Dr. Antonio Piccolomini D'Aragona - The Proof-Theoretic Square

Recorded as part of the Carl Friedrich von Weizsäcker Colloquium on June 22nd

Dr. Antonio Piccolomini D'Aragona (University of Siena and Aix-Marseille University) - The Proof-Theoretic Square

Abstract:
In this talk, I focus on the interaction between two dichotomies in Prawitz’s proof-theoretic semantics, i.e. the dichotomy between monotonicity and non- monotonicity of validity of arguments over a specific atomic base (call this the local level), and the dichotomy between schematicity and non-schematicity of validity of arguments over all atomic bases (call this the global level). I argue that these dichotomies undergo some conceptual symmetries, both internally - i.e. the opposition at the local level is in a way conceptually analogous to the opposition at the global level - and externally - i.e. the alternative at the local level is somehow conceptually mirrored by the alternative at the global level. These symmetries may be understood as imposing a quite strict constraint on the overall semantic framework, i.e. one requires non-monotonicity at the local level iff one requires schematicity at the global level and, vice versa, one requires monotonicity at the local level iff one requires non-schematicity at the global level. This returns two conceptually and extensionally distinct proof-theoretic semantics, which both seem to be compatible with Prawitz’s philosophical tenets. However, I also argue that the aforementioned symmetries stem from a deeper interaction at play in Prawitz’s semantics, namely, the interaction between non-logical meanings and meaning - i.e. justification - of non-primitive inference rules. Based on this deeper interaction, two further “mixed” readings (monotonicity/ schematicity and non-monotonicity/non-schematicity) can be said to be compatible with Prawitz’s intentions - and are actually found in the literature. I finally claim that further combinations given by the interaction between non-logical meanings and meaning of rules are either void or equivalent to one of the four possibilities above. Thus, we are left with a group of four Prawitz-compatible semantics, forming a diagram whose arrows are “harmonically” oriented by the interaction between non-logical meanings and meaning of rules.