Defining Truth

Is there any more to the concept of truth than what is captured by Tarski’s TSchema: ‘s’ is true if and only if p, where ‘“s’” is a name of the sentence in the object language that is translated by ‘p’ in the metalanguage?

The concept of truth can appear to be simple at first glance. The normal idea of truth is that which actually exists as it does in reality. That what is said is also the way things are. This intuitive idea of truth is called correspondence, in that the proposition corresponds with actual reality. One of the best versions of this idea is the T-schema from Alfred Tarski. I'll offer a few objections to this theory of truth, along with responses.

An often used way to express the T-schema is: "Snow is white" is true if and only if snow is white. The same principle can be written as: "s" is true iff p. The simple structure of this formulation is deceptively complex in addressing the issue of truth.

The most noticeable thing about the T-schema is that it is recursive, it states the same thing twice, seemingly referring back to itself. This represents two levels of analysis. The first time that "'Snow is white'" is stated is analyzed by the second time it is stated. Language is used to talk about, analyze, and even construct language. These uses are metalanguage.

One problem with this is that to then support this, another level is needed, such that: "'Snow is white' is true if and only if snow is white" is true if and only if snow is white. This tendency toward expansion of the definition of truth can go on indefinitely, the predicate doing the analysis continually becoming part of the term that is being analyzed.

The T-schema contains an answer to this problem, although it may not be adequately expressed in the original formulation. To make it more clear I'll restate the natural language example as: "Snow is white" is true if and only if snow is actually white. The point of adding actually is that it indicates a referent outside of propositional language. The referent is reality. It's language designed to pull us out of language and into referent reality, to make abstractions point to that which is actually real. To say that the end definition must be an ostensive definition.

If we accept this we move on to our next problem, the issue of actually pointing at snow and saying "Snow is white." It may be that the snow is white, but it may be that the snow is not white. Here we have an issue of identity and the attributes of that identity. This problem does not contradict the use of Tarski's schema. If the snow is not white the original schema stands: "Snow is white" is not true if and only if snow is not white. Or, "Snow is not white" is true if and only if snow is not white. Or, the first instance as previously amended, "Snow is white" is not true if and only if snow is not actually white. Or, the second instance as previously amended, "Snow is not white" is true if and only if snow is actually not white.

We also encounter an issue with vagueness, for determining when one color becomes another color on a sliding scale is no simple problem. Uncertainty can be added to the schema in that it could be made to read: "Snow may be white" is true if and only if snow may be white. This same problem can be encountered with many adjectives other than color. For instance, "Steel is hard" is true if and only if steel is hard. Or, "Wet cement is heavy" is true if and only if wet cement is heavy. This vagueness issue when applied to the T-schema can be handled in any way that vagueness is otherwise handled, with a gap or a glut or many values. The issue here then is with vagueness and not the schema itself.

One way of utilizing examples that omit this vagueness problem while assessing the T-schema would be to use objective attributes. For instance, a specific measure for weight or length. At first this seems like a solution, but on further inspection it has its own difficulties. Without a shared ostensive referent we are unable to say "This weighs 21 grams" is true if and only if this weighs 21 grams. Thus we must say "The flower that weighs 21 grams weighs 21 grams" is true if and only if the flower that weighs 21 grams weighs 21 grams weighs 21 grams. The measurement adds no clarity to the statement. Plus, the exact measurement also falls into the sorites paradox about precise versus vague measurements as well. So, we may leave vagueness to discussions about vagueness and realize that their bearing is not special in the case of the T-schema.

The final problem that I'm going to address is the liar's paradox. This can be given in a number of different forms. For instance, if we take as our term "This sentence is not true" and put it into the schema we end up with: "This sentence is not true" is true if and only if this sentence is not true. The problem here is the self-reference, in this case specifically the word "this". The first time that "this" is used it refers to the sentence within the quotation marks. The second time the word "this" is used in the schema it can appear at first glance to refer to the entire schema. It's important to remember that both uses of "this" refer to the sentence inside of the quotation marks.

There are several different positions to take on the liar's paradox. One is that a language cannot contain its own truth predicate. That's why a metalanguage ends up being used, and I've previously addressed certain problems with that approach as well as one proposed solution. Another take is that the sentence is both truth and false. Another is that the sentence is neither true nor false.

It is possible to put a term into the schema with no content. For instance, "This" is true if and only if this is true. This helps to show that the self-referential nature of the paradox is the paradox itself. A self-referential sentence can never fulfill its own referential content, and is therefore an infinite regression in itself, and ends up referring to nothing. The emptiness of a self-referential statement means that it is nonsensical and can have no truth value.

An objection to this solution is to use two statements that refer to each other. In this case both sentences should be included as the term in the schema since they are the referenced content of each other. The solution remains the same.

Tarski's T-schema seems like a simplistic account of truth and that there must therefore be something beyond this statement. And, it is true that many objections can be raised to the schema. It is also true that there is the potentiality for solutions to those objections. The T-schema will continue to remain both relevant and prominent into future discussions of logic and truth.

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