8,560 research outputs found

    On Nested Sequents for Constructive Modal Logics

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    We present deductive systems for various modal logics that can be obtained from the constructive variant of the normal modal logic CK by adding combinations of the axioms d, t, b, 4, and 5. This includes the constructive variants of the standard modal logics K4, S4, and S5. We use for our presentation the formalism of nested sequents and give a syntactic proof of cut elimination.Comment: 33 page

    Modal Logics with Hard Diamond-free Fragments

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    We investigate the complexity of modal satisfiability for certain combinations of modal logics. In particular we examine four examples of multimodal logics with dependencies and demonstrate that even if we restrict our inputs to diamond-free formulas (in negation normal form), these logics still have a high complexity. This result illustrates that having D as one or more of the combined logics, as well as the interdependencies among logics can be important sources of complexity even in the absence of diamonds and even when at the same time in our formulas we allow only one propositional variable. We then further investigate and characterize the complexity of the diamond-free, 1-variable fragments of multimodal logics in a general setting.Comment: New version: improvements and corrections according to reviewers' comments. Accepted at LFCS 201

    The Complexity of Reasoning with Boolean Modal Logics: Extended Version

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    Since Modal Logics are an extension of Propositional Logic, they provide Boolean operators for constructing complex formulae. However, most Modal Logics do not admit Boolean operators for constructing complex modal parameters to be used in the box and diamond operators. This asymmetry is not present in Boolean Modal Logics, in which box and diamond quantify over arbitrary Boolean combinations of atomic model parameters.This is an extended version of the article in: Advances in Modal Logic (AiML), Volume

    Moving up and down in the generic multiverse

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    We give a brief account of the modal logic of the generic multiverse, which is a bimodal logic with operators corresponding to the relations "is a forcing extension of" and "is a ground model of". The fragment of the first relation is called the modal logic of forcing and was studied by us in earlier work. The fragment of the second relation is called the modal logic of grounds and will be studied here for the first time. In addition, we discuss which combinations of modal logics are possible for the two fragments.Comment: 10 pages. Extended abstract. Questions and commentary concerning this article can be made at http://jdh.hamkins.org/up-and-down-in-the-generic-multiverse

    From Linear to Branching-Time Temporal Logics: Transfer of Semantics and Definability

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    This paper investigates logical aspects of combining linear orders as semantics for modal and temporal logics, with modalities for possible paths, resulting in a variety of branching time logics over classes of trees. Here we adopt a unified approach to the Priorean, Peircean and Ockhamist semantics for branching time logics, by considering them all as fragments of the latter, obtained as combinations, in various degrees, of languages and semantics for linear time with a modality for possible paths. We then consider a hierarchy of natural classes of trees and bundled trees arising from a given class of linear orders and show that in general they provide different semantics. We also discuss transfer of definability from linear orders to trees and introduce a uniform translation from Priorean to Peircean formulae which transfers definability of properties of linear orders to definability of properties of all paths in tree

    Bi-Logic and Multi-Modal Argumentation: Understanding Emotional Arguments

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    According to Bi-logic theory, there are two logics operating in the mind. One is traditional logic, and the other one is called “symmetrical”, because it does not respect asymmetrical relations. Bi-logic assumes that mental processes involve combinations of both logics in different proportions. From that perspective, Michael Gilbert’s theory of Multi-Modal argumentation is discussed focusing upon emotional arguments. It is claimed that these arguments are bi-logical, that is, they contain a combination of traditional and symmetrical logics

    Designing Normative Theories for Ethical and Legal Reasoning: LogiKEy Framework, Methodology, and Tool Support

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    A framework and methodology---termed LogiKEy---for the design and engineering of ethical reasoners, normative theories and deontic logics is presented. The overall motivation is the development of suitable means for the control and governance of intelligent autonomous systems. LogiKEy's unifying formal framework is based on semantical embeddings of deontic logics, logic combinations and ethico-legal domain theories in expressive classic higher-order logic (HOL). This meta-logical approach enables the provision of powerful tool support in LogiKEy: off-the-shelf theorem provers and model finders for HOL are assisting the LogiKEy designer of ethical intelligent agents to flexibly experiment with underlying logics and their combinations, with ethico-legal domain theories, and with concrete examples---all at the same time. Continuous improvements of these off-the-shelf provers, without further ado, leverage the reasoning performance in LogiKEy. Case studies, in which the LogiKEy framework and methodology has been applied and tested, give evidence that HOL's undecidability often does not hinder efficient experimentation.Comment: 50 pages; 10 figure

    Efficient Constraints on Possible Worlds for Reasoning about Necessity

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    Modal logics offer natural, declarative representations for describing both the modular structure of logical specifications and the attitudes and behaviors of agents. The results of this paper further the goal of building practical, efficient reasoning systems using modal logics. The key problem in modal deduction is reasoning about the world in a model (or scope in a proof) at which an inference rule is applied—a potentially hard problem. This paper investigates the use of partial-order mechanisms to maintain constraints on the application of modal rules in proof search in restricted languages. The main result is a simple, incremental polynomial-time algorithm to correctly order rules in proof trees for combinations of K, K4, T and S4 necessity operators governed by a variety of interactions, assuming an encoding of negation using a scoped constant ┴. This contrasts with previous equational unification methods, which have exponential performance in general because they simply guess among possible intercalations of modal operators. The new, fast algorithm is appropriate for use in a wide variety of applications of modal logic, from planning to logic programming
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