Saturday, December 15, 2012

Liar, Liar, Theory on Fire (Part Two)

This a follow-up to Part One, which noted that examples (1-2) present a challenge for (D).
(1)  Lari is false.
(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
To review, it seems that (2)—a.k.a. Lari—is neither true nor false. So one might hope for a truth theory that generates (3) rather than (4); 
(3)  Legit(Lari) --> [True(‘Lari is false.’)  False(Lari)]
(4)  True(‘Lari is false.’)  False(Lari)
where ‘Legit(_)’ is some paradox-avoiding restriction. But given “Foster’s Problem,” (D) seems to require truth theories that don’t generate (3) or (5),
(5)  True(‘Ernie snores.’)  Snores(Ernie) & [True(‘Bert yells.’)  Yells(Bert)]
which is true but not meaning-specifying. So one might seek a theory with a very weak system of deduction, reflecting a hypothesized competence that lets humans apply semantic axioms—lexical and compositional—only as required to compute the semantic properties of complex expressions. But can such a theory be a theory of truth of the sort required by (D)?
The worry is not that (4) is false. If Lari is neither true nor false, and in this respect like my dog Bode, then both sides of (4) are false. But if a true theory specifies a truth condition for sentence S, then in one fine sense, S—unlike Bode—has a truth condition. So one wonders: what special property does Lari lack (or have), in contrast with linguistic entities that are allegedly true-or-false? It’s no answer to say that Lari induces paradox. But following Davidson, advocates of (D) might say that truth theories for Human Languages specify truth conditions for utterances, not expressions relativized to contexts. And channeling Strawson, they might say that while utterances of (6) are typically true-or-false, utterances of (7) need not be.
            (6)  I am hungry.                             (7) Vulcan is a rocky planet.
If a user of (7) presupposes that Vulcan exists, in order to say that it is a rocky planet, that user’s utterance of (7) might fail to be false. Falsity may well require more than grammaticality and absence of truth; the world may have to “cooperate with,” or at least not frustrate, certain communicative intentions. This familiar point extends to at least some utterances of (8) and (9),
                        (8)  I saw that.                                  (9)  He is bald. 
since attempts to demonstrate an object can fail, and a speaker can wrongly assume that someone is not a vague case with respect to ‘bald’ (or ‘hungry’). So perhaps uses of (1), along with many uses of (2), fail to meet certain conditions for being true-or-false; where these conditions may themselves be determined in part by contextual factors (see, e.g., Michael Glanzberg's work).
I am happy to say that speech acts—in particular, attempts to make truth-evaluable claims by using sentences—are governed by norms of truth that go beyond any conditions specified by theories of Human Languages. It would be nice to have an ideal language whose sentences are themselves true-or-false in suitable contexts. But if Human Languages are I-languages that generate expressions in a biologically natural way, why think that theories of meaning for such languages specify truth conditions for utterances that need not be true-or-false? If truth is downstream of linguistic meaning—in that acts of using I-language sentences are candidates for being true-or-false, subject to review—why think that good theories of meaning for Human Languages will deploy the predicate ‘True(_)’? If truth is a property of utterances, it’s hard to see how specifications of truth conditions can be derived from a specification of a constrained capacity to generate meaningful expressions. (Saying that meaning is use doesn’t make it so.)
Moreover, Davidson did not discuss quantificational examples like (10) in detail.
                        (10)  I saw something.
But a Tarski-style theory of truth is specified in terms of expressions being satisfied by (or true of) sequences that assign values to variables. Given (11), T-theorems follow trivially.
                        (11) for each sentence S: True(S for each assignment A, Satisfies(A, S)
The trick is to show how “S-theorems,” of the form indicated in (12), can be derived.
(12) for each assignment A: Satisfies(A, ‘I saw something.’)  F(A)
For example, ‘F(A)’ might replaced with ‘for some assignment A* that differs from A at most with regard to A* what assigns to the variable x, A*(speaker) saw A*(x)’; where ‘A*(...)’ stands for whatever A* assigns to ‘...’, and speaker indexes a dimension of assignments that is (a la Kaplan) related to utterance interpretation via some pragmatic constraint—e.g., that assignment A* is germane to a conversational situation s only if A*(speaker) is the speaker in s.
For today, grant that instances of (12) can be meaning-specifying, despite the technicalia. The important point here is that a theory of truth for English will need to have S-theorems like these: Satisfies(A, ‘Lari is false.’)  False(Lari); Satisfies(A, ‘Lari is true.’)  True(Lari); Satisfies(A, ‘Lari is not true.’)  ~Satisfies(A, ‘Lari is true.’); and Satisfies(A, ‘Lari is not false.’)  ~Satisfies(A, ‘Lari is false.’). Such a theory will imply that no assignment satisfies (1) or (13),
            (1)  Lari is false.                                    (13)  Lari is true.
while each assignment satisfies (14) and (15). So (16) must be rejected.
                        (14)  Lari is not true.                              (15)  Lari is not false.
            (16)  for each sentence S: False(S for each assignment A, ~Satisfies(A, S).
This is not yet a contradiction. One can say (14-15) are true, along with (17-18).
                        (17) It is not true that Lari is false.            
                        (18) It is not true that Lari is true.
Drawing on Kleene/Kripke, one can also say that False(S True(not-S). But then (19)
                        (19) The 19th numbered example in “Liar, Liar, Theory on Fire” is not true.
is true if it isn’t true; so it isn’t true, and hence it is true. That’s not good. If each assignment satisfies (19) if and only if (19) isn’t true, then since (19) isn’t true, each assignment satisfies (19); in which case, (19)—a.k.a. Linus—is true. So it doesn’t help to say that (1) and (13), unlike (14-15) and (17-18), fail to meet certain conditions on being true-or-false. One can try to deny (20),
                        (20)  Satisfies(A, ‘Linus is not true.’)  ~Satisfies(A, ‘Linus is true.’)
or at least offer a theory, perhaps formulated in terms of a hierarchy of types, that does not imply (20). But even if some such theory avoids analogous (“revenge”) paradoxes, remember that (D)
 (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
requires truth theories whose theorems are meaning specifying. Moreover, as Parsons and Kripke remind us, contingent facts can make apparently innocuous claims into trouble-makers. It can seem that (21) is true in a context if and only if more than half of the relevant examples are false.
 (21) Most of the examples were false.
But imagine twenty-one examples: ten true (e.g., ‘2 >1’, ten false (e.g., ‘1 > 2’), and (21).
Like most things, many Human Language sentences are not true-or-false relative to each assignment of values to variables. So why think that any Human Language sentences have this remarkable property? I can’t prove that (10) is like (1) in being unlike Tarskian sentences. Perhaps there is “something about Lari” that makes it unusually unsatisfiable, and not merely unsatisfied—and not unsatisfiable in the ways that my dog is. But that hypothesis needs defense. Prima facie, satisfaction has its place in theories of truth, not in specifying linguistic meanings. And (D) seems to be a massive simplification: useful for certain purposes, but not true.

No comments:

Post a Comment