Foundational Studies in Proof-theoretic Semantics - Barroso Nascimento, Victor Luis (2024)
This thesis investigates the technical and conceptual foundations of multibase semantics, a new kind of proof-theoretic semantics. Proof-theoretic semantics are frameworks in which the tools of Proof Theory are used for the semantic analysis of logics, bridging the gap between formal syntax and semantics. The first part of the thesis focuses on conceptual aspects of the notions of truth and proof, arguing that their distinct philosophical characteristics naturally lead to differences in their formal characterizations. It is also argued that, even though such proposals are usually presented by defenders of intuitionistim, proof-theoretic semantics should not be made for the intuitionist alone. The second part focuses on the technical aspects of multibase semantics. Multibases are presented in a standard and a focused version; standard multibases are no different from Kripke models for minimal logic, but focused multibases are shown to have many other interesting properties. In particular, focused multibases allow a generalization of the notion of S-validity, one of the main selling points of another semantics called proof-theoretic validity. Proof-theoretic validity, originally proposed by Prawitz and later championed by Dummett, was one of the first proposed proof-theoretic semantics, but the interest initially surrounding it was partially lost after a plethora of negative results were discovered (including incompleteness ones). Generalized S-validity is shown to have almost all of the properties originally expected to hold for S-validity, including completeness with respect to minimal logic. In particular, generalized S-validity is shown to be completely reducible to atomic derivability, and it is shown that multibases for predicate logic can be obtained without the aid of any model-theoretic notions. We also show that, as is expected of proof-theoretic semantics, it is possible to use methods characteristic of Proof Theory to obtain results that are semantic in nature.