It is widely accepted that classical logic is trivialized in the presence of a transparent truth-predicate. In this paper, we will explain why this point of view must be given up. The hierarchy of metainferential logics defined in Barrio et al. and Pailos recovers classical logic, either in the sense that every classical inferential validity is valid at some point in the hierarchy ), or because a logic of a transfinite level defined in terms of the hierarchy shares its (...) validities with classical logic. Each of these logics is consistent with transparent truth—as is shown in Pailos —, and this suggests that, contrary to standard opinions, transparent truth is after all consistent with classical logic. However, Scambler presents a major challenge to this approach. He argues that this hierarchy cannot be identified with classical logic in any way, because it recovers no classical antivalidities. We embrace Scambler’s challenge and develop a new logic based on these hierarchies. This logic recovers both every classical validity and every classical antivalidity. Moreover, we will follow the same strategy and show that contingencies need also be taken into account, and that none of the logics so far presented is enough to capture classical contingencies. Then, we will develop a multi-standard approach to elaborate a new logic that captures not only every classical validity, but also every classical antivalidity and contingency. As a€truth-predicate can be added to this logic, this result can be interpreted as showing that, despite the claims that are extremely widely accepted, classical logic does not trivialize in the context of transparent truth. (shrink)

In a rather general setting, we prove a number of basic theorems concerning computational complexity of derivability in adaptive logics. For that setting, the so-called standard format of adaptive logics is suitably adopted, and the corresponding completeness results are established in a very uniform way.

Unlike the standard Quantum Computational Logic, where the carrier of information is conventionally assumed to be only the last qubit over a sequence of many qubits, here we propose an extended version of the QCL where the number and the position of the target qubits are arbitrary.

In this paper we analyze how the semantics of the Alternating-time Temporal Logic ATL$^*$ deals with agents' commitments to strategies in the process of formula evaluation. In (\acro{atl}$^*$), one can express statements about the strategic ability of an agent (or a coalition of agents) to achieve a goal $\phi$ such as: ``agent $i$ can choose a strategy such that, if $i$ follows this strategy then, no matter what other agents do, $\phi$ will always be true''. However, strategies in \acro{atl} are (...) \emph{revocable} in the sense that in the evaluation of the goal $\phi$ the agent $i$ is no longer restricted by the strategy she has chosen in order to reach the state where the goal is evaluated. -/- Here we discuss some alternatives leading to amendments of that semantics. In particular, we consider variants of \acro{atl}$^*$ where strategies, on the contrary, are \emph{irrevocable}. Unlike in the standard semantics of \acro{atl}, memory plays an essential role in the semantics based on irrevocable strategies. -/- Further, we propose and discuss various syntactic and semantics mechanisms for handling commitments to strategies and release from such commitments in the semantics of ATL$^*$, leading to more expressive and semantically refined versions of that logic. (shrink)

_T__S_ is a logic that has no valid inferences. But, could there be a logic without valid metainferences? We will introduce _T__S_ _ω_, a logic without metainferential validities. Notwithstanding, _T__S_ _ω_ is not as empty—i.e., uninformative—as it gets, because it has many antivalidities. We will later introduce the two-standard logic [_T__S_ _ω_, _S__T_ _ω_ ], a logic without validities and antivalidities. Nevertheless, [_T__S_ _ω_, _S__T_ _ω_ ] is still informative, because it has many contingencies. The three-standard logic [ (...) \(\mathbf {TS}_{\omega }, \mathbf {ST}_{\omega }, [{\overline {\emptyset }}{\emptyset }, {\emptyset } {\overline {\emptyset }}]\) ] that we will further introduce, has no validities, no antivalidities and also no contingencies whatsoever. We will also present two more validity-empty logics. The first one has no supersatisfiabilities, unsatisfabilities and antivalidities ∗. The second one has no invalidities nor non-valid-nor-invalid (meta)inferences. All these considerations justify thinking of logics as, at least, three-standard entities, corresponding, respectively, to what someone who takes that logic as correct, accepts, rejects and suspends judgement about, just because those things are, respectively, validities, antivalidities and contingencies of that logic. Finally, we will present some consequences of this setting for the monism/pluralism/nihilism debate, and show how nihilism and monism, on one hand, and nihilism and pluralism, on the other hand, may reconcile—at least according to how Gillian Russell understands nihilism, and provide some general reasons for adopting a multi-standard approach to logics. (shrink)

. I give a systematic presentation of a fairly large family of multiple-conclusion modal logics that are paraconsistent and/or paracomplete. After providing motivation for studying such systems, I present semantics and tableau-style proof theories for them. The proof theories are shown to be sound and complete with respect to the semantics. I then show how the “standard” systems of classical, single-conclusion modal logics fit into the framework constructed.

This paper describes axiomatic theories SA and SAR, which are versions of second order arithmetic with countably many sorts for sets of natural numbers. The theories are intended to be applied in reverse mathematics because their multi-sorted language allows to express some mathematical statements in more natural form than in the standard second order arithmetic. We study metamathematical properties of the theories SA, SAR and their fragments. We show that SA is mutually interpretable with the theory of arithmetical (...) truth PATr obtained from the Peano arithmetic by adding infinitely many truth predicates. Corresponding fragments of SA and PATr are also mutually interpretable. We compare the proof-theoretical strengths of the fragments; in particular, we show that each fragment SAs with sorts <=s is weaker than next fragment SAs+1. (shrink)

In this paper1, we develop an epistemic logic to specify and reason about the information flow on the underlying communication channels. By combining ideas from Dynamic Epistemic Logic (DEL) and Interpreted Systems (IS), our semantics offers a natural and neat way of modelling multi-agent communication scenarios with different assumptions about the observational power of agents. We relate our logic to the standard DEL and IS..

Current dynamic epistemic logics for analyzing effects of informational events often become cumbersome and opaque when common knowledge is added for groups of agents. Still, postconditions involving common knowledge are essential to successful multi-agent communication. We propose new systems that extend the epistemic base language with a new notion of ‘relativized common knowledge’, in such a way that the resulting full dynamic logic of information flow allows for a compositional analysis of all epistemic postconditions via perspicuous ‘reduction axioms’. (...) We also show how such systems can deal with factual alteration, rather than just information change, making them cover a much wider range of realistic events. After a warm-up stage of analyzing logics for public announcements, our main technical results are expressivity and completeness theorems for a much richer logic that we call LCC. This is a dynamic epistemic logic whose static base is propositional dynamic logic (PDL), interpreted epistemically. This system is capable of expressing all model-shifting operations with finite action models, while providing a compositional analysis for a wide range of informational events. This makes LCC a serious candidate for a standard in dynamic epistemic logic, as we illustrate by analyzing some complex communication scenarios, including sending successive emails with both ‘cc’ and ‘bcc’ lines, and other private announcements to subgroups. Our proofs involve standard modal techniques, combined with a new application of Kleene’s theorem on finite automata, as well as new Ehrenfeucht games of model comparison. (shrink)

There are several formal systems for persuasive dialogue. Dialogue systems are multi-Agent systems, and this contrasts with the general lack of any agency in standard logics other than in the case of epistemic and deontic logics. Dialogue systems have been called logics. A logic usually has a semantics and a proof system, and questions of soundness and completeness arise. Any dialogue conducted according to the rules of a dialogue logic is a complex process. Dynamic Logic (...) is a logic of processes, with a possible world semantics. This paper is a preliminary exploration of transforming the dialogues of dialogue logic into complex Dynamic Logic processes with emphasis on semantic models. The transformation gives rise to many questions, three of which are discussed. Dialogue logic includes commitment sets which behave in a way similar to belief sets and undergo changes during any dialogue. There are issues as to the relationship between belief and commitment, and whether the logics of belief change apply to commitment stores. There is also the issue of the logical nature of the evaluation of dialogues as legal or illegal. (shrink)

Modal logics, originally conceived in philosophy, have recently found many applications in computer science, artificial intelligence, the foundations of mathematics, linguistics and other disciplines. Celebrated for their good computational behaviour, modal logics are used as effective formalisms for talking about time, space, knowledge, beliefs, actions, obligations, provability, etc. However, the nice computational properties can drastically change if we combine some of these formalisms into a many-dimensional system, say, to reason about knowledge bases developing in time or moving objects. (...) To study the computational behaviour of many-dimensional modal logics is the main aim of this book. On the one hand, it is concerned with providing a solid mathematical foundation for this discipline, while on the other hand, it shows that many seemingly different applied many-dimensional systems (e.g., multi-agent systems, description logics with epistemic, temporal and dynamic operators, spatio-temporal logics, etc.) fit in perfectly with this theoretical framework, and so their computational behaviour can be analyzed using the developed machinery. We start with concrete examples of applied one- and many-dimensional modal logics such as temporal, epistemic, dynamic, description, spatial logics, and various combinations of these. Then we develop a mathematical theory for handling a spectrum of 'abstract' combinations of modal logics - fusions and products of modal logics, fragments of first-order modal and temporal logics - focusing on three major problems: decidability, axiomatizability, and computational complexity. Besides the standard methods of modal logic, the technical toolkit includes the method of quasimodels, mosaics, tilings, reductions to monadic second-order logic, algebraic logic techniques. Finally, we apply the developed machinery and obtained results to three case studies from the field of knowledge representation and reasoning: temporal epistemic logics for reasoning about multi-agent systems, modalized description logics for dynamic ontologies, and spatio-temporal logics. The genre of the book can be defined as a research monograph. It brings the reader to the front line of current research in the field by showing both recent achievements and directions of future investigations (in particular, multiple open problems). On the other hand, well-known results from modal and first-order logic are formulated without proofs and supplied with references to accessible sources. The intended audience of this book is logicians as well as those researchers who use logic in computer science and artificial intelligence. More specific application areas are, e.g., knowledge representation and reasoning, in particular, terminological, temporal and spatial reasoning, or reasoning about agents. And we also believe that researchers from certain other disciplines, say, temporal and spatial databases or geographical information systems, will benefit from this book as well. Key Features: Integrated approach to modern modal and temporal logics and their applications in artificial intelligence and computer science Written by internationally leading researchers in the field of pure and applied logic Combines mathematical theory of modal logic and applications in artificial intelligence and computer science Numerous open problems for further research Well illustrated with pictures and tables. (shrink)

We consider the problem of axiomatizing various natural "successor" logics for 2-dimensional integral spacetime. We provide axiomatizations in monomodal and multimodal languages, and prove completeness theorems. We also establish that the irreflexive successor logic in the "standard" modal language (i.e. the language containing □ and ◊) is not finitely axiomatizable.

Pit is a multi-player card game that simulates the commodities trading market, and where actions consist of bidding and of swapping cards. We present a formal description of the knowledge and change of knowledge in that game. The description is in a standard language for dynamic epistemics expanded with assignment. Assignment is necessary to describe that cards change hands. The formal description is a prerequisite to model Pit in game theory. The main contribution of this paper should be (...) seen as the rigorous formalization of all knowledge in Pit. (shrink)

Pit is a multi-player card game that simulates the commodities trading market, and where actions consist of bidding and of swapping cards. We present a formal description of the knowledge and change of knowledge in that game. The description is in a standard language for dynamic epistemics expanded with assignment. Assignment is necessary to describe that cards change hands. The formal description is a prerequisite to model Pit in game theory. The main contribution of this paper should be (...) seen as the rigorous formalization of all knowledge in Pit. (shrink)

We present an abstract model of rationality that focuses on structural properties of attitudes. Rationality requires coherence between your attitudes, such as your beliefs, values, and intentions. We define three 'logical' conditions on attitudes: consistency, completeness, and closedness. They parallel the familiar logical conditions on beliefs, but contrast with standard rationality conditions like preference transitivity. We establish a formal correspondence between our logical conditions and standard rationality conditions. Addressing John Broome's programme 'rationality through reasoning', we formally characterize how (...) you can (not) become more logical by reasoning. Our analysis connects rationality with logic, and enables logical talk about multi-attitude psychology. (shrink)

We develop a conceptually clear, intuitive, and feasible decision procedure for testing satisfiability in the full multi\-agent epistemic logic \CMAELCD\ with operators for common and distributed knowledge for all coalitions of agents mentioned in the language. To that end, we introduce Hintikka structures for \CMAELCD\ and prove that satisfiability in such structures is equivalent to satisfiability in standard models. Using that result, we design an incremental tableau-building procedure that eventually constructs a satisfying Hintikka structure for every satisfiable input (...) set of formulae of \CMAELCD\ and closes for every unsatisfiable input set of formulae. (shrink)

This paper is dedicated to extending and adapting to modal logic the approach of fractional semantics to classical logic. This is a multi-valued semantics governed by pure proof-theoretic considerations, whose truth-values are the rational numbers in the closed interval $[0,1]$. Focusing on the modal logic K, the proposed methodology relies on three key components: bilateral sequent calculus, invertibility of the logical rules, and stability (proof-invariance). We show that our semantic analysis of K affords an informational refinement with respect to (...) the standard Kripkean semantics (a new proof of Dugundji’s theorem is a case in point) and it raises the prospect of a proof-theoretic semantics for modal logic. (shrink)

A many-valued (aka multiple- or multi-valued) semantics, in the strict sense, is one which employs more than two truth values; in the loose sense it is one which countenances more than two truth statuses. So if, for example, we say that there are only two truth values—True and False—but allow that as well as possessing the value True and possessing the value False, propositions may also have a third truth status—possessing neither truth value—then we have a many-valued semantics in (...) the loose but not the strict sense. A many-valued logic is one which arises from a many-valued semantics and does not also arise from any two-valued semantics [Malinowski, 1993, 30]. By a ‘logic’ here we mean either a set of tautologies, or a consequence relation. We can best explain these ideas by considering the case of classical propositional logic. The language contains the usual basic symbols (propositional constants p, q, r, . . .; connectives ¬, ∧, ∨, →, ↔; and parentheses) and well-formed formulas are defined in the standard way. With the language thus specified—as a set of well-formed formulas—its semantics is then given in three parts. (i) A model of a logical language consists in a free assignment of semantic values to basic items of the non-logical vocabulary. Here the basic items of the non-logical vocabulary are the propositional constants. The appropriate kind of semantic value for a proposition is a truth value, and so a model of the language consists in a free assignment of truth values to basic propositions. Two truth values are countenanced: 1 (representing truth) and 0 (representing falsity). (ii) Rules are presented which determine a truth value for every proposition of the language, given a model. The most common way of presenting these rules is via truth tables (Figure 1). Another way of stating such rules—which will be useful below—is first to introduce functions on the truth values themselves: a unary function ¬ and four binary functions ∧, ∨, → and ↔ (Figure 2).. (shrink)

Any set of truth-functional connectives has sequent calculus rules that can be generated systematically from the truth tables of the connectives. Such a sequent calculus gives rise to a multi-conclusion natural deduction system and to a version of Parigot’s free deduction. The elimination rules are “general,” but can be systematically simplified. Cut-elimination and normalization hold. Restriction to a single formula in the succedent yields intuitionistic versions of these systems. The rules also yield generalized lambda calculi providing proof terms for (...) natural deduction proofs as in the Curry-Howard isomorphism. Addition of an indirect proof rule yields classical single-conclusion versions of these systems. Gentzen’s standard systems arise as special cases. (shrink)

As noted in 1962 by Timothy Smiley, if Aristotle’s logic is faithfully translated into modern symbolic logic, the fit is exact. If categorical sentences are translated into many-sorted logic MSL according to Smiley’s method or the two other methods presented here, an argument with arbitrarily many premises is valid according to Aristotle’s system if and only if its translation is valid according to modern standard many-sorted logic. As William Parry observed in 1973, this result can be proved using my (...) 1972 proof of the completeness of Aristotle’s syllogistic. (shrink)

This paper describes an axiomatic theory BT, which is a suitable formal theory for developing constructive mathematics, due to its expressive language with countable number of set types and its constructive properties such as the existence and disjunction properties, and consistency with the formal Church thesis. BT has a predicative comprehension axiom and usual combinatorial operations. BT has intuitionistic logic and is consistent with classical logic. BT is mutually interpretable with a so called theory of arithmetical truth PATr and with (...) a second-order arithmetic SA that contains infinitely many sorts of sets of natural numbers. We compare BT with some standard second-order arithmetics and investigate the proof-theoretical strengths of fragments of BT, PATr and SA. (shrink)

The paper explains how Gettier’s conclusion can be reached on general theoretical grounds within the framework of epistemic logic, without reliance on thought experiments. It extends the argument to permissive conceptions of justification that invalidate principles of multi-premise closure and require neighbourhood semantics rather than semantics of a more standard type. The paper concludes by recommending a robust methodology that aims at convergence in results between thought experimentation and more formal methods. It also warns against conjunctive definitions as (...) sharing several of the drawbacks of disjunctive definitions. (shrink)

We introduce quantificational modal operators as dynamic modalities with (extensions of) Henkin quantifiers as indices. The adoption of matrices of indices (with action identifiers, variables and/or quantified variables as entries) gives an expressive formalism which is here motivated with examples from the area of multi-agent systems. We study the formal properties of the resulting logic which, formally speaking, does not satisfy the normality condition. However, the logic admits a semantics in terms of (an extension of) Kripke structures. As a (...) consequence, standard techniques for normal modal logic become available. We apply these to prove completeness and decidability, and to extend some standard frame results to this logic. (shrink)

We present a propositional and a first-order logic for reasoning about higher-order upper and lower probabilities. We provide sound and complete axiomatizations for the logics and we prove decidability in the propositional case. Furthermore, we show that the introduced logics generalize some existing probability logics.

We present a propositional and a first-order logic for reasoning about higher-order upper and lower probabilities. We provide sound and complete axiomatizations for the logics and we prove decidability in the propositional case. Furthermore, we show that the introduced logics generalize some existing probability logics.

In a multiagent and multi-cultural world, the fine-grained analysis of agents’ dynamic behaviour, i.e. of their activities, is essential. Dynamic activities are actions that are characterized by an agent who executes the action and by other participants of the action. Wh-questions on the participants of the actions pose a difficult particular challenge because the variability of the types of possible answers to such questions is huge. To deal with the problem, we propose the analysis and classification of Wh-questions apt (...) for agents’ communication in a multiagent system (MAS). Our proposal of such a system consists of agents who communicate with their fellow agents by messaging so that each autonomous agent, though resource-bounded, can make less or more rational decisions to meet its own and collective goals. In addition, by communicating with other fellow agents and their environment, agents can learn new concepts and enrich their ontology so that their behaviour is dynamic. We aim to make a general proposal of the system so that the ‘envelope’ of agents’ messages can be formalized in any MAS standard, be it The Foundation for Intelligent Physical Agents - Agent Communication Language (FIPA-ACL) or Knowledge Query and Manipulation Language (KQML). Yet, the content of messages is encoded in a formalized natural language. To this end, we apply Transparent Intensional Logic (TIL) with its procedural semantics which is particularly apt for a fine-grained analysis in which all the semantically salient features of natural language can be plausibly formalized. In this paper, we concentrate on analysing the content of query messages, particularly the content of those that encode Wh-questions and the answers to them. We also summarize TIL deduction system that makes it possible to answer such questions in an intelligent way. Linguists distinguish several subtypes of Wh-questions. Though the linguistic classification is helpful, it is not always suitable for agents’ communication and reasoning. We need the classification based on a logical analysis of Wh-questions so that the agents can infer possible answers to such questions rather than only looking for them by keywords. This paper aims to apply an appropriate classification of the logical types of Wh-questions and the analysis of such questions; we concentrate in particular on questions concerning the participants of activities. The application of these results to the analysis of processes and events based on verb valency frames is another novelty of the paper. (shrink)

We propose a novel interpretation of natural-language questions using a modal predicate logic of knowledge. Our approach brings standard model-theoretic and proof-theoretic techniques from modal logic to bear on questions. Using the former, we show that our interpretation preserves Groenendijk and Stokhof's answerhood relation, yet allows an extensional interpretation. Using the latter, we get a sound and complete proof procedure for the logic for free. Our approach is more expressive; for example, it easily treats complex questions with operators that (...) scope over questions. We suggest a semantic criterion that restricts what natural-language questions can express. We integrate and generalize much previous work on the semantics of questions, including Beck and Sharvit's families of subquestions, non-exhaustive questions, and multi-party conversations. (shrink)

A sequent root-first proof-search procedure for intuitionistic propositional logic is presented. The procedure is obtained from modified intuitionistic multi-succedent and classical sequent calculi, making use of Glivenko’s Theorem. We prove that a sequent is derivable in a standard intuitionistic multi-succedent calculus if and only if the corresponding prefixed-sequent is derivable in the procedure.

Although there are a few books available that give brief surveys of a variety of nonstandard logics, there is a growing need for a critical presentation providing both a greater depth and breadth of insight into these logics. This book assembles a wider and deeper view of the many potentially applicable logics. Three appendixes provide short tutorials on classical logic and modal logics, and give a brief introduction to the existing literature on the logical aspects of (...) probability theory. These tutorials and the bibliography included in the appendixes provide useful reference material for the reader. (shrink)

Logic; Predicate calculus; Axiomatic systems; Knowledge representation and reasoning; Logic and revisable reasoning; Formal grammars and logic programming; Prolog and logic programming.

Conceived by Johan van Benthem and Yde Venema, arrow logic started as an attempt to give a general account of the logic of transitions. The generality of the approach provided a wide application area ranging from philosophy to computer science. The book gives a comprehensive survey of logical research within and around arrow logic. Since the natural operations on transitions include composition, inverse and identity, their logic, arrow logic can be studied from two different perspectives, and by two (complementary) methodologies: (...) modal logic and the algebra of relations. Some of the results in this volume can be interpreted as price tags. They show what the prices of desirable properties, such as decidability, (finite) axiomatisability, Craig interpolation property, Beth definability etc. are in terms of semantic properties of the logic. The research program of arrow logic has considerably broadened in the last couple of years and recently also covers the enterprise to explore the border between decidable and undecidable versions of other applied logics. The content of this volume reflects this broadening. The editors included a number of papers which are in the spirit of this generalised research program. (shrink)

We study structural rules in the context of multi-valued logics with finitely-many truth-values. We first extend Gentzen’s traditional structural rules to a multi-valued logic context; in addition, we propos some novel structural rules, fitting only multi-valued logics. Then, we propose a novel definition, namely, structural rules completeness of a collection of structural rules, requiring derivability of the restriction of consequence to atomic formulas by structural rules only. The restriction to atomic formulas relieves the need to (...) concern logical rules in the derivation. (shrink)

The closed world assumption plays a fundamental role in the theory of deductive databases. On the other hand, multi-valued logics occupy a vast field in non-classical logics. Some questions are better explained and expressed in terms of such logics. To enhance the expressive power and the declarative ability of a deductive database, we extend various CWA formalizations, including the naive CWA, the generalized CWA and the careful CWA, to multi-valued logics. The basic idea is (...) to embed logic formulae into some polynomial ring. The extensions can be applied in a uniform manner to any finitely multi-valued logics. Therefore they have also computational interest. (shrink)

We formally introduce a novel, yet ubiquitous, category of norms: norms of instrumentality. Norms of this category describe which actions are obligatory, or prohibited, as instruments for certain purposes. We propose the Logic of Agency and Norms (LAN) that enables reasoning about actions, instrumentality, and normative principles in a multi-agent setting. Leveraging LAN , we formalize norms of instrumentality and compare them to two prevalent norm categories: norms to be and norms to do. Last, we pose principles relating the (...) three categories and evaluate their validity vis-à-vis notions of deliberative acting. On a technical note, the logic will be shown decidable via the finite model property. (shrink)

The aim of this contribution is to trace the transformation of the meanings of certain terms as the order of the logics in which they appear is raised. By “order of the logic” we simply refer to the number of truth-values characterizing the logic, so that if the number of truth values shows symptoms of traveling to infinity we may speak of the goal as a logic of infinite order.

In some presentations of classical and intuitionistic logics, the objectlanguage is assumed to contain (two) truth-value constants: ⊤ (verum) and ⊥ (falsum), that are, respectively, true and false under every bivalent valuation. We are interested to define and study analogical constants ‡, 1 ≤ i ≤ n, that in an arbitrary multi-valued logic over truth-values V = {v1,..., vn} have the truth-value vi under every (multi-valued) valuation. As is well known, the absence or presence of such constants (...) has a significant deductive impact on the logics studied. We define such constants proof-theoretically via their associated I/E-rules in a natural-deduction proof system. In particular, we propose a generalization of the notions of contradiction and explosiveness of a logic to the context of multi-valued logics. (shrink)

Compositional verification aims at managing the complexity of theverification process by exploiting compositionality of the systemarchitecture. In this paper we explore the use of a temporal epistemiclogic to formalize the process of verification of compositionalmulti-agent systems. The specification of a system, its properties andtheir proofs are of a compositional nature, and are formalized within acompositional temporal logic: Temporal Multi-Epistemic Logic. It isshown that compositional proofs are valid under certain conditions.Moreover, the possibility of incorporating default persistence ofinformation in a system, (...) is explored. A completion operation on aspecific type of temporal theories, temporal completion, is introducedto be able to use classical proof techniques in verification withrespect to non-classical semantics covering default persistence. (shrink)

The rearrangement inequality states that the sum of products of permutations of 2 sequences of real numbers are maximized when the terms are similarly ordered and minimized when the terms are ordered in opposite order. We show that similar inequalities exist in algebras of multi-valued logic when the multiplication and addition operations are replaced with various T-norms and T-conorms respectively. For instance, we show that the rearrangement inequality holds when the T-norms and T-conorms are derived from Archimedean copulas.

ABSTRACT The two main directions pursued in the present paper are the following. The first direction was started by Pigozzi in 1969. In [Mak 91] and [Mak 79] Maksimova proved that a normal modal logic has the Craig interpolation property iff the corresponding class of algebras has the superamalgamation property. In this paper we extend Maksimova's theorem to normal multi-modal logics with arbitrarily many, not necessarily unary modalities, and to not necessarily normal multi-modal logics with modalities (...) of ranks smaller than 2. To extend the characterization beyond multi-modal logics, we look at arbitrary algebraizable logics. We will introduce an algebraic property equivalent with the Craig interpolation property in algebraizable logics, and prove that the superamalgamation property implies the Craig interpolation property. The problem of extending the characterization result to non-normal non-unary modal logics also will be discussed. In the second direction pursued herein: for non-normal modal logic with one unary modality Lemmon [Lem 66] gave a possible worlds semantics. Here we give a more general possible worlds semantics for not necessarily normal multi-modal logics with arbitrarily many not necessarily unary modalities. Strongly related to the above is the theorem, proved, e.g., in Jóns son-Tarski [JT 52] and Henkin-Monk-Tarski [HMT 71], that every normal Boolean algebra with operators can be represented as a subalgebra of the complex algebra of some relational structure. We extend this result to not necessarily normal BAO's as follows. We define partial relational structures and show that every not necessarily normal BAO is embeddable into the complex algebra of a partial relational structure. This gives a possible worlds semantics for not necessarily normal multi-modal logics. (shrink)