A compositional theory of truth

Tarski's disquotational theory of truth in a structure is a compositional theory of truth, by which the truth of a complex assertion reduces to the truth of its constituent simpler pieces.

Another installment from A Panorama of Logic, an introduction to topics in logic for philosophers, mathematicians, and computer scientists.

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Truth in a structure

Having introduced the formal language, let us now define the accompanying semantics, a notion of truth for first-order logic. We aim to define what it means to say that a given sentence σ is true in a given structure M, which we write as follows:

\( M ⊨ \sigma.\)

We shall define truth by recursion on σ, but in doing so, the recursion will reduce from sentences to the slightly more general case of assertions with free variables, with temporary assignments of those variables to individuals of the model.

Namely, for any assertion φ(x₁,...,x_n) and individuals a₁, ..., a_n from M, we shall define the satisfaction relation

\( M ⊨ φ[a_1,...,a_n], \)

which expresses that assertion φ is true in M of those individuals. This relation is commonly read in various ways:

• “M satisfies φ(a₁,...,a_n )”

• “M satisfies φ at (a₁,...,a_n )”

• “M is a model of φ(a₁,...,a_n )”

• “M thinks that φ(a₁,...,a_n )”

• “φ(a₁,...,a_n) is true in M”

The definition will thus make use of the concept of a valuation in a model M of variable symbols x₁, ..., x_n, which is simply a function, conveniently denoted by [a₁,...,a_n], assigning each variable symbol x_i to a particular individual a_i of M. Such a valuation is sometimes written more elaborately as

\(\left[{x_1\, x_2\ \cdots\ x_n}\atop{ a_1\, a_2\ \cdots\ a_n}\right]\)

in order to make the functional relation x_i ↦ a_i more clear, and it is good to keep in mind the functional meaning of this more elaborate notation, which is what we intend implicitly with the simpler notation [a₁,...,a_n].

In effect, a valuation amounts to a temporary expansion of the language treating those particular variable symbols as constant symbols, interpreted in M according to the valuation. Ultimately we shall define what it means for an assertion φ to be true by appealing recursively to the truth of the direct subformulas of φ, and it is naturally the quantifier case that will require us to make adjustments to the valuation, making new assignments to the variable of quantification.

Disquotational theory of truth

The recursive definition of truth that we shall provide, due to Alfred Tarski, instantiates his disquotational theory of truth, according to which an assertion is true exactly when the proposition that it asserts is the case.

Thus,

“Snow is white” is true if and only if snow is white.

This disquotational idea is the foundation of Tarski’s definition of truth in a model—it is a compositional theory of truth, by which the truth of a compound assertion is defined in terms of the truth of the constituents of which it is composed. Let me explain.