arXiv subjects
Publications and source records attributed to Alessandra Palmigiano.
We introduce proper display calculi for basic monotonic modal logic,the conditional logic CK and a number of their axiomatic extensions. These calculi are sound, complete, conservative and enjoy cut elimination and subformula property. Our proposal applies the multi-type methodology in the design of display calculi, starting from a semantic analysis based on the translation from monotonic modal logic to normal bi-modal logic.
In recent years, unified correspondence has been developed as a generalized Sahlqvist theory which applies uniformly to all signatures of normal and regular (distributive) lattice expansions. This includes a general definition of the Sahlqvist and inductive formulas and inequalities in every such signature, based on order theory. This definition covers in particular all (bi-)intuitionistic modal logics. The theory of these logics has been intensively studied over the past seventy years in connection with classical polyadic modal logics, using suitable versions of Goedel-McKinsey-Tarski translations as main tools. It is therefore natural to ask (1) whether a general perspective on Goedel-McKinsey-Tarski translations can be attained, also based on order-theoretic principles like those underlying the general definition of Sahlqvist and inductive formulas and inequalities, which accounts for the known Goedel-McKinsey-Tarski translations and applies uniformly to all signatures of normal (distributive) lattice expansions; (2) whether this general perspective can be used to transfer correspondence and canonicity theorems for Sahlqvist and inductive formulas and inequalities in all signatures described above under Goedel-McKinsey-Tarski translations. In the present paper, we set out to answer these questions. We answer (1) in the affirmative; as to (2), we prove the transfer of the correspondence theorem for inductive inequalities of arbitrary signatures of normal distributive lattice expansions. We also prove the transfer of canonicity for inductive inequalities, but only restricted to arbitrary normal modal expansions of bi-intuitionistic logic. We also analyze the difficulties involved in obtaining the transfer of canonicity outside this setting, and indicate a route to extend the transfer of canonicity to all signatures of normal distributive lattice expansions.
RS-frames were introduced by Gehrke as relational semantics for substructural logics. They are two-sorted structures, based on RS-polarities with additional relations used to interpret modalities. We propose an intuitive, epistemic interpretation of RS-frames for modal logic, in terms of categorization systems and agents' subjective interpretations of these systems. Categorization systems are a key to any decision-making process and are widely studied in the social and management sciences. A set of objects together with a set of properties and an incidence relation connecting objects with their properties forms a polarity which can be `pruned' into an RS-polarity. Potential categories emerge as the Galois-stable sets of this polarity, just like the concepts of Formal Concept Analysis. An agent's beliefs about objects and their properties (which might be partial) is modelled by a relation which gives rise to a normal modal operator expressing the agent's beliefs about category membership. Fixed-points of the iterations of the belief modalities of all agents are used to model categories constructed through social interaction.
Taking an algebraic perspective on the basic structures of Rough Concept Analysis as the starting point, in this paper we introduce some varieties of lattices expanded with normal modal operators which can be regarded as the natural rough algebra counterparts of certain subclasses of rough formal contexts, and introduce proper display calculi for the logics associated with these varieties which are sound, complete, conservative and with uniform cut elimination and subformula property. These calculi modularly extend the multi-type calculi for rough algebras to a `nondistributive' (i.e. general lattice-based) setting.
In the present paper, we endow a family of axiomatic extensions of semi De Morgan logic with proper multi-type display calculi which are sound, complete, conservative, and enjoy cut elimination and subformula property. Our proposal builds on an algebraic analysis of semi De Morgan algebras and its subvarieties and applies the guidelines of the multi-type methodology in the design of display calculi.
We prove a uniform version of the Goldblatt-Thomason theorem for logics algebraically captured by normal lattice expansions (normal LE-logics).
In the present paper, we endow the logics of topological quasi Boolean algebras, topological quasi Boolean algebras 5, intermediate algebras of types 1-3, and pre-rough algebras with proper multi-type display calculi which are sound, complete, conservative, and enjoy cut elimination and subformula property. Our proposal builds on an algebraic analysis and applies the principles of the multi-type methodology in the design of display calculi.
The present paper contributes to the development of the mathematical theory of epistemic updates using the tools of duality theory. Here we focus on Probabilistic Dynamic Epistemic Logic (PDEL). We dually characterize the product update construction of PDEL-models as a certain construction transforming the complex algebras associated with the given model into the complex algebra associated with the updated model. Thanks to this construction, an interpretation of the language of PDEL can be defined on algebraic models based on Heyting algebras. This justifies our proposal for the axiomatization of the intuitionistic counterpart of PDEL, of which prove soundness and completeness with respect to algebraic probabilistic epistemic models.
In the present paper, we introduce a multi-type calculus for the logic of measurable Kleene algebras, for which we prove soundness, completeness, conservativity, cut elimination and subformula property. Our proposal imports ideas and techniques developed in formal linguistics around the notion of structural control.
The present paper provides an analysis of the existing proof systems for dynamic epistemic logic from the viewpoint of proof-theoretic semantics. Dynamic epistemic logic is one of the best known members of a family of logical systems which have been successfully applied to diverse scientific disciplines, but the proof theoretic treatment of which presents many difficulties. After an illustration of the proof-theoretic semantic principles most relevant to the treatment of logical connectives, we turn to illustrating the main features of display calculi, a proof-theoretic paradigm which has been successfully employed to give a proof-theoretic semantic account of modal and substructural logics. Then, we review some of the most significant proposals of proof systems for dynamic epistemic logics, and we critically reflect on them in the light of the previously introduced proof-theoretic semantic principles. The contributions of the present paper include a generalisation of Belnap's cut elimination metatheorem for display calculi, and a revised version of the display-style calculus D.EAK. We verify that the revised version satisfies the previously mentioned proof-theoretic semantic principles, and show that it enjoys cut elimination as a consequence of the generalised metatheorem.
In the present paper, we introduce a multi-type display calculus for dynamic epistemic logic, which we refer to as Dynamic Calculus. The display-approach is suitable to modularly chart the space of dynamic epistemic logics on weaker-than-classical propositional base. The presence of types endows the language of the Dynamic Calculus with additional expressivity, allows for a smooth proof-theoretic treatment, and paves the way towards a general methodology for the design of proof systems for the generality of dynamic logics, and certainly beyond dynamic epistemic logic. We prove that the Dynamic Calculus adequately captures Baltag-Moss-Solecki's dynamic epistemic logic, and enjoys Belnap-style cut elimination.
We introduce a multi-type display calculus for Propositional Dynamic Logic (PDL). This calculus is complete w.r.t. PDL, and enjoys Belnap-style cut-elimination and subformula property.
We introduce the logic LRC, designed to describe and reason about agents' abilities and capabilities in using resources. The proposed framework bridges two - up to now - mutually independent strands of literature: the one on logics of abilities and capabilities, developed within the theory of agency, and the one on logics of resources, motivated by program semantics. The logic LRC is suitable to describe and reason about key aspects of social behaviour in organizations. We prove a number of properties enjoyed by LRC (soundness, completeness, canonicity, disjunction property) and its associated analytic calculus (conservativity, cut elimination and subformula property). These results lay at the intersection of the algebraic theory of unified correspondence and the theory of multi-type calculi in structural proof theory. Case studies are discussed which showcase several ways in which this framework can be extended and enriched while retaining its basic properties, so as to model an array of issues, both practically and theoretically relevant, spanning from planning problems to the logical foundations of the theory of organizations.
We employ the theory of canonical extensions to study residuation algebras whose associated relational structures are functional, i.e., for which the ternary relations associated to the expanded operations admit an interpretation as (possibly partial) functions. Providing a partial answer to a question of Gehrke, we demonstrate that no universal first-order sentence in the language of residuation algebras is equivalent to the functionality of the associated relational structures.
We introduce a proper multi-type display calculus for bilattice logic (with conflation) for which we prove soundness, completeness, conservativity, standard subformula property and cut-elimination. Our proposal builds on the product representation of bilattices and applies the guidelines of the multi-type methodology in the design of display calculi.
Categorization systems are widely studied in psychology, sociology, and organization theory as information-structuring devices which are critical to decision-making processes. In the present paper, we introduce a sound and complete epistemic logic of categories and agents' categorical perception. The Kripke-style semantics of this logic is given in terms of data structures based on two domains: one domain representing objects (e.g. market products) and one domain representing the features of the objects which are relevant to the agents' decision-making. We use this framework to discuss and propose logic-based formalizations of some core concepts from psychological, sociological, and organizational research in categorization theory.
We generalize Venema's result on the canonicity of the additivity of positive terms, from classical modal logic to a vast class of logics the algebraic semantics of which is given by varieties of normal distributive lattice expansions (normal DLEs), aka `distributive lattices with operators'. We provide two contrasting proofs for this result: the first is along the lines of Venema's pseudo-correspondence argument but using the insights and tools of unified correspondence theory, and in particular the algorithm ALBA; the second closer to the style of Jónsson. Using insights gleaned from the second proof, we define a suitable enhancement of the algorithm ALBA, which we use prove the canonicity of certain syntactically defined classes of DLE-inequalities (called the meta-inductive inequalities), relative to the structures in which the formulas asserting the additivity of some given terms are valid.
We introduce a proper display calculus for (non-distributive) Lattice Logic which is sound, complete, conservative, and enjoys cut-elimination and sub-formula property. Properness (i.e. closure under uniform substitution of all parametric parts in rules) is the main interest and added value of the present proposal, and allows for the smoothest Belnap-style proof of cut-elimination. Our proposal builds on an algebraic and order-theoretic analysis of the semantic environment of lattice logic, and applies the guidelines of the multi-type methodology in the design of display calculi.