A Note on Parameterised Knowledge Operations in Temporal Logic

We consider modeling the conception of knowledge in terms of temporal logic. The study of knowledge logical operations is originated around 1962 by representation of knowledge and belief using modalities. Nowadays, it is very good established area. However, we would like to look to it from a bit another point of view, our paper models knowledge in terms of linear temporal logic with {\em past}. We consider various versions of logical knowledge operations which may be defined in this framework. Technically, semantics, language and temporal knowledge logics based on our approach are constructed. Deciding algorithms are suggested, unification in terms of this approach is commented. This paper does not offer strong new technical outputs, instead we suggest new approach to conception of knowledge (in terms of time).Comment: 10 page

Classical BI: Its Semantics and Proof Theory

We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O'Hearn and Pym's logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including in particular a multiplicative version of classical negation). At the semantic level, CBI-formulas have the normal bunched logic reading as declarative statements about resources, but its resource models necessarily feature more structure than those for other bunched logics; principally, they satisfy the requirement that every resource has a unique dual. At the proof-theoretic level, a very natural formalism for CBI is provided by a display calculus \`a la Belnap, which can be seen as a generalisation of the bunched sequent calculus for BI. In this paper we formulate the aforementioned model theory and proof theory for CBI, and prove some fundamental results about the logic, most notably completeness of the proof theory with respect to the semantics.Comment: 42 pages, 8 figure

The complexity of admissible rules of {\L}ukasiewicz logic

We investigate the computational complexity of admissibility of inference rules in infinite-valued {\L}ukasiewicz propositional logic (\L). It was shown in [13] that admissibility in {\L} is checkable in PSPACE. We establish that this result is optimal, i.e., admissible rules of {\L} are PSPACE-complete. In contrast, derivable rules of {\L} are known to be coNP-complete.Comment: 14 pages, 2 figures; to appear in Journal of Logic and Computatio

Inference Rules in some temporal multi-epistemic propositional logics

Multi-modal logics are among the best tools developed so far to analyse human reasoning and agents’ interactions. Recently multi-modal logics have found several applications in Artificial Intelligence (AI) and Computer Science (CS) in the attempt to formalise reasoning about the behavior of programs. Modal logics deal with sentences that are qualified by modalities. A modality is any word that could be added to a statement p to modify its mode of truth. Temporal logics are obtained by joining tense operators to the classical propositional calculus, giving rise to a language very effective to describe the flow of time. Epistemic logics are suitable to formalize reasoning about agents possessing a certain knowledge. Combinations of temporal and epistemic logics are particularly effective in describing the interaction of agents through the flow of time. Although not yet fully investigated, this approach has found many fruitful applications. These are concerned with the development of systems modelling reasoning about knowledge and space, reasoning under uncertainty, multi-agent reasoning et c. Despite their power, multi modal languages cannot handle a changing environment. But this is exactly what is required in the case of human reasoning, computation and multi-agent environment. For this purpose, inference rules are a core instrument. So far, the research in this field has investigated many modal and superintuitionistic logics. However, for the case of multi-modal logics, not much is known concerning admissible inference rules. In our research we extend the investigation to some multi-modal propositional logics which combine tense and knowledge modalities. As far as we are concerned, these systems have never been investigated before. In particular we start by defining our systems semantically; further we prove such systems to enjoy the effective finite model property and to be decidable with respect to their admissible inference rules. We turn then our attention to the syntactical side and we provide sound and complete axiomatic systems. We conclude our dissertation by introducing the reader to the piece of research we are currently working on. Our original results can be found in [9, 4, 11] (see Appendix A). They have also been presented by the author at some international conferences and schools (see [8, 10, 5, 7, 6] and refer to Appendix B for more details). Our project concerns philosophy, mathematics, AI and CS. Modern applications of logic in CS and AI often require languages able to represent knowledge about dynamic systems. Multi-modal logics serve these applications in a very efficient way, and we would absorb and develop some of these techniques to represent logical consequences in artificial intelligence and computation

A new calculus for intuitionistic Strong L\"ob logic: strong termination and cut-elimination, formalised

We provide a new sequent calculus that enjoys syntactic cut-elimination and strongly terminating backward proof search for the intuitionistic Strong L\"ob logic $\sf{iSL}$, an intuitionistic modal logic with a provability interpretation. A novel measure on sequents is used to prove both the termination of the naive backward proof search strategy, and the admissibility of cut in a syntactic and direct way, leading to a straightforward cut-elimination procedure. All proofs have been formalised in the interactive theorem prover Coq.Comment: 21-page conference paper + 4-page appendix with proof

Multi-agent non-linear temporal logic with embodied agent describing uncertainty

© Springer International Publishing Switzerland 2014 We study multi-agent non-linear temporal Logic TEm , Int Knwith embodied agent. Our approach models interaction of the agents and various aspects for computation of uncertainty in multi-agent environment. We construct algorithms for verification satisfiability and truth statements in the logic TEm , Int Kn. Found computational algorithms are based at refutability of rules in reduced form at special finite frames of effectively bounded size. We show that our chosen framework is rather flexible and it allows to express various approaches to uncertainty and formalizing meaning of the embodied agent

Negation in natural language

Negation is ubiquitous in natural language, and philosophers have developed plenty of different theories of the semantics of negation. Despite this, linguistic theorizing about negation typically assumes that classical logic's semantics for negation---a simple truth-functional toggle---is adequate to negation in natural language, and philosophical discussions of negation typically ignore vital linguistic data. The present document is thus something of an attempt to fill a gap, to show that careful attention to linguistic data actually militates {\\em against} using a classical semantics for negation, and to demonstrate the philosophical payoff that comes from a nonclassical semantics for natural-language negation. I present a compositional semantics for natural language in which these questions can be posed and addressed, and argue that propositional attitudes fit into this semantics best when we use a nonclassical semantics for negation. I go on to explore several options that have been proposed by logicians of various stripes for the semantics of negation, providing a general framework in which the options can be evaluated. Finally, I show how taking non-classical negations seriously opens new doors in the philosophy of vagueness

Abstraktní studium úplnosti pro infinitární logiky

V této dizertační práci se zabýváme studiem vlastností úplnosti infinitárních výrokových logik z pohledu abstraktní algebraické logiky. Cílem práce je pochopit, jak lze základní nástroj v důkazech uplnosti, tzv. Lindenbaumovo lemma, zobecnit za hranici finitárních logik. Za tímto účelem studujeme vlastnosti úzce související s Lindenbaumovým lemmatem (a v důsledku také s vlastnostmi úplnosti). Uvidíme, že na základě těchto vlastností lze vystavět novou hierarchii infinitárních výrokových logik. Také se zabýváme studiem těchto vlastností v případě, kdy naše logika má nějaké (případně hodně obecně definované) spojky implikace, disjunkce a negace. Mimo jiné uvidíme, že přítomnost daných spojek může zajist platnost Lindenbaumova lemmatu. Keywords: abstraktní algebraická logika, infinitární logiky, Lindenbau- movo lemma, disjunkce, implikace, negaceIn this thesis we study completeness properties of infinitary propositional logics from the perspective of abstract algebraic logic. The goal is to under- stand how the basic tool in proofs of completeness, the so called Linden- baum lemma, generalizes beyond finitary logics. To this end, we study few properties closely related to the Lindenbaum lemma (and hence to com- pleteness properties). We will see that these properties give rise to a new hierarchy of infinitary propositional logic. We also study these properties in scenarios when a given logic has some (possibly very generally defined) connectives of implication, disjunction, and negation. Among others, we will see that presence of these connectives can ensure provability of the Lin- denbaum lemma. Keywords: abstract algebraic logic, infinitary logics, Lindenbaum lemma, disjunction, implication, negationKatedra logikyDepartment of LogicFaculty of ArtsFilozofická fakult

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established