Modal Logics for Qualitative Possibility and Beliefs

Modal logics for qualitative possibility theory

The Local Computation Framework has been used to improve the efficiency of computation in various uncertainty formalisms. This paper shows how it can be applied to logical deduction in first-order logic, modal and conditional logics, circumscription and possibilistic logic.

Emotion is a cognitive mechanism that directs an agent's thoughts and attention to what is relevant, important, and significant. Such a mechanism is crucial for the design of resource-bounded agents that must operate in highly-dynamic, semi-predictable environments and which need mechanisms for allocating their computational resources efficiently. The aim of this work is to propose a logical analysis of emotions and their influences on an agent's behavior. We focus on four emotion types (viz., hope, fear, joy, and distress) and provide their logical characterizations in a modal logic frame-work. As the intensity of emotion is essential for its influence on an agent's behavior, the logic is devised to represent and reason about graded beliefs, graded goals and intentions. The belief strength and the goal strength determine the intensity of emotions. Emotions trigger different types of coping strategy which are aimed at dealing with emotions either by forming or revising an intention to act in the world, or by changing the agent's interpretation of the situation (by changing beliefs or goals).

This paper proposes a concise overview of the role of possibility theory in logical approaches to reasoning under uncertainty. It shows that three traditions of reasoning under or about uncertainty (set-functions, epistemic logic and three-valued logics) can be reconciled in the setting of possibility theory. We offer a brief presentation of basic possibilistic logic, and of its generalisation that comes close to a modal logic albeit with simpler more natural epistemic semantics. Past applications to various reasoning tasks are surveyed, and future lines of research are also outlined.

Realism for agents with unconditional beliefs, desires and intentions (BDI agents) has been analyzed in modal logic. This paper provides a logical analysis of realism for agents with conditional beliefs and desires in a rule based approach analogous to Reiter's default logic. We distinguish two types of realism, which we call ‘a priori' and ‘a posteriori' realism. We analyze whether these two new properties are compatible with other properties discussed in the literature, such as existence of extensions. We show that Reiter's default logic is too strong, in the sense that a weaker notion of maximality of extensions is needed to satisfy realism. Finally we show that several existing approaches do not satisfy the new realism properties, and we introduce a new construction that does satisfy them.

We present a framework for understanding abduction within modal logic and Kripke semantics; worlds of a Kripke frame will represent possible theories, and a change in theory will be understood as a passage from one world to an adjacent possible world. Further, these steps may agree with the accessibility relation or may ‘backtrack’, accordingly as new information refutes or reinforces our present theory. Our formalism can be used to model not only abduction, but also to talk about the inner structure of theories as well as relations between them, allowing us to interpret many ideas from philosophy of science within the well-understood framework of modal logic.

We introduce here the study of generalnonmonotonic rule systems. These deal with situations where a conclusion is drawn from a system of beliefsS (and seen to be inS), basedboth on some being inS and on some restraints not being inS. In the monotone systems of traditional logic there are no restraints, conclusions are drawn solely based on premises being inS. Nonmonotonic rule systems capture the essential syntactic, semantic, and algorithmic features of many nonmonotone systems such as default logic, negation as failure, truth maintenance, autoepistemic logic, and also important combinatorial questions from mathematics such as the marriage problem. This reveals semantics and syntax and proof procedures and algorithms for computing belief sets in many cases where none were previously available and entirely uniformly. In particular, we introduce and study deductively closed sets, extensions and weak extensions. Semantics of nonmonotonic rule systems is studied in part II of this paper and extensions to predicate classical, intuitionistic, and modal logics are left to a later paper.

In this paper we present a modal approach to “contrastive logic”, the logic of contrasts as these appear in natural language conjunctions such as ‘but’. We use a simple modal logic, which is an extension of the well-knownS5 logic, and base the contrastive operators proposed by Francez in [2] on the basic modalities that appear in this logic. We thus obtain a logic for contrastive operators that is more in accord with the tradition of intensional logic, and that, moreover — we argue — has some more natural properties. Particularly, attention is paid to nesting contrastive operators. We show that nestings of ‘but’ give quite natural results, and indicate how nestings of other contrastive operators can be done adequately. Finally, we discuss the example of the Hangman's Paradox and some similarities (and differences) with default reasoning.

It has been well recognized that common-sense reasoning is nonmonotonic. Nonmonotonic reasoning is prototypically described by the reasoning about normality and exception. In this paper, we firstly propose a modal logic about normality and exception. We then present a nonmonotonic modal logic which formalizes the intuitions of nonmonotonic reasoning directly, rather than go around to indirect criterion. The results can be seen by comparing our logic with major nonmonotonic logics in recent literature. In particular, we provide a first-order extension of default logic and autoepistemic logic, and the capability of circumscription to infer universal statement. As a result, the relations among various existing nonmonotonic logics coincide in our logic. >

The logic program formalism is commonly viewed as a modal or default logic. In this paper, we propose an alternative interpretation of the formalism as a terminological logic. A terminological logic is designed to represent two different forms of knowledge. A TBox represents definitions for a set of concepts. An ABox represents the assertional knowledge of the expert. In our interpretation, a logic program is a TBox providing definitions for all predicates; this interpretation is present already in Clark's completion semantics. We extend the logic program formalism such that some predicates can be left undefined and use classical logic as the language for the ABox. The resulting logic can be seen as an alternative interpretation of abductive logic program formalism. We study the expressivity of the formalism for representing uncertainty by proposing solutions for problems in temporal reasoning, with null values and open domain knowledge.

Default logic is the most popular non monotonic formalism. We present a general framework, defining the classical default extensions as “justified pre-extensions” having a particular property, “stability”. These pre-extensions have a constructive definition, contrary to classical extensions, and they give rise to a notion of “formula provable by default” which has appeared yet under several formulations in the literature. Also, we show how a less known proposal of this kind gives a very interesting “semi-constructive” definition of default extensions. Finally our framework applies to some modal translations of defaults, making precise the relations between the stability in default logic and in autoepistemic modal logic.

In this paper, we provide a re-interpretation of qualitative and quantitative modeling from a representationalist perspective. In this view, both approaches attempt to construct abstract representations of empirical relational structures. Whereas quantitative research uses variable-based models that abstract from individual cases, qualitative research favors case-based models that abstract from individual characteristics. Variable-based models are usually stated in the form of quantified sentences (scientific laws). This syntactic structure implies that sentences about individual cases are derived using deductive reasoning. In contrast, case-based models are usually stated using context-dependent existential sentences (qualitative statements). This syntactic structure implies that sentences about other cases are justifiable by inductive reasoning. We apply this representationalist perspective to the problems of generalization and replication. Using the analytical framework of modal logic, we argue that the modes of reasoning are often not only applied to the context that has been studied empirically, but also on a between-contexts level. Consequently, quantitative researchers mostly adhere to a top-down strategy of generalization, whereas qualitative researchers usually follow a bottom-up strategy of generalization. Depending on which strategy is employed, the role of replication attempts is very different. In deductive reasoning, replication attempts serve as empirical tests of the underlying theory. Therefore, failed replications imply a faulty theory. From an inductive perspective, however, replication attempts serve to explore the scope of the theory. Consequently, failed replications do not question the theory per se, but help to shape its boundary conditions. We conclude that quantitative research may benefit from a bottom-up generalization strategy as it is employed in most qualitative research programs. Inductive reasoning forces us to think about the boundary conditions of our theories and provides a framework for generalization beyond statistical testing. In this perspective, failed replications are just as informative as successful replications, because they help to explore the scope of our theories.

Strong equivalence is an important property for nonmonotonic formalisms, allowing safe local changes to a nonmonotonic theory. This paper considers strong equivalence for nonmonotonic causal theories of the kind introduced by McCain and Turner. Causal theories T 1 and T 2 are strongly equivalent if, for every causal theory T, T 1 ∪ T and T 2 ∪ T are equivalent (that is, have the same causal models). The paper introduces a convenient characterization of this property in terms of so-called SE-models, much as was done previously for answer set programs and default theories. A similar result is provided for the nonmonotonic modal logic UCL. The paper also introduces a reduction from the problem of deciding strong equivalence of two causal theories to the problem of deciding equivalence of two sets of propositional formulas.

This paper put forward constrained epistemic action model logic (EAML), which extended the action modal logic and reason epistemic default logic in the dynamic epistemic theory framework.In this paper, the constrained epistemic default logic and action model logic was combined to model the dynamic cognitive update process by introducing a cognitive operator.As constrained epistemic default logic restricts the current possible epistemic state and action model logic can describe epistemic actions, the new proposed logic can better portray the cognitive operations and the restriction of the corresponding accessibility relations.The description of the logic framework was given in this study, and the related theorems were proved to show the interpretation of the constrained epistemic action model logic.

The ability to “define” propositions using default assumptions about the same propositions is identified as a significant source of computational complexity in nonmonotonic reasoning. If such constructs are not allowed, i.e. the knowledge base is stratified, a notable computational advantage is obtained. This is demonstrated by developing an iterative algorithm for propositional stratified autoepistemic theories the complexity of which is dominated by required classical reasoning. Thus efficient subclasses of stratified nonmonotonic reasoning can be obtained by further restricting the form of sentences in the knowledge base. As an example quadratic and linear time algorithms are derived for specific subclasses of stratified autoepistemic theories. The results are shown to imply efficient reasoning methods for stratified cases of default logic, logic programs, truth maintenance systems, and nonmonotonic modal logics.