A Two-Tiered Propositional Framework for Handling Multisource Inconsistent Information

The two main uncertainty representations in the literature that tolerate imprecision are possibility distributions and random disjunctive sets. This chapter devotes special attention to the theories that have emerged from them. The first part of the chapter discusses epistemic logic and derives the need for capturing imprecision in information representations. It bridges the gap between uncertainty theories and epistemic logic showing that imprecise probabilities subsume modalities of possibility and necessity as much as probability. The second part presents possibility and evidence theories, their origins, assumptions and semantics, discusses the connections between them and the general framework of imprecise probability. Finally, chapter points out the remaining discrepancies between the different theories regarding various basic notions, such as conditioning, independence or information fusion and the existing bridges between them.

An active set is a unifying space being able to act as a “bridge” for transferring information, ideas and results between distinct types of uncertainties and different types of applications. An active set is a set of agents who independently deliver true or false values for a given proposition. It is not a simple vector of logic values for different propositions, the results are a vector but the set is not. The difference between an ordinary set and active set is that the ordinary set has passive elements with values of the attributes defined by an external agent, in the active set any element is an agent that internally defines the value of a given attribute for a passive element. So agents in many cases are in a logic conflict and this generates semantic uncertainty on the evaluation. Criteria and agents are the two variables by which we give different logic values to the same attribute or proposition. In modal logic, given a proposition, we can evaluate the proposition only when we know the worlds where the proposition is located. In epistemic logic any world is an agent that knows that the proposition is true or false. The active set is a set of agents as in epistemic logic but the difference with modal logic is that all the agents (worlds) are not separate but are joined in the evaluation of the given proposition. In active set for one agent and one criteria we have one logic value but for many agents and criteria the evaluation is not true or false but is a matrix of true and false. This matrix is not only a logic evaluation as in the modal logic but give us the conflicting structure of the active set evaluation. The agent multi dimensional space to evaluate active set also includes the Hilbert multidimensional space where is possible to simulate quantum logic gate. New logic operation are possible as fuzzy gate operations and more complex operations as conflicting solving, consensus operations, syntactic inconsistency, semantic inconsistency and knowledge integration. In the space of the agents evaluations morphotronic geometric operations are a new frontier to model new types of computers. In this paper we show the connection between classical, fuzzy, evidence theory and active sets.

A formal language of two-valued logic is developed, whose terms are formulas of the language of Kleene's three-valued logic. The atomic formulas of the former language are pairs of formulas of the latter language joined by “consequence” operators. These operators correspond to the three “sensible” types of consequence (strong-strong, strong-weak and weak-weak) in Kleene's logic in analogous way as the implication connective in the classical logic corresponds to the classical consequence relation. The composed formulas of the considered language are built from the atomic ones by means of the classical connectives and quantifiers. A deduction system for the developed language is given, consisting of a set of decomposition rules for sequences of formulas. It is shown that the deduction system is sound and complete.

This article is devoted to applications of fuzzy set theory, possibility theory and evidence theory in civil engineering, presenting current work of a group or researchers at the University of Innsbruck. We argue that these methods are well suited for analyzing and processing the parameter uncertainties arising in soil mechanics and construction management. We address two specific applications here: finite element computations in foundation engineering and a queueing model in earth work.

Mathematical theories of uncertainty in which measures of uncertainty are now well established are fuzzy set theory, evidence theory, and possibility theory, in addition to classical set theory and probability theory. Some of these theories are connected. Classical set theory is subsumed under fuzzy set theory. Probability theory and possibility theory are branches of evidence theory, while evidence theory is, in turn, a branch of fuzzy measure theory. Although information aspects of fuzzy measure theory have not been investigated as yet, the theory is briefly introduced in this chapter because it represents a broad framework for future research.

CONCEPTUAL FOUNDATIONS OF QUANTUM MECHANICS: THE ROLE OF EVIDENCE THEORY, QUANTUM SETS, AND MODAL LOGIC

An annotated survey of approaches to design optimization based on possibility theory and evidence theory is presented and prominent characteristics are described in this paper, especially addressing epistemic uncertainty for large-scale and complex systems when statistical data is scarce or incomplete. We first analyze the uncertainties encountered in design and the limitations of probabilistic approach. Then possibility-based design optimization (PBDO), evidence-based design optimization (EBDO) and their integrated approaches are comparatively investigated respectively, form the viewpoints of computational development and performance improvement. After that, this paper discusses the fusion technologies and an example of integrated approach in reliability to reveal the qualitative value and efficiency. In the end, technical difficulties, application areas and future directions are summarized.

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.

Background: Uncertainty is a common factor in every real-life decision-making problem. Possibility theory is one uncertainty theory in Fuzzy sets (FS). The possibility-based decision-making under a fuzzy environment is a significant multi-criteria decision-making (MCDM) method. Methods: A bipolar FS is an extension of a fuzzy set. With the bipolarity concept, we can handle both positive and negative thoughts. In this study, we have provided a possibility mean of a bipolar fuzzy number. We have developed a ranking method for bipolar fuzzy numbers using this pos-sibility concept. A novel possibility MCDM method is suggested for solving the water resources management (WRM) in the Nagpur area, Maharashtra State, India. Results: The MCDM technique is an effective tool for solving WRM problems in an area. Many uncertainties and bipolarities occur together in Nagpur water resources. WRM technique with fuzzy is one approach that can be used to solve the area's water problem. We have used the pro-posed MCDM to address the water-related issues of this district. With this proposed MCDM method, numerically, we employed water resource problems under a bipolar fuzzy environ-ment. Conclusion: The Nagpur area is covered by Basaltic rock and faces water shortage. The district is experiencing severe water shortages. Groundwater, surface water, and rainfall are three water resources considered as alternatives. According to the proposed MCDM technique in Nagpur district, Groundwater is the best water source from three flanks: quality of water, affordability, and availability.

Parametric families of continuous belief functions based on generalized Gaussian random fuzzy numbers

An active set is a unifying space being able to act as a bridge for transferring information, ideas and results between distinct types of uncertainties and different types of applications. An active set is a set of agents who independently deliver true or false values for a given proposition. An active set is not a simple vector of logic values for different propositions, the results are a vector but the set is not. The difference between an ordinary set and active set is that the ordinary set has passive elements with values of the attributes defined by an external agent, in the active set any element is an agent that internally defines the value of a given attribute for a passive element. In this paper we show the connection between classical, fuzzy, evidence theory and active sets. In conclusion at one external agent we substitute a set of experts or agents that evaluate in a conflicting way the logic value of a given proposition or attribute. Under the same meta level of active sets we can discover analogy and similarity among distinct theories of uncertainty and types of many valued logics.

Following a proposal by Kooi and Tamminga, we introduce a conservative translation manual for every four-valued truth-functional propositional logic into a modal logic. However, the application of this translation does not preserve the intuitive reading of the truth-values for every four-valued logic. In order to solve this problem, we modify the translation manual and prove its conservativity by exploiting the method of generalized truth-values.

In recent years it has become apparent that an important part of the theory of Artificial Intelligence is concerned with reasoning on the basis of uncertain, incomplete or inconsistent information. Classical logic and probability theory are only partially adequate for this, and a variety of other formalisms have been developed, some of the most important being fuzzy methods, possibility theory, belief function theory, non monotonic logics and modal logics. The aim of this workshop was to contribute to the elucidation of similarities and differences between the formalisms mentioned above.

The study is to show the similarities between J. Fodor's Language of Thought hypothesis and epistemic modal logic. According to the J. Fodor's hypothesis there is the language of thought (also called Mentalese) that is the meta-language in which mental representations of attitudes of organism to propositions expressed in object-language (for example: knowledge, belief, hope, desire, statement) are formulated. These attitudes are called attitudes. In the hypothesis propositional attitudes are thoughts and relations between organism and proposition. Propositional attitudes are of interest for epistemic modal logics. In epistemic logics propositional attitudes are relations considered as modalities, i.e. propositions with modal operators. On certain conditions J. Fodor's Language of Thought hypothesis may be considered as a sort of epistemic modal logic, only it would require certain updating as a formal logical system. The similarities between J. Fodor's Language of Thought and epistemic modal logics give two opportunities. First opportunity is to make special version of epistemic modal logic as an instrument of research for cognitive psychology based on J. Fodor's Language of Thought hypothesis. Second opportunity is to accept the idea of J. Fodor's Language of Thought as an innate language and to attain comprehension that epistemic modal logics are not especially constructed formal languages (formal systems) but innate cognitive programs. It might enable to extrapolate the idea of innate languages onto all formal languages including logical languages and to attempt verifying that idea experimentally within cognitive sciences.

The paper proposes a parallel survey of revision and updating operations available in the probability theory and in the possibility theory frameworks. In these two formalisms the current state of knowledge is generally represented by a [0,1]-valued function whose domain is an exhaustive set of mutually exclusive possible states of the world. However, in possibility theory, the unit-interval can be viewed as a purely ordinal scale. Two general kinds of operations can be defined on this assignment function: conditioning, and imaging (or “projection”). the difference between these two operations is analogous to the one made between belief revision à la Gärdenfors and updating à la Katsuno and Mendelzon in the logical framework. In the probabilistic framework these two operations are respectively Bayesian conditioning and Lewis' imaging. Counterparts to these operations are presented for the possibilistic framework including the case of conditioning upon uncertain observations, and justifications are given which parallel the ones existing for the probabilistic operations. More particularly, it is recalled that possibilistic conditioning satisfies all the postulates proposed by Alchourrón, Gärdenfors and Makinson for belief revision (stated in possibilistic terms), and it is proved that possibilistic imaging satisfies all the postulates proposed by Katsuno and Mendelzon. the situation where our current knowledge is stated in terms of weighted logical propositions is discussed in connection to possibility theory. Revision in other more complex numerical formalisms, namely belief and plausibility functions, and upper and lower probabilities is also surveyed. Recent results on the revision of conditional knowledge bases are also reviewed. the frameworks of belief functions, upper and lower probabilities and conditional bases are more sophisticated than the previous ones because they enable to distinguish between factual evidence and generic knowledge in a cognitive state. This, framework leads to two forms of belief revision respectively taking care of the revision of evidence and the revision of knowledge. © 1994 John Wiley & Sons, Inc.