Modal operator

About: Modal operator is a research topic. Over the lifetime, 1151 publications have been published within this topic receiving 22865 citations. The topic is also known as: modal connective.

Man Muss Immer Umkehren

Abstract: The 19th century geometrist Jacobi famously said that one should always try to invert every geometrical theorem. But his advice applies much more widely! Choose any class of relational frames, and you can study its valid modal axioms. But now turn the perspective around, and fix some modal axiom beforehand. You can then find the class of frames where the axiom is guaranteed to hold by 'modal correspondence' analysis ‐ and we all know the famous examples of that. It may look as if this style of analysis is tied to one particular semantics, say relational frames: but it is not. Correspondence analysis also works on neighbourhood models, telling us, e.g., just which modal axioms collapse these to binary relational frames. We will show how this same style of inverse thinking also applies to modern dynamic logics of information change. Basic axioms for knowledge after information update !A tell us what sort of operation must be used for updating a given model M to a new one incorporating A. Likewise, we will show how modal axioms for (conditional) beliefs that hold after revision actions *A actually fix one particular operation of changing the relative plausibility orderings which agents have on the universe of possible worlds. And finally, going back to the traditional heartland of logic, we show how we can read standard predicate-logical axioms as constraints on the sort of abstract 'process models' that lie at the heart of first-order semantics, properly understood. In all these cases, in order for the inversion to work and illuminate a given subject, we need to step back and reconsider our standard modeling. But that, I think, is what Shahid Rahman is all about. 2 Standard modal frame correspondences One of the most attractive features of the semantics of modal logic is the match between modal axioms and corresponding patterns in the accessibility relation between worlds. This can be seen by giving a class of models, say temporal or epistemic, and then axiomatizing its set of modal validities. On top of the minimal modal logic which holds under all circumstances, one gets additional axioms reflecting more specific structure. For general background to modal completeness theory, as well as the rest of this paper, we refer to the Handbook of Modal Logic (P. Blackburn, J. van Benthem & F. Wolter, eds.) which has just come out with Elsevier Science Publishers, Amsterdam, 1997.

Duality Results for (Co)Residuated Lattices

Abstract: We present dualities (discrete duality, duality via truth and Stone duality) for implicative and (co)residuated lattices. In combination with our recent article on a discrete duality for lattices with unary modal operators, the present article contributes in filling in a gap in the development of Orlowska and Rewitzky’s research program of discrete dualities, which seemed to have stumbled on the case of non-distributive lattices with operators. We discuss dualities via truth, which are essential in relating the non-distributive logic of two-sorted frames with their sorted, residuated modal logic, as well as full Stone duality for (co)residuated lattices. Our results have immediate applications to the semantics of related substructural (resource consious) logical calculi.

Algebraic Approach to Tense Operators

Abstract: Propositional logics usually do not incorporate the dimension of time. However, even Aristotle already mentioned that time plays an important role in the evaluation of truth values of propositions. After Aristotle's time, a lot was created by men and, nowadays, logic is not an exceptional area for human reasoning. From the 1940's on, computers were built and the era of the Artificial Intelligence gently started. Nowadays, practically any more advanced product contains some kind of processor which decides situations in a way similar to that of a human being. However, for such technical devices the forecast for truth values of propositions in the future is not only a speculation. Due to the constructions and the technical possibilities, we can often compute these values, and propositions concerning the near future are of great importance. This has motivated many authors to investigate the so-called temporal logic, i.e., the logic where time is considered as a variable of the propositional formula. The logical language of tense logic contains, in addition to the ususal truth-functional operators, four the so-called tense operators. The aim of the monograph is to present an algebraic approach to an axiomatization of tense operators which are the most powerful tools in every tense logic.The authors hope that their monograph will be an incentive for further research.