1. This is how Tarski describes the T-Schema in The concept of truth in formalized languages, in the volume Logic, semantics, metamathematics, Oxford at the Clarendon Press, 1956:
1) a true sentence is one which says that the state of affairs is so and so, and the state of affairs is indeed so and so.
(2) x is a true sentence if and only if p.
In order to obtain concrete definitions we substitute in the place of the symbol ”p” in this scheme any sentence, and in the place of ”x” any individual name of this sentence.
The most important common names for which the above condition is satisfied are the so-called quotation-mark names. We denote by this term every name if a sentence (or of any other, even meaningless, expression) which consists of quotation mars, left- and right-hand, and the expression which lies between them, and which (expression) is the object denoted by the name in question.
(3) ”snow is white” is a true sentence if and only if snow is white. In defining the correspondence between a sentence and a state of affairs, one must specify what particular state of affairs the sentence is corresponding to. In order to do this, we have to use the same sentence whose correspondence relation we are defining to name the state of affairs it corresponds to. Thus, we already presuppose the correspondence relation we are trying to define.
In saying that "snow is white" is a true sentence iff snow is white we implicitly presuppose that we understand the second occurrence of "snow is white" in the right side of the biconditional and that it can be true or false.
This way, the T-schema is either a vicious circle or it just says that "snow is white" is true has no different meaning than "snow is white" simpliciter (i.e. it is a deflationary theory of truth).
2. My supervisor said that the T-Schema is only apparently circular because the object-language sentence and the meta-language sentence coincide; but the meta-language may differ, for example:
"snow is white" is a true sentence iff snow is green.
But then one would have to state the truth condition for the meta-language sentence "snow is green" either in a meta-meta-language the coincides with the object-meta-language (a), or not (b):
(a) "snow is green" is a true sentence iff snow is green.
(b) "snow is green" is a true sentence iff snow is black.
And for each expression of meta-language, its truth conditions will have to be defined in a higher-order language ad infinitum, without ever obtaining a final truth condition. One may say that, even though the truth of an expression cannot be completely defined in this way, the T-Schema is still somewhat functional because of this recursion. But in the case that I do not already know the truth condition of any of the expressions used as truth conditions for the ones in the lower-order language, this recursion tells me nothing.
3. I take the T-schema to intend to say that a sentence like "snow is white" is true if and only if a fact like snow being white is the case. It tries to bypass the fact that "snow being white is the case" is still alinguistic expression, requiring its own definition of meaning or truth.
4. The truth predicate describes the relation of language and the world, but it itself is a part of language, and thus it can refer to itself; it is by nature a part of meta-language.
To make it even more obvious, let's replace Tarski's quotation-mark names with a different way of referring to a sentence:
The last three words in this phrase make up a true sentence if and only if snow is white.
Just as with the T-Schema, the same sentence is used here as (part of) its own truth condition.
5. In other words, to speak about the truth of a sentence is to mention what it is already being used (the correspondence relation). Mentioning this relation in regards to a certain sentence is trivial, because one always already understands it once it understands the sentence; and this knowledge cannot have a meaningful verbal expression.
6. So the question is, can the correspondence of language to the world be described within language?
7. The same as with Grelling's paradox, to define the truth of a sentence is to say something about a relation between the sentence and something exterior to it, while the sentence already relates to something exterior to itself. Also in common with Grelling's paradox, defining truth seems to lead to recursion.
B. Tarski's treatment of the Liar paradox
6. This is how Tarski analyzes the Liar paradox in The semantic conception of truth and the foundations of semantics, pp. 65-66, in the volume The Philosophy of Language, third edition, edited by A. P. Martinich, Oxford University Press 1996:
To obtain this antinomy in a perspicuous form, consider the following sentence:
The sentence written in this post on line 71 from the top is not true.
For brevity we shall replace the sentence just stated by the letter ‘s.’
According to our convention concerning the adequate usage of the term ‘true’. We assert the following equivalence of the form (T):
(1) ‘s’ is true if, and only if, the sentence written in this post on line 71 from the top is not true.
On the other hand, keeping in mind the meaning of the symbol ‘s,’ we establish empirically the following fact:
(2) 's’ is identical with the sentence written in this post on line 71 from the top.
Now, by a familiar law from the theory of identity (Leibniz’s law), it follows from (2) that we may replace in (1) the expression “the sentence printed in this post on line 71 from the top” by the symbol “ ‘s.’ “ We thus obtain what follows:
(3) ‘s’ is true if, and only if, ‘s’ is not true.
In this way we have arrived at an obvious contradiction. (I have changed the naming of the paradoxical sentence to something relevant in this context; I counted the lines of text from the top and I don't know if they coincide for every screen, but the point should be obvious.)
8. It seems that initially 's' replaces the sentence written in this post on line 71 from the top, which is: "the sentence written in this post on line 71 from the top is not true".
(2) says that 's' is identical with the sentence written in this post on line 71 from the top -- so 's' is identical with "the sentence written in this post on line 71 from the top is not true". It is not identical with the expression "the sentence written in this post on line 71 from the top". Thus in (3) 's' seems to stand for two different things: the first occurrence in the left side of the biconditional stands for "the sentence written in this post on line 71 from the top is not true", while the second occurrence of 's', in the right side of the biconditional stands for "the sentence written in this post on line 71 from the top" to which is added the predicate "is not true".
Now (3) doesn't even follow the T-Schema, because it adds a truth predicate to the second occurrence of the sentence in the right side of the biconditional; while (1) does -- the sentence "s" is true if and only if s.
9. Nevertheless, the expression "the sentence written in this post on line 71 from the top" is a name for "the sentence written in this post on line 71 from the top is not true", and the name itself is contained in the sentence. So the subject of the paradoxical sentence is actually a name for the whole sentence and replacing it with that which it names yields:
"the sentence written in this post on line 71 from the top is not true" is not true.
And, again, the replacement can be repeated infinitely.
No comments:
Post a Comment