Symmetric Logic of Proofs

On the complexity of the reflected logic of proofs

What do epistemic logic and cognitive science have to do with each other?

Epistemic Logic and Information Update

Justification logics are epistemic logics that explicitly include justifications for the agents' knowledge. We develop a multi-agent justification logic with evidence terms for individual agents as well as for common knowledge. We define a Kripke-style semantics that is similar to Fitting's semantics for the Logic of Proofs LP. We show the soundness, completeness, and finite model property of our multi-agent justification logic with respect to this Kripke-style semantics. We demonstrate that our logic is a conservative extension of Yavorskaya's minimal bimodal explicit evidence logic, which is a two-agent version of LP. We discuss the relationship of our logic to the multi-agent modal logic S4 with common knowledge. Finally, we give a brief analysis of the coordinated attack problem in the newly developed language of our logic.

This handbook provides a systematic survey of the central areas of philosophical logic. It is divided into four volumes, each devoted to a major sub-field within the disciplines. The articles should be of value to general philosophers, linguists, logicians, mathematicians and computer scientists. Volume 1, Elements of Classical Logic, deals with the background to what has come to be considered the standard formulation of predicate logic - both as far as its semantics and proof theory are concerned. The central chapter on predicate logic is followed by chapters outlining various alternative, but essentially equivalent ways of constructing the semantics for first-order logic as well as its proof theory. In addition, this volume contains a discussion of higher-order extensions of first-order logic and a compendium of the algorithmic and decision-theoretic prerequisites in the study of logical systems. Volume 2, Extensions of Classical Logic, surveys the most significant extensions of predicate logic and their applications to various philosophical fields of inquiry. The 12 chapters in this volume together provide a succinct introduction to a variety of intensional frameworks, a discussion of the most well-known logical systems, as well as an overview of major applications and of the open problems in the respective fields. Volume 3, Alternatives to Classical Logic, consists of a series of surveys of some of the alternatives to the basic assumptions of classical logic. These include many-valued logic, partial logic, free logic, relevance and entailment logics, dialogue logic, quantum logic and intuitionism. Volume 4, Topics in the Philosophy of Language, presents a panorama of the applications of logical tools and methods in the formal analysis of natural language. Since a number of developments in philosophical logic were originally stimulated by concern arising in the semantic analysis of natural language discourse, the chapters in this volume provide some criteria of evaluation of the applications of work in philosophical logic. In revealing both the adequacies and inadequacies of logical investigations in the semantic structures of natural discourse, these chapters also point the way to future developments in philosophical logic in general and thus close again the circle of inquiry relating logic and language.

In this paper, we exploit epistemic logic (or the modal logic of knowledge) for multiagent systems to discuss the compositionality of several privacy-related information-hiding/disclosure properties. The properties considered here are anonymity, privacy, onymity, and identity. Our initial observation reveals that anonymity/privacy properties are not necessarily sequentially compositional. This means that even though a system comprising several sequential phases satisfies a certain unlinkability property in each phase, the entire system does not always enjoy a desired unlinkability property. We show that the compositionality can be guaranteed provided that the phases of the system satisfy what we call independence assumptions. More specifically, we develop a series of theoretical case studies of what assumptions are sufficient to guarantee the sequential compositionality of various degrees of anonymity, privacy, onymity, and/or identity properties. Similar results for parallel composition are also discussed. Further, we use the probabilistic extension of epistemic logic to consider the compositionality of probabilistic anonymity/privacy. We show that the compositionality can also be guaranteed in the probabilistic setting, provided that the phases of the system satisfy a probabilistic independence assumption.

This paper presents a logic of questions developed as an extension of (S5) epistemic logic. We discuss many features that are important for erotetic logic (formalization and semantics of questions, answerhood conditions, and inferential structures with questions). The aim is to introduce an erotetic system which corresponds well with epistemic terms and can form an appropriate background for dynamic approaches in epistemic logic.

We investigate two notions of independence—(usual) independence and complete independence—applied to the Peano axioms for the sequence of natural numbers. We review the results that, although they are independent, the Peano axioms are not completely independent. The standard proof that the Peano axioms are not completely independent is algebraic, in the sense that it makes essential reference to the relationship between several mathematical structures that satisfy, or do not satisfy, these axioms. We then present an alternative logical proof, which makes no essential references to the relationship between mathematical structures. There is a completely independent set of axioms for the sequence of natural numbers, but it is based on primitives different from those originally adopted by Peano. Therefore, we present a new completely independent set of axioms based on the same set of primitives as the one originally adopted by Peano.

Journal Article Arithmetical Completeness of the Intuitionistic Logic of Proofs Get access Evgenij Dashkov Evgenij Dashkov Department of Mathematical Logic and Theory of Algorithms, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow 119992, Russia.E-mail: edashkov@gmail.com Search for other works by this author on: Oxford Academic Google Scholar Journal of Logic and Computation, Volume 21, Issue 4, August 2011, Pages 665–682, https://doi.org/10.1093/logcom/exp041 Published: 17 August 2009 Article history Received: 03 February 2009 Published: 17 August 2009

We show that dynamic epistemic logic (DEL) is a substructural logic and that it is an extension of the update logic introduced in the companion article [12]. We identify axioms and inference rules that completely characterize the DEL product update, and we provide a sequent calculus for DEL. Finally, we show that DEL with a finite number of atomic events is as expressive as epistemic logic. In parallel, we provide a sequent calculus for update logic which turns out to be a generalization of the non-associative Lambek calculus.

With the amount of information available to us today, making sense of potentially conflicting information is a problem that we are bound to run into. We aim to study this problem from the perspective of epistemic logic. So far, we have studied logics for reasoning about information distributed across groups of agents in the epistemic logic framework, focusing on the notion of distributed belief. We have introduced the notions of cautious and bold distributed belief. These are intended as alternatives to the standard distributed belief, behaving better when agents have conflicting beliefs. With standard distributed belief, such situations lead to explosion: everything becomes distributedly believed. The idea behind the new notions is that we look at maximal consistent subgroups, allowing us to meaningfully express that a group collectively possesses information supporting a belief, also in cases where some group members disagree. Going forward, we are interested in continuing to explore the aggregation of potentially conflicting information in the setting of epistemic logic, e.g. by looking at distributed belief in richer extensions of epistemic logic and in relation to other areas that deal with similar problems.

Epistemic logic has, in the past few decades, grown beyond its origins in philosophy, to be embraced in several other disciplines, including economics, linguistics and computer science. Within computer science, its application can be found in several subdisciplines: artificial intelligence, distributed computing and computer security.

In its many guises and variations, propositional dynamic logic (PDL) plays an important role in various areas of computer science such as databases, artificial intelligence, and computer linguistics. One relevant and powerful variation is ICPDL, the extension of PDL with intersection and converse. Although ICPDL has several interesting applications, its computational properties have never been investigated. In this paper, we prove that ICPDL is decidable by developing a translation to the monadic second order logic of infinite trees. Our result has applications in information logic, description logic, and epistemic logic. In particular, we solve a long-standing open problem in information logic. Another virtue of our approach is that it provides a decidability proof that is more transparent than existing ones for PDL with intersection (but without converse).

The emergence of Semantic Web (SW) and the related technologies promise to make the Web a meaningful experience. Conversely, success of SW and its applications depends largely on utilization and interoperability of well-formulated ontology bases in an automated heterogeneous environment. This creates a need to investigate utilization of an (materialized) ontology view as an alternative version of an ontology. However, high level modeling, design and querying techniques still proves to be a challenging task for SW paradigm, as, unlike classical database systems, ontology view definitions and querying have to be done at high-level abstraction. In order to address such an issue, in this chapter, we describe an abstract view formalism forSW (SW-view) with conceptual and logical extensions. SW-views provides the needed conceptual and logical semantics to engineer ontology bases using three levels of abstraction, namely (1) conceptual, (2) logical/schema and (3) instance levels. We first provide the view model and it formal properties including a set of conceptual operators, which enable us to do ontology extraction at the conceptual level. Later, we provide a schemata transformation methodology to materialize SW-views under the Ontology Extraction Methodology (OEM) framework.

This lecture is supposed to be a mini–review of the recent results obtained by the Dubna-MainzTubingen Collaboration, see Refs. [1, 2, 3]. The research is aiming to study the semileptonic, rare and radiative decays of the Λb-baryon by using the covariant quark model previously developed by us. The decay Λb → Λ l+l− (l = e, μ, τ) is a rare b − s favor-changing neutral current process that in the Standard Model proceeds through electroweak loop (penguin andW−box) diagrams. This decay can be considered to be a welcome complement to the well–analyzed rare meson decays B → K(∗) l+l−, Bs → φ l+l− etc. to study the short– and long–distance dynamics of rare decays induced by the transition b → s l+l−. However, the study of baryon decays is of more interest because the Λb baryon has spin one half compare with zero spin of the B−meson. Therefore, the matrix element of the baryon decay possesses more rich helicity structure. For the first time, the CDF Collaboration has reported on the measurement of the Λb → Λ+μ+μ− total branching ratio: B(Λb → Λ+μ+μ−) = (1.73± 0.42± 0.55) · 10−6 [4]. Recently, the LHCb Collaboration [5] has measured the differential branching fraction of this decay as a function of the square of the dimuon invariant mass. Integrating the differential branching fraction gives a branching fraction of B(Λb → Λ+μ+μ−) = (0.96±0.16±0.13±0.21)·10−6. Here, the uncertainties are statistical, systematic and due to the normalisation mode, Λb → ΛJ/ψ, respectively. The physics of heavy baryon decays appears to have entered a new era with these experimental results. There have been a number of theoretical papers on the rare Λb → Λ baryon decays involving the one-photon mode Λb → Λγ and the dilepton modes Λb → Λ l+l− (l = e, μ, τ). They use the same set of (penguin) operators or their non–Standard Model extensions to describe the short distance dynamics but differ in their use of theoretical models to calculate the nonperturbative transition matrix element 〈Λ|Oi |Λb〉. We use the covariant constituent quark model (for short: covariant quark model) as dynamical input to calculate the nonperturbative transition matrix elements. In the covariant quark model the current–induced transitions between baryons are calculated from two–loop Feynman diagrams with free quark propagators in which the divergent high energy behavior of the loop integrations is tempered by Gaussian vertex functions. Quark confinement has incorporated in an effective way, first, by introducing the scale integration in the space of α-parameters, and, second, by cutting this scale integration on the upper limit which corresponds to an infrared