On Philosophy

November 2, 2006

Essay: T Theories and Their Relationship to Meaning

Filed under: Language — Peter @ 12:13 am

In their paper “Knowledge of Meaning and Theories of Truth” Larson and Segal claim that T theories are not necessarily interpretive, meaning that even given a complete T theory we could not conclude what sentences in the metalanguage have the same meaning as a sentence in the object language, which implies that T theories don’t capture meaning. In spite of this T theories seem to be part of the psychological basis for meaning, since people seem to acquire T theories and treat them as interpretive. And so T theories, along with a few facts about human psychology, may really be the mechanism behind meaning, or so Larson and Segal claim. Certainly these claims seem plausible, but perhaps the reason that we are unable to conclude that T theories are interpretive is because we have been too loose in defining what counts as a T theory. I claim that it is possible for a T theory, properly constructed, to be necessarily interpretive, and thus be a good model of meaning. But this is all I think that it is, a model of meaning; unlike the authors I doubt that T theories are really part of the psychological basis of meaning.

If a T theory is to be interpretive, and thus a good theory of meaning, it must overcome two potential problems, which are called by Larson and Segal the problem of extension and the problem of information [1]. The problem of extension is to construct a T theory that yields T sentences of the form “‘Q’ is true if and only if R” only when “‘Q’ means R” holds. Clearly, if the T theory for a language doesn’t address the problem of extension then we can’t draw conclusions about meaning from the T sentences it yields, since there would be some cases in which we would either be unable to conclude that ‘Q’ means R when it does, or would erroneously conclude that ‘Q’ means R when it doesn’t. As the authors point out this problem can be broken down into two smaller problems: ensuring that the T theory yields a T sentence for every sentence in the language (it is sufficiently productive), and ensuring that the T theory yields only a single T sentence from each sentence in the language (it is not over productive). Of these two smaller problems it is overproduction that seems the most likely to be a problem, since T theories employ the classical biconditional. The classical biconditional holds whenever both of its objects have the same truth value, meaning that T sentences such as “‘snow is white’ is true if and only if grass is green” are true even though sentences such as “‘snow is white’ means that grass is green” are not. We might hope that even if such sentences are true they might not be derivable from our T theory, but, as Larson and Segal point out, if the classical biconditional is employed anywhere in the rules for the derivation of T sentences within our T theory the same problems will arise.

A possible solution to this dilemma is to abandon the classical biconditional. We might, for example, produce T sentences in a modal logic, in the form “It is necessary that ‘Q’ is true if and only if Q”. However, even a modal version of the T theory might still erroneously produce sentences such as “It is necessary that ‘Jill knows Kate’ is true if and only if Jill knows Kate and 2+2=4”, since the statement 2+2=4 is itself a necessary truth. To eliminate even these cases we could define the biconditional in some relevant logic. Certainly it seems like such a move would eliminate almost all cases of overproduction [2]. But would doing so be begging the question, since we might understand the conditional, as defined in relevant logic, as “is derivable from”? I think not, for two reasons. First the conditional in relevant logic can be defined in terms of some structure of possible worlds, in a completely formal manner, without making reference to our intuitions about derivability. Secondly, what the conditional means is not exactly significant to the T theory itself, only the derivation rules that employ the conditional in transforming object language sentences into T sentences are. No matter how we define the conditional, the T theory may have some primitive T sentences as part of it, as the example T theory provided by Larson and Segal does, from which the T sentences for more complicated statements are be derived. Since the conditional, no matter how it is defined, is already incorporated into the T theory, we don’t have to show that it somehow emerges as a result of the derivations, and hence don’t have to worry about what it means any more than we do about other primitive connectives such as “and” and “or”.

With the potential problem of over production set aside we can now turn to the problem of underproduction. Larson and Segal think that underproduction isn’t a problem for T theories, and if we allow some additional transformation rules that can turn statements that are not plain assertions (such as questions and exclamations) into assertions, then I am inclined to agree with them. Except for the small problem of tautologies. If the T theory in question is supposed to be yielding truth conditions for sentences then strictly speaking it will be unable to provide a T sentence for tautologous statements, since the truth of a tautology is not conditional upon anything. But if we resolve the problem of over production by employing some kind of relevant logic then this problem too may disappear. In such a theory the T sentences are, perhaps, more accurately viewed as stating what can be concluded from the object language sentence and what facts would allow us to conclude that the object language sentence was true. And if we are working outside of classical logic then even for tautologies there is something that plays this role (at least in some systems).

Let us assume then that the problem of extension can be dealt with satisfactorily. This means that for any sentence Q there is exactly one T sentence of the form “‘Q’ is true if and only if P” that will produced by a proper T theory. Given this, the problem of information, whether sentences in the form “‘Q’ is true if and only if P” imply that ‘Q’ means P, can be dealt with. Larson and Segal claim that the problem of information can’t be resolved, that one can’t conclude that ‘Q’ means P from the corresponding T sentence. However, their argument as to why this can’t be so rests upon the assumption that for any sentence in the object language there are many T sentences about it that will be yielded by the T theory, which is not necessarily the case if we can resolve the problem of extension. To see if we can really conclude that Q means P from the appropriate T theorem let us assume that we can’t, which implies that for some Q there is a T sentence yielded by our T theory for that Q of the form “‘Q’ is true if and only if R”, that Q means P, and that R does not mean the same thing as P. But this is impossible, because if Q means P then we can truthfully assert Q whenever P holds, but by the T theorem we can conclude that we can truthfully assert Q whenever R holds. This means that whenever we can assert Q when both R and P hold, and when we can’t assert Q neither R nor P hold, in all possible worlds. This in turn implies that R and P have the same intension, since our T theorems hold in all possible worlds (under a relevant logic adaptation), and thus that R and P mean the same thing, which contradicts our premise [3]. Therefore given a T theorem of the form “‘Q’ is true if and only if R” we must be able to conclude that Q means R [4].

But, as Larson and Segal point out, whether T theories can meet all the formal requirements is somewhat irrelevant if it turns out that people in practice acquire a T theory and treat it as interpretive. The authors see this as a likely possibility, which mitigates the concerns they raised earlier about T theories, since even if the T theories by themselves don’t define meaning those same T theories plus a few psychological facts might. And admittedly if we do use a T theory it is almost certainly unconscious, and so no amount of consideration without experiment will really provide a definitive answer. However, there are some reasons to doubt that we actually acquire and use a T theory. For one we know that people acquire a new language word by word, but the T theories that we have been considering begin with complete sentences as fundamental axioms. It might be possible to construct a T theory that was built on axioms such as “‘Rabbit’ refers to rabbit”, but given that words acquire different meanings in different contexts (such as “run”, which can refer to either physical activity or proper operation) it seems doubtful. A second reason to doubt that we unconsciously employ a T theory is simply a matter of simplicity. Certainly if the mind could use a T theory it could also develop a theory that yields sentences of the form “‘X’ means Y”, without bringing in the notion of truth, which would certainly put to rest worries whether the theory is properly interpretive. Of course this wouldn’t explain the notion of meaning, but it could very well be a concept that is “hard-wired” into us. Thus I judge it to be unlikely that we actually employ a T theory. This doesn’t mean that T theories can’t illuminate meaning. Certainly it seems like they might provide a good formal model of how meaning works, and thus help us understand what meaning is, but I doubt that they themselves are behind meaning in the human mind, just as I doubt that the law of gravitation is itself behind the movement of the planets, even though it is a good description of that motion.

Notes:

1. Actually they call them the extension question and the information question, but it is cumbersome to say that T theories need to “answer the extension question affirmatively” instead of “solve the problem of extension”.

2. By which I mean: I can’t think of a case that it wouldn’t eliminate.

3. Here I am assuming that the intension can be defined by a set of objects from all possible worlds. Even if we don’t think that this set is the intension it still seems reasonable to assume that two words sharing this same set have the same meaning, since there doesn’t seem to be any features or properties that we could use to base a distinction in meaning upon.

4. We can also show, if we accept the T theory as a theory about meaning, that the T theorem relating Q to P and the statement “Q means P” have the same meaning. Let us call the appropriate T theorem QT and the assertion that they have the same meaning QM. These have the same meaning, according to the T theory if “‘QT’ is true if and only if QM”. And since we have settled the extension question we know that this statement is true, and thus that they mean the same thing. Of course you can’t prove that a theory is true from within the theory, but to simply assert that they don’t mean the same thing is begging the question, since it is equivalent to saying that the T theorem fails in some way (by not producing all the T theorems, or by producing too many), and thus Larson and Segal’s claim that the impossibility of addressing the problem of extension is a reason to believe that the T theory can’t be interpretive by itself is misleading, since they are really the same claim.

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