Tarski's Semantic Theory of Truth

The Challenge: Defining Truth

For much of the early 20th century, the concept of 'truth' was viewed with suspicion by many logically-minded philosophers. It seemed vague, metaphysical, and prone to paradoxes. The most famous of these is the Liar Paradox. Consider the sentence:

"This sentence is false."

If the sentence is true, then what it says must be the case, which means it must be false. But if it's false, then what it says is not the case, which means it must be true. This leads to a flat-out contradiction. For a logician, this is a disaster, as a system that allows contradictions is trivial—it can prove anything.

Tarski's goal was to rescue the concept of truth from this predicament. He argued that any adequate definition of 'true' must satisfy two fundamental conditions:

1. Material Adequacy: The definition must align with our common-sense understanding of truth. It must capture the intuition that, for example, the sentence "Snow is white" is true if and only if snow is, in fact, white.
2. Formal Correctness: The definition must be structurally sound and, above all, must not lead to contradictions like the Liar Paradox.

Let's watch a video that introduces Tarski's project and explains these two conditions in more detail.

As the video explains, Tarski formalized the material adequacy condition with his Convention T. A definition of truth is materially adequate only if it entails all sentences of the form:

(T) X is true if, and only if, p.

Here, 'p' stands for a sentence (e.g., snow is white), and 'X' stands for the name of that sentence (e.g., "snow is white"). The challenge is to find a single definition of "is true" that allows us to derive every possible instance of this schema for a given language, without falling into the trap of formal incorrectness revealed by the Liar Paradox.

Tarski's Solution: Object Language and Metalanguage

Tarski's diagnosis of the Liar Paradox was that it arises in what he called "semantically closed" languages. A language is semantically closed if it contains its own truth predicate—that is, if it can be used to talk about the truth or falsity of its own sentences. Natural languages like English appear to be semantically closed, which is why the paradox can be so easily stated.

Tarski's brilliant solution was to insist that when we define truth, we must always use two distinct languages:

1. The Object Language (L): This is the language for which we are defining truth. It is the object of our study.
2. The Metalanguage (M): This is the language in which we formulate the definition of truth. It is the language we use to talk about the object language.

The crucial rule is that the truth predicate, is-true-in-L, is part of the metalanguage M, and it can only apply to sentences of the object language L. This structure makes it impossible for a sentence to refer to its own truth value.

The Liar sentence, "This sentence is false," is therefore rendered ungrammatical. To express it, you would have to say something like:

"This sentence is false in language L."

However, this sentence itself cannot be in language L, because it uses the predicate "is false in language L," which belongs to the metalanguage M. The sentence is in M, but it purports to talk about itself as if it were in L. The paradoxical self-reference is blocked.

This solution implies that there can be no single, universal definition of "Truth." We can only ever define "truth-in-L" for some specific, formally specified object language L. If we then want to discuss the truth of sentences in our metalanguage M, we need to ascend to a new meta-metalanguage M', and so on, creating an infinite hierarchy of languages.

Let's watch the next segment of the video, which explains this core distinction.

To reinforce this fundamental concept, let's read a short, precise description from the Stanford Encyclopedia of Philosophy.

The Structure of the Definition

Now that we have the conceptual framework (the OL/ML distinction) and the success criteria (Convention T), how did Tarski actually construct the definition of "truth-in-L"?

He couldn't just list all the true sentences, because any interesting language contains infinitely many of them. Instead, he used a recursive definition. The structure should be familiar from your work in mathematical logic. The definition proceeds in two main stages:

1. Base Clause: Define a semantic property for the simplest parts of the language. Tarski used the notion of satisfaction. Satisfaction relates objects in the domain of discourse to the open formulas of the language (formulas with free variables). For example, the number 2 satisfies the open formula x > 1. For a language of predicate logic, Tarski would give a list-like definition of which objects satisfy the basic predicates of the language.

2. Recursive Clauses: Define satisfaction for complex formulas in terms of the satisfaction of their simpler parts. For example:

   • A sequence of objects satisfies A ∧ B if and only if it satisfies A and it satisfies B.
   • A sequence of objects satisfies ¬A if and only if it does not satisfy A.
   • A sequence satisfies ∀x F(x) if and only if F(x) is satisfied by every sequence that differs from the original only in what it assigns to x.

With the notion of satisfaction fully defined, the definition of truth becomes elegantly simple:

A sentence S (a formula with no free variables) is true in L if and only if S is satisfied by all sequences of objects.

This recursive approach allows a finite definition in the metalanguage to generate the truth conditions for all of the infinite sentences in the object language, thus satisfying Convention T.

The final part of the video you've been watching gives a good intuitive sketch of this recursive method.

Conclusion

Tarski's semantic theory of truth was a landmark achievement in 20th-century logic and philosophy. It demonstrated how the seemingly elusive concept of truth could be given a rigorous, formal definition for specified languages, finally taming the paradoxes that had plagued it.

Key Takeaways:

• Tarski's Goal: To provide a formally correct (paradox-free) and materially adequate (intuitive) definition of truth for formal languages.
• The Liar Paradox: This paradox served as the primary motivation, demonstrating the danger of "semantically closed" languages that can discuss their own truth.
• Object Language vs. Metalanguage: Tarski's core innovation was to insist on a strict separation between the language being analyzed (object language) and the language of the analysis (metalanguage). The truth predicate for a language L must reside in its metalanguage M.
• Hierarchy of Languages: This distinction creates an infinite hierarchy of languages, as defining truth for a metalanguage requires a meta-metalanguage.
• Recursive Definition: The definition of "truth-in-L" is recursive. It is built upon the more fundamental notion of satisfaction, which is defined first for atomic formulas and then extended to all complex formulas. A sentence is then defined as true if it is satisfied by all sequences.

Preview of the Next Lesson:

Tarski's theory of truth was not an end in itself. Its most significant application was as a foundation for a new, powerful definition of logical consequence. Now that we have a rigorous way to talk about truth, we can define a valid argument as one that preserves truth under all possible interpretations. In the next lesson, we will explore Tarski's model-theoretic definition of logical consequence and analyze its profound philosophical implications.