• The heart of this account appears in subsections 7-11. The first two of these are negative, noting the paradoxical consequences of doing all that we might wish. Tarski then sketches what is possible in 9-11. Along the way, he discusses the idea of a “metalanguage,” which is probably his most influential contribution to broader culture.

Tarski doesn’t provide many details of his actual definition of ‘is true’, partly to avoid technicalities and partly because the actual definition would depend on the language for which truth is being defined. To get a sense of what the definition might be like, suppose we are considering a language including an operations of conjunction, which forms a sentence A & B from sentences A and B, and negation, which forms a sentence not: A from a sentence A. Then we can say the following about the application of ‘is true’ to sentences of these sorts:

with the full definition of ‘is true’ consisting a series of analogous clauses (though, in any language containing quantifiers, they would be stated for the satisfaction relation rather than for truth). (You should compare these examples with Tarski’s example of disjunction on p. 353.) The fact that the application of ‘is true’ is defined for complex sentences in terms of its application to simpler sentences is what makes the definition “recursive” (i.e., it repeatedly recurs to the predicate being defined).

• The second part of the paper, Tarski’s “polemical remarks,” addresses a miscellaneous group of issues. The most important for our purposes are the concerns about metaphysics addressed in subsection 19 and the beginning of 21. Because of the ties between ‘truth’ and ‘reality’, many of the logical positivists regarded accounts of truth as metaphysics and thus not legitimate. Tarski’s work changed the direction of their thinking about this; and Rudolph Carnap, who is historically the most important of the group, put semantics at the core of his later thinking.

Tarski’s subsection 22 only hints at the possible applications of semantics to mathematics, but Tarski became one of the leaders of a new field of “model theory,” which came to provide what were probably the most influential applications of logic to mathematics. And when, several decades after this paper, Tarski’s ideas were applied in semantic theory as this is understood by linguists, this application came by way of the extension of ideas from mathematical model theory to natural languages.