Algebras and Input/Output Logic

The present thesis focuses on the study of the mathematical structures underlying input/output logic. Building on insights from Abstract Algebraic Logic and on the recent introduction of subordination algebras and related structures as a semantic environment for input/output logic, the present thesis establishes a strong and systematic bridge between areas of logic which have been investigated for decades independently of each other, and by doing so, it opens new possibilities of interpretations and areas of applications for these formal frameworks. Motivated by a research program investigating ‘logic at work’ rather than ‘logic in isolation’, input/output logic has been introduced for modelling the interaction between logical inferences and other agency-related notions such as conditional obligations, goals, ideals, preferences, actions, and beliefs, in the context of the formalization of normative systems in philosophical logic and AI, but also in connection and combination with a very diverse range of issues, also relevant to present-day AI, spanning from formal argumentation, to causal reasoning and non-monotonic reasoning. The first contribution of this thesis is a generalization of the framework of input/output logic from classical propositional logic to selfextensional logics, a large class of logics which prominently includes those logical systems the consequence relation of which corresponds to a partial order on the algebras canonically associated with it. The various notions of permission systems, namely, negative permission, positive static permission and dynamic permission, are generalized and studied uniformly in the context of selfextensional logics, both in themselves, and in connection with normative systems. In the same context, the notion of dual negative permission system is introduced and studied. The second contribution of this thesis is a systematic study of the algebraic counterparts of normative, permission, and dual permission systems based on the algebras of the classes canonically associated with (fully) selfextensional logics. In particular, the proof of the main characterization result of the previous chapter is refined and streamlined, and further extended to characterize properties of various notions of permission systems also in relation to normative systems, by means of modal axioms in an expanded slanted signature which includes negative modal operators (intuitively representing prohibitions). Moreover, these characterization results are then applied to resolve different issues: from the logical characterizations of output operators for both normative and permission systems, to the dual characterizations of conditions on subordination algebra (resp. precontact algebras) and on their associated spaces; the algebraization of positive static permissions; the modal characterization of the notion of cross-coherence; the clarification of the relation between certain conditions on positive bi-subordination lattices and Dunn’s axioms for positive modal logic. The third contribution of this thesis generalizes the modal characterizations, achieved in the previous chapter, of finitely many conditions on normative and permission systems, as well as on subordination algebras and related structures, to those conditions corresponding to the infinite class of clopen-analytic inequalities. Conversely, Kracht (first-order) formulas are introduced as the class of those first order conditions on algebras with subordination, precontact, and dual precontact relations which correspond to clopen-analytic axioms; finally, as an application of these results, Celani’s dual characterization results between subordination lattices and subordination spaces is generalized to the environment of algebras with relations based on distributive lattices and the infinite class of inductive Kracht formulas.