Sabine Frittella

Multi-type display calculus for propositional dynamic logic

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.

Multi-type display calculus for dynamic epistemic logic

In the present paper, we introduce a multi-type display calculus for dynamic epistemic logic, which we refer to as Dynamic Calculus. The displayapproach 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.

A proof-theoretic semantic analysis of dynamic epistemic logic

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 [30]. 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.

Dual characterizations for finite lattices via correspondence theory for monotone modal logic

We establish a formal connection between algorithmic correspondence theory and certain dual characterization results for finite lattices, similar to Nations characterization of a hierarchy of pseudovarieties of finite lattices, progressively generalizing finite distributive lattices. This formal connection is mediated through monotone modal logic. Indeed, we adapt the correspondence algorithm ALBA to the setting of monotone modal logic, and we use a certain duality-induced encoding of finite lattices as monotone neighbourhood frames to translate lattice terms into formulas in monotone modal logic.

A Multi-type Calculus for Inquisitive Logic

In this paper, we define a multi-type calculus for inquisitive logic, which is sound, complete and enjoys Belnap-style cut-elimination and subformula property. Inquisitive logic is the logic of inquisitive semantics, a semantic framework developed by Groenendijk, Roelofsen and Ciardelli which captures both assertions and questions in natural language. Inquisitive logic adopts the so-called support semantics also known as team semantics. The Hilbert-style presentation of inquisitive logic is not closed under uniform substitution, and some axioms are sound only for a certain subclass of formulas, called flat formulas. This and other features make the quest for analytic calculi for this logic not straightforward. We develop a certain algebraic and order-theoretic analysis of the team semantics, which provides the guidelines for the design of a multi-type environment accounting for two domains of interpretation, for flat and for general formulas, as well as for their interaction. This multi-type environment in its turn provides the semantic environment for the multi-type calculus for inquisitive logic we introduce in this paper.

Fixed-Point Theory in the Varieties Dn

The varieties of lattices \(\mathcal{D}_n\), n ≥ 0, were introduced in [Nat90] and studied later in [Sem05]. These varieties might be considered as generalizations of the variety of distributive lattices which, as a matter of fact, coincides with \(\mathcal{D}_{0}\). It is well known that least and greatest fixed-points of terms are definable on distributive lattices; this is an immediate consequence of the fact that the equation \(\phi^{2}(\bot) = \phi(\bot)\) holds on distributive lattices, for any lattice term φ(x). In this paper we propose a generalization of this fact by showing that the identity φ n + 2(x) = φ n + 1(x) holds in \(\mathcal{D}n\), for any lattice term φ(x) and for \(x \in \{\top,\bot\}\). Moreover, we prove that the equations φ n + 1(x) = φ n (x), \(x = \bot,\top\), do not hold in the variety \(\mathcal{D}_{n}\) nor in the variety \(\mathcal{D}_{n} \cap \mathcal{D}_{n}^{op}\), where \(\mathcal{D}_{n}^{op}\) is the variety containing the lattices L op , for \(L \in \mathcal{D}_{n}\).

Software Tool Support for Modular Reasoning in Modal Logics of Actions

We present a software tool for reasoning in and about propositional sequent calculi for modal logics of actions. As an example, we implement the display calculus D.EAK of dynamic epistemic logic. The tool generates embeddings of the calculus in the theorem prover Isabelle/HOL for formalising proofs about D.EAK. Integrating propositional reasoning in D.EAK with inductive reasoning in Isabelle/HOL, we verify the solution of the muddy children puzzle for any number of muddy children. There also is a set of meta-tools that allows us to adapt the software for a wide variety of user defined calculi.

Probabilistic Epistemic Updates on Algebras

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.

Fixed-Point Theory in the Varieties \mathcal{D}_{n}

The varieties of lattices \(\mathcal{D}_n\), n ≥ 0, were introduced in [Nat90] and studied later in [Sem05]. These varieties might be considered as generalizations of the variety of distributive lattices which, as a matter of fact, coincides with \(\mathcal{D}_{0}\). It is well known that least and greatest fixed-points of terms are definable on distributive lattices; this is an immediate consequence of the fact that the equation \(\phi^{2}(\bot) = \phi(\bot)\) holds on distributive lattices, for any lattice term φ(x). In this paper we propose a generalization of this fact by showing that the identity φ n + 2(x) = φ n + 1(x) holds in \(\mathcal{D}n\), for any lattice term φ(x) and for \(x \in \{\top,\bot\}\). Moreover, we prove that the equations φ n + 1(x) = φ n (x), \(x = \bot,\top\), do not hold in the variety \(\mathcal{D}_{n}\) nor in the variety \(\mathcal{D}_{n} \cap \mathcal{D}_{n}^{op}\), where \(\mathcal{D}_{n}^{op}\) is the variety containing the lattices L op , for \(L \in \mathcal{D}_{n}\).