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Latticebased relation algebras and their representability II
 In Theory and Applications of Relational Structures as Knowledge Instruments, de
, 2003
"... www.cosc.brocku.ca Lattice–based relation algebras and their representability ⋆ ..."
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www.cosc.brocku.ca Lattice–based relation algebras and their representability ⋆
Relational representation theorems for some latticebased structures
 Journal on Relational Methods in Computer Science
"... The major elements of the method of proving relational representation theorems presented in this paper are, on the one hand, Urquhart representation theorem for lattices [17] and Allwein and Dunn developments on Kripke semantics for linear logic (see [1] and also [4]), and on the other hand, a gener ..."
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The major elements of the method of proving relational representation theorems presented in this paper are, on the one hand, Urquhart representation theorem for lattices [17] and Allwein and Dunn developments on Kripke semantics for linear logic (see [1] and also [4]), and on the other hand, a generalisation of Jon
Metaepistemic logic: A minimal logic for reasoning about revealed beliefs
, 2010
"... Reasoning about knowledge described in classical propositional logic is usually handled either in the metalanguage as in belief revision, considering the dynamics of belief bases, or at the object level by means of modal logic. In the latter case, modalities express knowledge, belief, or absence th ..."
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Reasoning about knowledge described in classical propositional logic is usually handled either in the metalanguage as in belief revision, considering the dynamics of belief bases, or at the object level by means of modal logic. In the latter case, modalities express knowledge, belief, or absence thereof, about the truth of formulae. But the semantics is described in terms of accessibility relations, whose expressive power seems to be too powerful to account for mere epistemic states of an agent. This paper proposes a simpler logic whose atoms express beliefs about formulae expressed in another basic propositional language, and that allows for conjunctions, disjunctions and negations of beliefs. The idea is to model an agent reasoning about some beliefs of another agent as revealed by the latter. This logic, called MetaEpistemic Logic (MEL), borrows its syntax and axioms from the modal logic KD. It can be be viewed as a fragment of KD, but it is an encapsulation of propositional logic rather than an extension thereof. Its semantics is given in terms of epistemic states understood as subsets of propositional interpretations. We prove soundness and completeness of this logic, and that any family of nonempty subsets 1 of propositional interpretations can be expressed as a single formula in MEL. Inference rules and normal forms in MEL are discussed. We show that this logic is very similar to the consensus logic of Pauly. It is also simpler than many previous formalisms for reasoning about knowledge, and it avoids paradoxes of truthfunctional accounts of incomplete information handling like partial logic. Our approach is in fact much closer to the logical rendering of uncertainty theories like possibilistic logic. MEL has indeed potential to be extended to deal with graded beliefs. For instance, we show that MEL can express a symbolic counterpart of the Möbius transform in the theory of belief functions. 1
Lattice–based relation algebras II ⋆
"... Abstract. We present classes of algebras which may be viewed as weak relation algebras, where a Boolean part is replaced by a not necessarily distributive lattice. For each of the classes considered in the paper we prove a relational representation theorem. 1 ..."
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Abstract. We present classes of algebras which may be viewed as weak relation algebras, where a Boolean part is replaced by a not necessarily distributive lattice. For each of the classes considered in the paper we prove a relational representation theorem. 1
IOS Press A possibilitytheoretic view of formal concept analysis
"... Abstract. The paper starts from the standard relational view linking objects and properties in formal concept analysis, here augmented with four modalstyle operators (known as sufficiency, dual sufficiency, necessity and possibility operators). Formal concept analysis is mainly based on the first o ..."
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Abstract. The paper starts from the standard relational view linking objects and properties in formal concept analysis, here augmented with four modalstyle operators (known as sufficiency, dual sufficiency, necessity and possibility operators). Formal concept analysis is mainly based on the first operator, while the others come from qualitative data analysis and can be also related to rough set theory. A possibilitytheoretic reading of formal concept analysis with these four operators is proposed. First, it is shown that four and only four operators are indeed needed in order to describe the nine situations that can occur when comparing a statement (or its negation) with a state of information. The parallel between possibility theory and formal concept analysis suggests the introduction of new notions such as normalization and conditioning in the latter framework, also leading to point out some meaningful properties. Moreover, the graded setting of possibility theory allows us to suggest the extension of formal concept analysis to situations with incomplete or uncertain information. 1.