Abstract: By adding a validity predicate Val to a propositional language, we develop a validity theory that fully and correctly captures its metalinguistic validity. This project faces two challenges. First, weak validity and logical principles are inconsistent because of the Validity Paradox. Second, all existing validity theories, even those that avoid the paradox, get some features of validity wrong. More precisely, we show that they do not prove that all and only valid inferences, metainferences, and higher-level metainferences are valid, or they do not prove that all and only invalid ones are invalid. We formalize these desiderata with the V-schemata. We say that a validity theory is naïve iff it satisfies all the V-schemata. We point out why existing validity theories cannot be naïve. We develop a logical framework, which we call non-binary, to overcome this limitation. We define a validity theory based on a non-binary logic and a fixed-point interpretation for the validity predicate. We prove that this non-binary validity theory is consistent and naïve: it avoids paradoxes and satisfies all the V-schemata.
Activities > Graduate Research Seminar (GRSem) > GRSem: A Non-binary Naïve Validity Theory
