On the "semantics" for Languages with Their Own Truth Predicates

In the 1920s, logical positivists were skeptical of the notion of truth. For one thing, the liar's paradox seemed to show that truth is inconsistent. Other sources of skepticism were, as Soames [26] notes, "the frequent use of truth in metaphysical discussions, the tendency to confuse truth with epistemological notions like certainty and confirmation, and the inability to see how acceptance of a truth predicate could be squared with the doctrine of physicalism and the unity of science." On the other hand, the notion of truth seemed useful, even for scientific purposes. We would like to be able to assert that all the axioms of arithmetic are true, without asserting each of them. We might want to state general principles such as "valid arguments lead from true premises to true conclusions." What was wanted was a tool for "semantic ascent": As Etchemendy [2] puts it, the power to "recover or reassert a proposition [or a set of propositions] but to do so indirectly, without actually using a sentence that would, in the more usual fashion, express that same proposition." In 1933, Tarski [27] provided a general method for giving a definition of truth for formalized languages. His method was widely regarded as a rehabilitation of truth: he had shown how to give a mathematical definition of a notion of truth that was suitable for scientific purposes. Famously, the liar's paradox prevented Tarski from extending his methods to languages that have their own truth predicates, or that express their own truth-concepts. Both Tarski's original definition of truth, and its successor, the definition of truth-in-a-model, were not carried out in the language under investigation: truth is defined in a metalanguage for sentences of an object language. Only in the mid 1970s do we see papers—notably, Martin and Woodruff [21] and Kripke [19]—that extend Tarski's methods to languages that do, in some sense, contain their own truth predicates, even languages that display liar-like phenomena. These papers generated an industry aimed at providing "semantics" for languages expressing their own truth-concepts.