This is the sixth part of a series developed through the Department of Philosophical Theology at the Institute of Lutheran Theology’s Christ School of Theology, exploring major results in modern logic and their significance for philosophical and systematic theology.
By the time modern logic had developed the resources of quantification, set theory, axiomatic systems, and model theory, a question that had always accompanied logic could no longer be postponed: What does it mean to say that a sentence is true? Although philosophers had, of course, asked about truth from antiquity onward, Alfred Tarski showed that once the question is raised within a sufficiently precise formal setting, one must distinguish matters that ordinary language easily allows us to run together. In particular, one must distinguish a sentence from the language in which we speak about that sentence, derivability from truth, and truth simpliciter from truth relative to an interpretation or structure.
This distinction became decisive because the formal languages developed by Frege, Russell, Hilbert, Gödel, and others were powerful enough to express increasingly complicated claims, while at the same time making possible new forms of semantic self-reference. Once a language becomes sufficiently expressive, allowing it unrestrictedly to contain its own truth predicate invites familiar paradoxes, most famously the liar: a sentence saying of itself that it is not true. Tarski's achievement was not merely to warn against such paradoxes, but to show how the notion of truth could nevertheless be defined rigorously for formal languages, provided that we carefully distinguish the language under investigation from the metalanguage in which its semantic properties are described.
The resulting conception of truth is often called the semantic conception of truth. Its most famous intuitive requirement is represented by what Tarski called Convention T. A satisfactory definition of truth should entail instances of the following form:
“Snow is white” is true if and only if snow is white.
The point is not the example, which is deliberately trivial, but the logical form. On the left we mention a sentence; on the right we use language to state the condition under which that sentence is true. A theory of truth must connect sentence and world without simply identifying the two, and it must do so from a standpoint in which the sentence itself can be referred to as an object of semantic investigation.
For first-order languages, however, truth is reached through the more basic notion of satisfaction. Suppose that M is a structure for a language L, with domain D, and that s is an assignment of objects in D to the variables of L. We then write
M ⊨ φ[s]
which is read:
The formula φ is satisfied in structure M under assignment s.
This notation matters because an open formula such as Px is not, strictly speaking, true or false independently of an assignment to x. It is satisfied in M under s when the object assigned by s to x belongs to the extension of P in M. Thus, if
s(x) = a,
then
M ⊨ Px[s]
just in case a belongs to the extension of P in M.
The recursive clauses then proceed through the logical structure of formulas. Negation, conjunction, disjunction, and the other connectives receive satisfaction conditions in terms of their component formulas, while quantifiers are handled by varying assignments. Thus,
M ⊨ ∃x φ[s]
if and only if there is some a in D such that
M ⊨ φ[s[x ↦ a]].
Likewise,
M ⊨ ∀x φ[s]
if and only if, for every a in D,
M ⊨ φ[s[x ↦ a]].
What initially looks like a technical device turns out to be philosophically important, because the semantic relation between language and structure is built up compositionally. We do not begin with an unexplained global notion of truth and then apply it indiscriminately. We define what it is for atomic formulas to be satisfied, specify how satisfaction behaves under the logical operations, and arrive finally at truth for sentences, which, because they contain no free variables, are satisfied or not satisfied independently of the particular assignment.
Accordingly, for a sentence σ we may write
M ⊨ σ
and read this:
σ is true in M,
or, equivalently,
M satisfies σ.
At this point an important distinction becomes unavoidable. To say that σ is true in M is not yet to say that σ is true simpliciter, unless M is being taken as the intended interpretation. Model theory deliberately allows many structures to interpret the same formal language, and therefore the same sentence may be true in one structure and false in another. The semantics tells us what follows once an interpretation has been fixed; it does not, merely by giving us the formal semantics, determine which interpretation is the one about which we intended to speak.
This point connects directly with the Löwenheim–Skolem and Compactness results considered in the preceding essays. Those theorems showed that first-order theories frequently possess models very different from the structures one might initially have intended. Tarski now gives us the semantic machinery for stating the matter precisely. If a theory T has many models, then
M₁ ⊨ T,
M₂ ⊨ T,
M₃ ⊨ T,
and so forth,
may all hold even though the structures M₁, M₂, and M₃ differ substantially. Satisfaction tells us whether a structure makes the sentences of the theory true; it does not by itself confer intendedness upon that structure.
The distinction between truth and provability is equally important. If T is a theory and φ a sentence, then
T ⊢ φ
says that φ is derivable from T by the formal proof rules, whereas
T ⊨ φ
says that every model of T satisfies φ.
Gödel's completeness theorem connects these two notions for first-order logic:
T ⊢ φ if and only if T ⊨ φ.
But the equivalence does not erase the conceptual distinction. The expression on the left concerns syntactic derivability; the expression on the right concerns semantic consequence. Indeed, the importance of Gödel's theorem lies precisely in the fact that two independently defined notions—proof and semantic consequence—turn out to coincide for first-order logic.
Here Tarski's work makes a contribution that theology ought to notice, although perhaps not in the way theologians sometimes suppose. The result does not establish that truth is ineffable, that human language cannot speak about God, that propositions fail before transcendence, or that theological language must finally dissolve into mystery. None of these claims follows from Tarski. What does follow is more disciplined and, for theology, more useful: whenever we speak about the truth of sentences belonging to a language, we must distinguish the sentences themselves from the semantic framework within which their truth conditions are being specified.
That distinction becomes especially important when theology moves between biblical language, doctrinal formulation, philosophical reconstruction, and formal representation. Suppose, for example, that a theological theory contains the sentence
∀x(Fx → Cx),
read:
Everything that is finite is created.
A theologian may ask whether the sentence follows from some theological theory T, whether it is satisfied in some model M of that theory, whether it expresses accurately what the theological sources intend, or whether it is in fact true of reality. Those are related questions, but they are not identical questions. Formal semantics can illuminate their relations precisely because it does not allow them simply to collapse into one another.
The distinction also bears upon theological metalanguage. Creeds, confessions, biblical propositions, and doctrinal assertions ordinarily occur within historically developed languages whose terms already bear substantial semantic weight. When the theologian begins to say what those sentences mean, under what conditions they are true, what follows from them, or what models satisfy them, the theologian has moved, whether explicitly or not, into a metalanguage. Once that movement is recognized, one can ask more carefully whether the metalanguage merely explicates the theological language, whether it transforms it, or whether it imports ontological and semantic commitments that the original language itself did not possess.
For philosophical theology, therefore, Tarski's importance lies not in providing a theological theory of truth, but in teaching us how much must already be distinguished before such a theory can responsibly be attempted. A sentence, its proof, its interpretation, the structure in which it is satisfied, and the reality about which it is intended to speak belong to different logical relations, even though theological discourse often moves rapidly among them. The semantic conception of truth disciplines that movement because it forces us to say, at each stage, what language we are using, what structure we have fixed, and what relation we are asserting between them.
The consequence is not skepticism but precision. Tarski does not tell us that truth escapes language; he shows us that language can speak rigorously about truth only when it respects the distinctions required by its own semantic functioning. For theology, which must continually speak both within its inherited language and about that language, this is no small achievement.
Why It Matters for Theology
Tarski's work gives philosophical theology at least four enduring lessons. First, truth must be distinguished from provability. Second, satisfaction in a model must be distinguished from truth under an intended interpretation. Third, theological object-language must be distinguished from the metalanguage by which theologians analyze its meaning and truth conditions. Fourth, formal adequacy does not by itself establish theological adequacy, since a formal structure may satisfy a theory without yet being the structure about which the theology intends to speak.
These distinctions become increasingly important as theology employs formal logic, model theory, possible-world semantics, or other forms of analytic reconstruction. The more powerful our formal languages become, the more necessary it becomes to know exactly which claims belong to the formal system and which claims concern the interpretation of that system. Tarski's achievement was to make that difference visible with a precision that modern theology can scarcely afford to ignore.
Bibliographical Note
Alfred Tarski's classic statement appears in “The Concept of Truth in Formalized Languages,” originally published in Polish in 1933 and later translated into English in Logic, Semantics, Metamathematics. His broader semantic conception is presented accessibly in “The Semantic Conception of Truth and the Foundations of Semantics,” Philosophy and Phenomenological Research 4 (1944): 341–376. For contemporary treatments, see standard introductions to model theory and philosophical logic under the topics of satisfaction, semantic truth, object language, metalanguage, and Tarski's undefinability theorem.
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