving it open whether examples are sentence-assignment pairs or utterances). If Lari is truth evaluable, then presumably, Lari is true if and only if the second numbered example in this post is false. In which case, if Lari is true, then: the second numbered example in this post is false; and since Lari is that example, Lari is false. But likewise, if Lari is false, then: since Lari is the second numbered example in this post, Lari is true if and only if Lari is false; hence, Lari is true. So prima facie, Lari is neither true nor false. By itself, thats not paradoxical. Many things are neither true nor false: dogs, numbers, etc. But if a theory implies that Lari is true or false, that tells against the theory.
To be sure, there are ways of stipulating consistent truth theories for languages that generate analogs of (2) and are like English in other interesting respects. Clever logicians can invent clever theories that help us avoid certain mistakes in our attempts to describe the world. But it doesnt follow that for each Human Language L, there is a theory of truth that is not too clever to be the core of a good theory of meaning for L.
As Davidson noted and Dummett stressed, the idea behind (D) is that given some assumptions about how truth is related to the use of expressionsincluding demonstratives, indexicals, and nondeclarative sentencesa suitably formulated theory of truth for L can specify what expressions of L mean by specifying how they determine the conditions in which more complex sentences of L are true. But such a theory must meet conditions that may not be jointly satisfiable. First, it has to be consistent, even given examples like (2). Second, as Foster noted and Davidson admitted, it seems that the theory will need to have theorems like (3) without having theorems like (4).
(3)  True(Ernie snores.) Snores(Ernie)
                        (4)  True(Ernie snores.) Snores(Ernie) & Precedes(Three, Seven)
Biconditionals like (3) may seem to "specify the meanings" of object language sentences, now ignoring assignment relativity for simplicity. But it is hard to see how a theory that yields biconditionals like (4) could serve as a theory of meaning for English, especially if theories of meaning are supposed to be theories of understanding. The derivability of (3) does not explain why Ernie snores. means what it doesmuch less explain how speakers understand that sentenceif (4) is equally derivable.
Now if you just want a theory that specifies truth conditions, it does no harm if the theory generates boundlessly many instances of (5), so long as [...] is always true.
(5) True(Ernie snores.) Snores(Ernie) & [...]
And of course, a theory that generates (3) may not generate (4). It depends on the axioms and background logic. But a theory that generates (3) and (6) will generate (7)
(6) True(Bert yells.) Yells(Bert)
(7) True(Ernie snores.) Snores(Ernie) & [True(Bert yells.) Yells(Bert)]
if the theory permits replacement of P with the conjunction of P and a theorem. So if you want a theory of truth to serve as a theory of meaning, you need (i) a very weak background logic that doesnt generate any overly intellectual theorems, or (ii) a way of identifying the meaning-specifying theorems. I see no plausible way of providing (ii). And I worry that adopting (i) is at odds with the logical sophisticationKripkes fixed points, Gupta-Belnap revision rules, or whateverrequired to describe truth consistently.
In Knowledge of Meaning, Larson and Segal do offer an initially attractive version of (i). In effect, their system for deriving T-theorems only permits replacement of established equivalents: given P Q and Q R, P R; given x = y, Fx Fy. This system, which doesnt license derivations of T-sentences like (4) or (7), reflects an interesting hypothesis about the human capacity to apply (lexical and compositional) semantic competence to particular expressions. But if derivability is logically blind, apart from replacement of established equivalents, then (8) will be as derivable as (3) and (6).
(8)  True(Lari is false.) False(Lari)
Of course, (8) can be true if neither Lari is false. nor Lari is true or false. Though if (8) follows from a true theory of meaning, then Lari is false. is still truth evaluable. Thats not a contradiction. Perhaps Lari is false. is truth evaluablewe know it to be true if and only if Lari is falseby virtue of a certain competence applying to it. But if this is the best defense of (D), I think that tells against the strategy of explaining meaning in terms of speakers knowing theories whose theorems specify how the world needs to be in order for sentences to be true. Developing this line of thought requires a second post. Stay tuned: the issues concern linguistic knowledge and closure under entailment.

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