On Philosophy

October 13, 2006

Why T Sentences Don’t Illuminate Meaning

Filed under: Language — Peter @ 1:20 am

Tarski, as part of his work with formal languages, created the T schema to describe truth in those languages. In general the T schema is as follows: “S” is T (true) if and only if P. The T scheme themselves are taken as being in a meta-language while the language they describe the truth of is the object language. In the T schema P is some propositional statement in the meta-language and “S” is the name of some sentence S, which is part of the object language. For example, if “‘snow is white’ is true if and only if snow is white” is a T sentence (an instance of the T schema) then if the meta-language is English “snow is white” must be the name of a sentence, which is itself in another object language, perhaps some limited subset of English. As you may have guessed, Tarski saw his own work as relevant only to the study of formal languages, and not natural languages such as English, which is not to say that Tarski’s work is somehow flawed, since this was all he set out to accomplish.

However, since knowing the truth conditions for a sentence is commonly accepted as being enough to know its meaning, some, such as Davidson, have proposed that meaning in natural languages can be understood through T sentences as well. The most generous understanding of this idea is that in the T schema “S” is some sentence in a natural language and P is some expression of states of affairs, not in a language, but as understood. So, for example, knowing the T sentence “‘snow is white’ is true if and only if snow is white” means that one has connected the truth of “snow is white” with the fact that the world is such a way that snow is white. Another way of putting this is that P is the pure propositional content. In any case, to understand P as being English would make the T schema useless. It is my claim that, even understood in this generous way, knowing T sentences isn’t enough to understand meaning or to be a theory of meaning, and that a real theory of meaning is to be found elsewhere.

Before I object to the possibility that T sentences can illuminate meaning, however, let me strengthen the position of T schema advocates even further. One criticism of the T schema is that there exist true T sentences such as “‘snow is white’ is true if and only if grass is green” (since snow is white and grass is green, and thus the biconditional holds), but that knowing sentences such as these obviously don’t help us understand the meaning of “snow is white”. Some proponents of the T schema would dismiss these as being irrelevant, as long as the “right” T sentences hold, but this seems like a poor defense, since what is special about the “right” T sentences which gives them the power to illuminate meaning that these “erroneous” ones lack? We can however make a small change to the T schema that will eliminate some of the erroneous sentences, by restating the T schema in a modal logic as Necessarily*(“S” is true if and only if P), which means that “‘S’ is true if and only if P” must hold in all possible worlds. “Necessarily(‘snow is white’ is true if and only if grass is green)” is false, since there are possible worlds in which snow is white but grass is not green. Still, there are still some “erroneous” T sentences that are true but don’t seem to convey meaning, such as “Necessarily(‘snow is snow’ is true if and only if grass is grass)”. These sentences too can be eliminated by the use of even more esoteric logics. Specifically, if we define the biconditional in the original T schema as the biconditional from some relevant logic (say R) then the only true T sentences are those in which S and P seem to have the same meaning, to the best of my knowledge.

Given this understanding of the T schema there are three objections to the possibility that T sentences provide an adequate theory of meaning. The first is that given a theory of meaning about some language, which should obviously be finite and comprehendible since people seem to unconsciously master some theory of meaning about their native language, we should be able to understand the meaning of all sentences in that language. However, there are some languages in which no finite collection of T sentences can provide an understanding of all sentences in that language, allowing us to conclude that T sentences are not a valid theory of meaning. The simplest way to demonstrate this is to give an example of such a language. Consider then a language that is encrypted (specifically a public-private key scheme based on prime number factorization). The aliens that use this language are born knowing the encryption and decryption keys. Internally they form their sentences in an English-like language but they encrypt them before publicly communicating them. Their listeners hear the encrypted sentence and decrypt it, which allows them to understand what is being said. Clearly the public, encrypted, version of the language has meaning, since the aliens are able to communicate using it, but no finite set of T sentences can allow us to understand public sentences in this language which we don’t have T sentence for**, and since there are an infinite number of possible sentences in almost every natural language T sentences are thus shown to be deficient.

The second problem for T sentence theory of meaning is that if they are to exist for all sentences that have meaning it seems to imply that there are some sentences that have no meaning, even though they seem meaningful to us, or that words such as “truth” have no meaning. The sentence “this sentence is false” demonstrates this problem, since its T sentence is “‘this sentence is false’ is true if and only if ‘this sentence is false’ is false”. Clearly this T schema is itself false (since it is a contradiction), and hence there is no T sentence for ‘this sentence is false’. Tarski escaped from this problem by stipulating that object languages don’t have semantic terms about themselves, which is all very well for formal languages, but doesn’t seem to be something we can stipulate for natural languages, since here we are discussing a natural language within a natural language. A T sentence theory of meaning, then, seems to deny that we understand “this sentence is false”, which should give us pause, since I think we both know what that sentence means even if we can’t provide a truth value for it.

Finally, there is the problem of subjective words, such as “cute”. Even though “‘bunnies are cute’ is true if and only if bunnies are cute” is a valid T sentence it doesn’t seem like cute is a property that can be properly said to correspond to some state of affairs (which is what we were assuming the object language was), as cuteness exists only in the mind, and is different for each individual. Davidson seems like he would deal with this problem by relativizing sentences like this to the speaker, so that the T sentence would become “‘bunnies are cute’ is true if and only if the speaker thinks that bunnies are ‘cute’”. No matter how we relativize such a sentence it would seem like would be left with no way to transform ‘cute’ into our object language. This can be seen more vividly when we consider a “new” value word, one we don’t already know the meaning for, say ‘nett’. Would explaining the meaning of ‘nett’ by stating that something is ‘nett’ when the speaker thinks it to be ‘nett’ or when the speaker is inclined to say it is ‘nett’ be informative? It seems like the only way to get at the meaning of ‘nett’ is to define it in terms of other subjective judgments (‘nett’ means cute in German), which is something T schema can’t do. Or, as above, we could deny that words like “cute” have meaning, but this too seems unreasonable.

I won’t deny that intuitively it does seem like T sentences are explaining meaning, by connecting sentences with their reference. Connecting sentences with their reference is exactly what a good theory of meaning should do, but it is not what T sentences are doing. Properly speaking, T sentences are something that result from already having a theory of meaning; having the T sentence is evidence that one has a theory of meaning, not a theory of meaning themselves. One way of construing a theory of meaning then is a way to transform sentences into “pure propositional” claims (or into claims about when the words are used, ect). I will leave an exploration of positive claims about what a theory of meaning is for another day, however (this post is long enough without them).

* Normally this is denoted by a square operator, but I don’t feel like finding that symbol in Unicode at the moment.

** It might be objected that given some enormously large number of T schema and enough time we might be able to decrypt the public language and thus come to understand all of the sentences in it. Even if this were possible it is besides the point. It was supposed to be the case that T schema alone were enough to provide a theory of meaning; I would argue that within the decryption and interpretation rules for sentences that we don’t have T schema for are where the real theory of meaning is.


1 Comment

  1. You should get UnicodeChecker.app. It’s quite nice for finding unicode symbols for all kinds of random things, such as hair spaces and cube roots. Of course, in OS X, when using a US English keyboard you can just type option-v to get √, the square root symbol, but with UnicodeChecker, you can look the more obscure and untypeable things like double prime ″ or two vertically aligned asterisks ⁑ or whatever else.

    Comment by Carl — October 17, 2006 @ 12:13 am

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