Algebraic and Proof-Theoretic Foundations of the Logics for Social Behaviour

Abstract

This thesis is part of a line of research aimed at providing a strong and modular mathematical backbone to a wide and inherently diverse class of logics, introduced to capture different facets of social behaviour. The contributions of this thesis are rooted methodologically in duality, algebraic logic and structural proof theory, pertain to and advance three theories (unified correspondence, multi-type calculi, and updates on algebras) aimed at improving the semantic and proof-theoretic environment of wide classes of logics, and apply these theories to the introduction of logical frameworks specifically designed to capture concrete aspects of social behaviour, such as agents’ coordination and planning concerning the transformation and use of resources, and agents’ decision-making under uncertainty. The results of this thesis include: the characterization of the axiomatic extensions of the basic DLE-logics which admit proper display calculi; an algorithm computing the analytic structural rules capturing these axiomatic extensions; the introduction of a multi-type environment to describe and reason about agents’ abilities and capabilities to use and transform resources; the introduction of a proper display calculus for firstorder logic; the introduction of the intuitionistic counterpart of Probabilistic Dynamic Epistemic Logic, specifically designed to address situations in which truth is socially constructed. The results and methodologies developed in this thesis pave the way to the logical modelling of the inner workings of organizations and their dynamics, and of social phenomena such as reputational Matthew effects and bank runs.