Logics for Rough Concept Analysis

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.https://digitalcommons.chapman.edu/scs_books/1060/thumbnail.jp

Logics for Rough Concept Analysis

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

Semi De Morgan Logic Properly Displayed

In the present paper, we endow semi De Morgan logic and a family of its axiomatic extensions 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 the variety of semi De Morgan algebras, and applies the guidelines of the multi-type methodology in the design of display calculi

Syntactic completeness of proper display calculi

A recent strand of research in structural proof theory aims at exploring the notion of analytic calculi (i.e. those calculi that support general and modular proof-strategies for cut elimination), and at identifying classes of logics that can be captured in terms of these calculi. In this context, Wansing introduced the notion of proper display calculi as one possible design framework for proof calculi in which the analiticity desiderata are realized in a particularly transparent way. Recently, the theory of properly displayable logics (i.e. those logics that can be equivalently presented with some proper display calculus) has been developed in connection with generalized Sahlqvist theory (aka unified correspondence). Specifically, properly displayable logics have been syntactically characterized as those axiomatized by analytic inductive axioms, which can be equivalently and algorithmically transformed into analytic structural rules so that the resulting proper display calculi enjoy a set of basic properties: soundness, completeness, conservativity, cut elimination and subformula property. In this context, the proof that the given calculus is complete w.r.t. the original logic is usually carried out syntactically, i.e. by showing that a (cut free) derivation exists of each given axiom of the logic in the basic system to which the analytic structural rules algorithmically generated from the given axiom have been added. However, so far this proof strategy for syntactic completeness has been implemented on a case-by-case base, and not in general. In this paper, we address this gap by proving syntactic completeness for properly displayable logics in any normal (distributive) lattice expansion signature. Specifically, we show that for every analytic inductive axiom a cut free derivation can be effectively generated which has a specific shape, referred to as pre-normal form.Comment: arXiv admin note: text overlap with arXiv:1604.08822 by other author

Labelled calculi for lattice-based modal logics

We introduce labelled sequent calculi for the basic normal non-distributive modal logic L and 31 of its axiomatic extensions, where the labels are atomic formulas of a first order language which is interpreted on the canonical extensions of the algebras in the variety corresponding to the logic L. Modular proofs are presented that these calculi are all sound, complete and conservative w.r.t. L, and enjoy cut elimination and the subformula property. The introduction of these calculi showcases a general methodology for introducing labelled calculi for the class of LE-logics and their analytic axiomatic extensions in a principled and uniform way

Zero-one laws with respect to models of provability logic and two Grzegorczyk logics

It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5 and for frames corresponding to S4 and S5. In this paper, we prove zero-one laws for provability logic and its two siblings Grzegorczyk logic and weak Grzegorczyk logic, with respect to model validity. Moreover, we axiomatize validity in almost all relevant finite models, leading to three different axiom systems