Alfred Tarski: Truth, Semantics & Logical Consequence

Biography

Alfred Tarski was born Alfred Tajtelbaum on January 14, 1901, in Warsaw, then under Russian rule. He studied at the University of Warsaw, where he came under the influence of the great logicians Stanislaw Lesniewski and Jan Lukasiewicz and of the philosopher Tadeusz Kotarbinski, in the intellectual atmosphere of the Lvov-Warsaw school — a circle remarkable for its commitment to clarity, rigor, and formal methods. Tarski changed his name to Tarski in the 1920s, and he was baptized a Catholic in 1923; both changes were practical responses to the antisemitism that would soon engulf Poland.

In 1924 Tarski earned his doctorate and began his groundbreaking work. In 1933 he published his masterpiece, "The Concept of Truth in Formalized Languages," which gave the first mathematically exact definition of truth. He traveled to Vienna in 1930, where he met Godel and Carnap, and to Paris in 1935, where he presented his related work on logical consequence. His situation in Poland deteriorated; he emigrated to the United States in 1939, arriving just before the outbreak of war — a journey that saved his life, though most of his family perished in the Holocaust. He joined the University of California, Berkeley, in 1942 and built it into the world center of model theory, supervising dozens of students and shaping the field. He died on October 26, 1983, in Berkeley.

Historical Background

Tarski worked in a golden age of logic. Frege had created modern logic; Russell and Whitehead had tried to reduce mathematics to it; Godel had just proved the incompleteness theorems, which used the arithmetization of syntax to show the limits of formal systems. What was missing was a precise account of the semantics of logic — what it means for a formula to be true or false in a structure. The logical positivists, especially Rudolf Carnap, urgently wanted such an account for their project of grounding science and meaning in formal languages. Tarski supplied it.

His 1933 monograph solved the problem by making truth relative to a language and a model: rather than asking what truth is in general, he showed how to define a truth predicate for any given formalized language, using only the resources of a stronger metalanguage. The result satisfied both mathematicians (who gained a rigorous tool) and philosophers (who gained a model of semantic analysis), and it set the agenda for the philosophy of language for the next half century.

Core Ideas

The semantic theory of truth. Tarski's adequacy condition, now called the T-schema or Convention T, says that an acceptable definition of truth for a language L must entail, for every sentence S of L, an equivalence of the form:

"S is true in L if and only if p"

where "S" is a name of the sentence and p is its translation. The paradigm: "'Snow is white' is true if and only if snow is white." This captures the intuition that truth is correspondence between a sentence and the world, but in a way that is mathematically exact: truth is defined by recursive satisfaction conditions over the structure of the language.

Truth is not definable in the language itself. By combining his methods with Godel's diagonalization, Tarski proved that no language rich enough to express its own syntax can define its own truth predicate without contradiction. This undefinability theorem is the semantic cousin of Godel's incompleteness theorems and explains why the liar paradox cannot be dissolved within the language: "This sentence is false" has no consistent truth value in a language that can express it.

Logical consequence. In 1936 Tarski defined logical consequence model-theoretically: a sentence φ is a logical consequence of a set of premises Γ if every model of Γ is a model of φ. This definition, refined by his later work with Robert Vaught, became the standard account of validity and founded model theory as a discipline.

Model theory. With his students at Berkeley, Tarski developed the tools — theories, elementary equivalence, definability, saturated models — that turned model theory into one of the central branches of mathematical logic.

Major Works

"The Concept of Truth in Formalized Languages" (1933) — the semantic theory of truth and the undefinability theorem; the single most important paper in the philosophy of logic of the century.

"On the Concept of Logical Consequence" (1936) — the model-theoretic definition of consequence.

"The Completeness of Elementary Algebra and Geometry" (1948, with Andrew Mostowski) — decision procedures and the model theory of algebra.

Undecidable Theories (1953, with Mostowski and Raphael Robinson) — the limits of decidability.

Logic, Semantics, Metamathematics (1956) — the collected papers that made his work available to a generation.

Philosophical Influence

Tarski's theory of truth changed philosophy. For the logical positivists, it showed how a scientifically respectable notion of truth could be defined without metaphysics — Carnap adopted it enthusiastically. For the philosophy of language, it supplied the model for compositional semantics and inspired Donald Davidson's program of a truth-theoretic theory of meaning. For metaphysics and epistemology, it reframed the ancient question "what is truth?" as the technical question "what does an adequate truth definition require?" — and answered it.

The undefinability theorem is among the deepest limits ever proved about language: a language cannot, from within, fully capture its own concept of truth. This result reverberates through the liar paradox, Godel's theorems, and the semantics of natural language, where Kripke and others have explored weakening the classical assumptions to allow self-referential truth talk. Tarski's model-theoretic account of logical consequence likewise set the terms for the modern debate — including John Etchemendy's influential critique — and remains the default definition in logic textbooks. Every student of predicate logic works within Tarski's semantic framework.

Related Concepts

Tarski's work defines the semantics of logical consequence and validity, and it is the natural home of truth and the liar paradox. It presupposes predicate logic and complements the incompleteness theorems of Godel. His model-theoretic approach grounds the philosophy of logic, the philosophy of language, and the philosophy of mathematics, and his successors include Quine and Kripke.