The logic of justified belief,explicit knowledge,and conclusive evidence

We present a complete, decidable logic for reasoning about a notion of completely trustworthy (“conclusive”) evidence and its relations to justifiable (implicit) belief and knowledge, as well as to their explicit justifications. This logic makes use of a number of evidence-related notions such as availability, admissibility, and “goodness” of a piece of evidence, and is based on an innovative modification of the Fitting semantics for Artemov?s Justification Logic designed to preempt Gettier-type counterexamples. We combine this with ideas from belief revision and awareness logics to provide an account for explicitly justified (defeasible) knowledge based on conclusive evidence that addresses the problem of (logical) omniscience.     

Evidential support logic programming

An expert system is a computer program which can act in a similar way to a human expert in a restricted domain of application from the point of view of solving problems, taking decisions, planning and giving advice. It consists of two parts. One part is a knowledge base consisting of that knowledge used by the expert in his performance. A second part is an inference engine which allows queries to be answered by asking questions of the environment and performing evidential reasoning.This paper is concerned with the knowledge representation and inference mechanism for evidential reasoning. Man's knowledge consists of statements which cannot be guaranteed to be true and is expressed in a language containing imprecise terms. Uncertainties, either of a probabilistic or fuzzy nature, cannot be ignored when modelling human expertise. Not all practical reasoning takes the form of deductive inference. For practical affairs we use inductive, abductive, analogical and plausible reasoning methods and for each of these the concept of the strength of evidence would seem to be important.We describe a support logic programming system which generalises logic programming to the case in which various forms of uncertainty can be included. In this system a conclusion does not logically follow from some axioms but is supported to a certain degree by means of evidence. The negation of the conclusion is also supported to a certain degree and the two supports do not necessarily add up to one.A calculus for such a support logic programming system is described and applications to its use in expert systems and its use in providing recursive definitions of fuzzy concepts are given.     

Visualizations of the Square of Opposition

In logic, diagrams have been used for a very long time. Nevertheless philosophers and logicians are not quite clear about the logical status of diagrammatical representations. Fact is that there is a close relationship between particular visual (resp. graphical) properties of diagrams and logical properties. This is why the representation of the four categorical propositions by different diagram systems allows a deeper insight into the relations of the logical square. In this paper I want to give some examples. I would like to thank the two anonymous reviewers for helpful comments and criticisms.     

Inhabitation of polymorphic and existential types

This paper shows that the inhabitation problem in the lambda calculus with negation, product, polymorphic, and existential types is decidable, where the inhabitation problem asks whether there exists some term that belongs to a given type. In order to do that, this paper proves the decidability of the provability in the logical system defined from the second-order natural deduction by removing implication and disjunction. This is proved by showing the quantifier elimination theorem and reducing the problem to the provability in propositional logic. The magic formulas are used for quantifier elimination such that they replace quantifiers. As a byproduct, this paper also shows the second-order witness theorem which states that a quantifier followed by negation can be replaced by a witness obtained only from the formula. As a corollary of the main results, this paper also shows Glivenko’s theorem, Double Negation Shift, and conservativity for antecedent-empty sequents between the logical system and its classical version.     

Logic of agreement: Foundations,semantic system and proof theory

In this paper a multi-valued propositional logic — logic of agreement — in terms of its model theory and inference system is presented. This formal system is the natural consequence of a new way to approach concepts as commonsense knowledge, uncertainty and approximate reasoning — the point of view of agreement. Particularly, it is discussed a possible extension of the Classical Theory of Sets based on the idea that, instead of trying to conceptualize sets as “fuzzy” or “vague” entities, it is more adequate to define membership as the result of a partial agreement among a group of individual agents. Furthermore, it is shown that the concept of agreement provides a framework for the development of a formal and sound explanation for concepts (e.g. fuzzy sets) which lack formal semantics. According to the definition of agreement, an individual agent agrees or not with the fact that an object possesses a certain property. A clear distinction is then established, between an individual agent — to whom deciding whether an element belongs to a set is just a yes or no matter — and a commonsensical agent — the one who interprets the knowledge shared by a certain group of people. Finally, the logic of agreement is presented and discussed. As it is assumed the existence of several individual agents, the semantic system is based on the perspective that each individual agent defines her/his own conceptualization of reality. So the semantics of the logic of agreement can be seen as being similar to a semantics of possible worlds, one for each individual agent. The proof theory is an extension of a natural deduction system, using supported formulas and incorporating only inference rules. Moreover, the soundness and completeness of the logic of agreement are also presented.     

Fuzzy processing in transitivity development

We review the literature on the development of transitive reasoning, and note three historical stages. Stage 1 was dominated by the Piagetian idea that transitive inference is logical reasoning in which relationships between adjacent terms figure as premises. Stage 2 was dominated by the information-processing view that memory for relationships between adjacent terms is determinative in transitivity performance. Stage 3 has produced data that are inconsistent with both the logic and memery positions, leading to a new theory that is designed to account for such findings, fuzzy-trace theory. The basic assumption of fuzzytrace theory is that reasoners rely on global patterns, or gist. We describe the tenets of fuzzytrace theory, and explore its implications for different theoretical conceptions of logical competence, concluding that young children possess transitivity competence. We discuss the connection between transitivity competence (cognition) and intransitive preferences (metacognition).     

Refined common knowledge logics or logics of common information

 In terms of formal deductive systems and multi-dimensional Kripke frames we study logical operations know, informed, common knowledge and common information. Based on [6] we introduce formal axiomatic systems for common information logics and prove that these systems are sound and complete. Analyzing the common information operation we show that it can be understood as greatest open fixed points for knowledge formulas. Using obtained results we explore monotonicity, omniscience problem, and inward monotonocity, describe their connections and give dividing examples. Also we find algorithms recognizing these properties for some particular cases. Received: 21 October 2000 / Published online: 2 September 2002 Key words or phrases: Multi-agent systems – Non-standard logic – Knowledge representation – Common knowledge – Common information – Fixed points, Kripke models – Modal logic     

On the complexity of the Leibniz hierarchy

We prove that the problem of determining whether a finite logical matrix determines an algebraizable logic is complete for EXPTIME. The same result holds for the classes of order algebraizable, weakly algebraizable, equivalential and protoalgebraic logics. Finally, the same problem for the class of truth-equational logic is shown to be hard for EXPTIME.     

What is a Logic Translation?

We study logic translations from an abstract perspective, without any commitment to the structure of sentences and the nature of logical entailment, which also means that we cover both proof- theoretic and model-theoretic entailment. We show how logic translations induce notions of logical expressiveness, consistency strength and sublogic, leading to an explanation of paradoxes that have been described in the literature. Connectives and quantifiers, although not present in the definition of logic and logic translation, can be recovered by their abstract properties and are preserved and reflected by translations under suitable conditions. In memoriam Joseph Goguen     

A constructive investigation of satisfiability

We present a constructive analysis of the logical notions of satisfiability and consistency for first-order intuitionistic formulae. In particular, we use formal topology theory to provide a positive semantics for satisfiability. Then we propose a “co-inductive” logical calculus, which captures the positive content of consistency.     

On the rules of intermediate logics

If the Visser rules are admissible for an intermediate logic, they form a basis for the admissible rules of the logic. How to characterize the admissible rules of intermediate logics for which not all of the Visser rules are admissible is not known. In this paper we give a brief overview of results on admissible rules in the context of intermediate logics. We apply these results to some well-known intermediate logics. We provide natural examples of logics for which the Visser rule are derivable, admissible but nonderivable, or not admissible. Supported by the Austrian Science Fund FWF under projects P16264 and P16539.     

Hybrid possibilistic networks

Possibilistic networks and possibilistic logic are two standard frameworks of interest for representing uncertain pieces of knowledge. Possibilistic networks exhibit relationships between variables while possibilistic logic ranks logical formulas according to their level of certainty. For multiply connected networks, it is well-known that the inference process is a hard problem. This paper studies a new representation of possibilistic networks called hybrid possibilistic networks. It results from combining the two semantically equivalent types of standard representation. We first present a propagation algorithm through hybrid possibilistic networks. This inference algorithm on hybrid networks is strictly more efficient (and confirmed by experimental studies) than the one of standard propagation algorithm.     

Formalization of implication based fuzzy reasoning method

Fuzzy reasoning includes a number of important inference methods for addressing uncertainty. This line of fuzzy reasoning forms a common logical foundation in various fields, such as fuzzy logic control and artificial intelligence. The full implication triple I method (a method only based on implication, TI method for short) for fuzzy reasoning is proposed in 1999 to improve the popular CRI method (a hybrid method based on implication and composition). The current paper delves further into the TI method, and a sound logical foundation is set for the TI method based on the monoidal t-norm based logical system MTL.     

Dependence of variables construed as an atomic formula

We define a logic D capable of expressing dependence of a variable on designated variables only. Thus D has similar goals to the Henkin quantifiers of [4] and the independence friendly logic of [6] that it much resembles. The logic D achieves these goals by realizing the desired dependence declarations of variables on the level of atomic formulas. By [3] and [17], ability to limit dependence relations between variables leads to existential second order expressive power. Our D avoids some difficulties arising in the original independence friendly logic from coupling the dependence declarations with existential quantifiers. As is the case with independence friendly logic, truth of D is definable inside D. We give such a definition for D in the spirit of [11] and [2] and [1].     

A modal logic framework for reasoning about comparative distances and topology

We propose and investigate a uniform modal logic framework for reasoning about topology and relative distance in metric and more general distance spaces, thus enabling the comparison and combination of logics from distinct research traditions such as Tarski’s S4 for topological closure and interior, conditional logics, and logics of comparative similarity. This framework is obtained by decomposing the underlying modal-like operators into first-order quantifier patterns. We then show that quite a powerful and natural fragment of the resulting first-order logic can be captured by one binary operator comparing distances between sets and one unary operator distinguishing between realised and limit distances (i.e., between minimum and infimum). Due to its greater expressive power, this logic turns out to behave quite differently from both S4 and conditional logics. We provide finite (Hilbert-style) axiomatisations and ExpTime-completeness proofs for the logics of various classes of distance spaces, in particular metric spaces. But we also show that the logic of the real line (and various other important metric spaces) is not recursively enumerable. This result is proved by an encoding of Diophantine equations.     

Reducing belief simpliciter to degrees of belief

Is it possible to give an explicit definition of belief (simpliciter) in terms of subjective probability, such that believed propositions are guaranteed to have a sufficiently high probability, and yet it is neither the case that belief is stripped of any of its usual logical properties, nor is it the case that believed propositions are bound to have probability 1? We prove the answer is ‘yes’, and that given some plausible logical postulates on belief that involve a contextual “cautiousness” threshold, there is but one way of determining the extension of the concept of belief that does the job. The qualitative concept of belief is not to be eliminated from scientific or philosophical discourse, rather, by reducing qualitative belief to assignments of resiliently high degrees of belief and a “cautiousness” threshold, qualitative and quantitative belief turn out to be governed by one unified theory that offers the prospects of a huge range of applications. Within that theory, logic and probability theory are not opposed to each other but go hand in hand.     

Modules over quantaloids: Applications to the isomorphism problem in algebraic logic and π-institutions

We solve the isomorphism problem in the context of abstract algebraic logic and of π-institutions, namely the problem of when the notions of syntactic and semantic equivalence among logics coincide. The problem is solved in the general setting of categories of modules over quantaloids. We introduce closure operators on modules over quantaloids and their associated morphisms. We show that, up to isomorphism, epis are morphisms associated with closure operators. The notions of (semi-)interpretability and (semi-)representability are introduced and studied. We introduce cyclic modules, and provide a characterization for cyclic projective modules as those having a g-variable. Finally, we explain how every π-institution induces a module over a quantaloid, and thus the theory of modules over quantaloids can be considered as an abstraction of the theory of π-institutions.     

Overriding subsuming rules

This paper is concerned with intelligent agents that are able to perform nonmonotonic reasoning, not only with, but also about general rules with exceptions. More precisely, the focus is on enriching a knowledge base Γ with a general rule that is subsumed by other rules already there. Such a problem is important because evolving knowledge needs not follow logic as it is well-known from e.g. the belief revision paradigm. However, belief revision is mainly concerned with the case that the extra information logically conflicts with Γ. Otherwise, the extra knowledge is simply doomed to extend Γ with no change altogether. The problem here is different and may require a change in Γ even though no inconsistency arises. The idea is that when a rule is to be added, it might need to override any rule that subsumes it: preemption must take place. A formalism dedicated to reasoning with and about rules with exceptions is introduced. An approach to dealing with preemption over such rules is then developed. Interestingly, it leads us to introduce several implicants concepts for rules that are possibly defeasible.     

Completeness of two systems of illative combinatory logic for first-order propositional and predicate calculus

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers 4 systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In a preceding paper, Barendregt, Bunder and Dekkers, 1993, we proved completeness of the two direct translations. In the present paper we prove completeness of the two indirect translations by showing that the corresponding illative systems are conservative over the two systems for the direct translations. In another version, DBB (1997), we shall give a more direct completeness proof. These papers fulfill the program of Church and Curry to base logic on a consistent system of -terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). Received: February 15, 1996