Categorical Abstract Algebraic Logic: Meet-Combination of Logical Systems

The widespread and rapid proliferation of logical systems in several areas of computer science has led to a resurgence of interest in various methods for combining logical systems and in investigations into the properties inherited by the resulting combinations. One of the oldest such methods isfibring. In fibring the shared connectives of the combined logics inherit properties frombothcomponent logical systems, and this leads often to inconsistencies. To deal with such undesired effects, Sernadas et al. (2011, 2012) have recently introduced a novel way of combining logics, calledmeet-combination, in which the combined connectives share only thecommonlogical properties they enjoy in the component systems. In their investigations they provide a sound and concretely complete calculus for the meet-combination based on available sound and complete calculi for the component systems. In this work, an effort is made to abstract those results to a categorical level amenable tocategorical abstract algebraic logictechniques.

Modal Logics for Reasoning about Multiagent Systems

It becomes evident in recent years a surge of interest to applications of modal logics for specification and validation of complex systems. It holds in particular for combined logics of knowledge, time and actions for reasoning about multiagent systems (Dixon, Nalon & Fisher, 2004; Fagin, Halpern, Moses & Vardi, 1995; Halpern & Vardi, 1986; Halpern, van der Meyden & Vardi, 2004; van der Hoek & Wooldridge, 2002; Lomuscio, & Penczek, W., 2003; van der Meyden & Shilov, 1999; Shilov, Garanina & Choe, 2006; Wooldridge, 2002). In the next paragraph we explain what are logics of knowledge, time and actions from a viewpoint of mathematicians and philosophers. It provides us a historic perspective and a scientific context for these logics. For mathematicians and philosophers logics of actions, time, and knowledge can be introduced in few sentences. A logic of actions (ex., Elementary Propositional Dynamic Logic (Harel, Kozen & Tiuryn, 2000)) is a polymodal variant of a basic modal logic K (Bull & Segerberg, 2001) to be interpreted over arbitrary Kripke models. A logic of time (ex., Linear Temporal Logic (Emerson, 1990)) is a modal logic with a number of modalities that correspond to “next time”, “always”, “sometimes”, and “until” to be interpreted in Kripke models over partial orders (discrete linear orders for LTL in particular). Finally, a logic of knowledge or epistemic logic (ex., Propositional Logic of Knowledge (Fagin, Halpern, Moses & Vardi, 1995; Rescher, 2005)) is a polymodal variant of another basic modal logic S5 (Bull & Segerberg, 2001) to be interpreted over Kripke models where all binary relations are equivalences.

Grafted frames and S1 -completeness

A grafted frame is a new kind of frame which combines a modal frame and some relevance frames. A grafted model consists of a grafted frame and a truth-value assignment. In this paper, the grafted frame and the grafted model are constructed and used to show the completeness of S1. The implications of S1-completeness are discussed.A grafted frame does not combine two kinds of frames simply by putting relations defined in the components together. That is, the resulting grafted frame is not in the form of <W,R,R′>, or more generally, in the form of <W, R, R′,R″>,…>, which consists of a non-empty set with several relations defined on it.1 Rather, it resembles the construction of fibering proposed by D. M. Gabbay and M. Finger (see [4] and [3]). On a grafted frame, some modal worlds, which belong to the initial modal frame, are attached by some relevance frames.However, these two semantics have important differences. Consider the combined semantics involving semantics of relevance logic and modal logic. A fibred model and a grafted model proposed in this paper differ in the following respects. First, a fibred model is constructed from a class of modal models and a class of relevance models. A grafted model consists of a grafted frame and a truth-value assignment, where the grafted frame is constructed from a modal frame and some relevance frames, and the assignment is a union of a modal truth-value assignment VM and some relevance truth-value assignments VR. VM (VR) defined in this paper is not the same as the assignment contained in a modal (relevance) model. Second, in a fibred model each relevance world is associated (or fibred) with a modal model and each modal world with a relevance model.2 To be the grafted frame on which a grafted model is based, it is enough to have some modal worlds attached by some relevance frames. Moreover, no relevance world is associated with a modal frame in the grafted frame. Third, fibred models are intended to provide an appropriate semantics to combined logics. Grafted frames and grafted models are inspired to characterize S1, which, containing only one modality □, is not a combined logic. It is shown in this paper that S1 is sort of a meta-logic of the intersection of S0.4 and F, where S0.4, a new system proposed in this paper, is in turn a meta-logic of the relevance logic.

Fibred semantics and the weaving of logics. Part 1: Modal and intuitionistic logics

This is Part 1 of a paper on fibred semantics and combination of logics. It aims to present a methodology for combining arbitrary logical systems Li, i ∈ I, to form a new system LI. The methodology ‘fibres’ the semantics i of Li into a semantics for LI, and ‘weaves’ the proof theory (axiomatics) of Li into a proof system of LI. There are various ways of doing this, we distinguish by different names such as ‘fibring’, ‘dovetailing’ etc, yielding different systems, denoted by etc. Once the logics are ‘weaved’, further ‘interaction’ axioms can be geometrically motivated and added, and then systematically studied. The methodology is general and is applied to modal and intuitionistic logics as well as to general algebraic logics. We obtain general results on bulk, in the sense that we develop standard combining techniques and refinements which can be applied to any family of initial logics to obtain further combined logics.The main results of this paper is a construction for combining arbitrary, (possibly not normal) modal or intermediate logics, each complete for a class of (not necessarily frame) Kripke models. We show transfer of recursive axiomatisability, decidability and finite model property.Some results on combining logics (normal modal extensions of K) have recently been introduced by Kracht and Wolter, Goranko and Passy and by Fine and Schurz as well as a multitude of special combined systems existing in the literature of the past 20–30 years. We hope our methodology will help organise the field systematically.