How truthlike can a predicate be? A negative result (Q1069920)

Tarski showed that a theory of truth for a language, formulated within that very language, that entails all instances of the schema \(^{\ulcorner \ulcorner}\phi^{\urcorner}\) is true iff \(\phi^{\urcorner}\) will be inconsistent with Peano arithmetic. This paper presents a further result along the same lines, showing that, if \(\Gamma\) is a theory of truth that contains Robinson's Q, is closed under first-order consequence and under the rule that from \(^{\ulcorner}\phi^{\urcorner}\) one may infer \(^{\ulcorner \ulcorner}\phi^{\urcorner}\) is \(true^{\urcorner}\), and contains all instances of the schemata asserting that a true conditional with a true antecedent has a true consequent, that a sentence and its negation are not both true, and that a generalization beginning ''for each natural number x'' is true if all of its numerical instances are, then \(\Gamma\) is \(\omega\)-inconsistent. Implications for recent theories of truth, such as those of Kripke, Gupta, and Herzberger, are discussed.