An algebra of conditional events

Schay, Géza

doi:10.1016/0022-247x(68)90035-8

Cited by 65 publications

(30 citation statements)

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“…Another point is that in his discussion concerning truth tables, Hailperin (p. 127) did realize that conditional events have three possible truth values. This fact was realized much earlier by DeFinetti (1964) and Schay (1968) who defined conditional events precisely this way, that is from a semantic viewpoint (or equivalently, by extending the concept of ordinary indicator functions of events or sets). Hamilton, 1978).…”

Section: Rb-rirv Ab-a + Rbmentioning

confidence: 92%

“…[Among the few who have attempted to define operations among conditional events with different antecedents -the identical antecedent case being similar to the funconditional case -ofily Schay (1968) has justified his choice of (two proposed systems of) operators, and that indirectly through an-abstract characterization theorem. (See Schay (1968), Theorem 5.)…”

Section: !mentioning

confidence: 99%

“…Independently of each other, Reichenbach (1948Reichenbach ( , 1949, Schay (1968), DeFinetti U(1972, 1974, Volumes 1 and2), and Dubois andPrade (1987, 1990) considered the modeling of conditional events from a logical viewpoint. They all viewed a conditional event as an object with three possible "truth" values.…”

Section: Three-valued Logicmentioning

confidence: 99%

“…These turn out to be precisely Schay's operations (Schay, 1968), which will be discussed in Chapter 3. The problem with assigning probabilities to compounds of conditionals is discussed using Lewis' triv.ality result, which says that one cannot model conditionals as elements of [the Boolean ring R, compatible with conditional probability (Adams, pp.…”

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confidence: 91%

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Conditional Inference and Logic for Intelligent Systems: A Theory of Measure-Free Conditioning

Goodman¹,

Nguyen²,

Walker³

1991

128

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Approved for public release; distribution is unlimited. ABSTRACT (Maximum 200 words)This work addresses an anomaly involving probability and logic relative to thtu interprutation of irni.., .ti,,e statements, and the evaluation of those btateamunts compatible wvith _unditionarl probability. One of our cieuf motivrations 1s thle need to for..,alize rigorously the connections betwecen conditional probablity and the "hidden" logit. Uf Implicative statements, such as production rules in expert systemb and defaults in CommoI~I-n uns reasoning. The pjurpove Is to provide theoretica! results for probabilistic reasoning that will be useful in the design and evaluation of inference rules of such systems.f -L Published by North-Holland, 1991 Publishes, Amsterdam, New York, Oxford, and Tokyo. PREFACE 1.This book is concerned with a systematic investigation of the concept of "measure-free" conditioning and its associated logic for intelligent systems. Its purpose is to provide a foundation for inference in such systems. The basic problem is the representation and evaluation of implicative statements in natural language, in a way compatible with conditional probability. This longstanding problem involves three distinct disciplines: natural language, logic, and probability. [The situation is different today. The problem is before us because of the need to provide a firm foundation for probabilistic reasoning in intelligent systems; in particular, V how to combine conditional information arising from disparate sources in expert systems and how to compute it probabilistically. This is in line with the Bayesian approach to probabilistic reasoning in intelligent systems (Pearl, 1988). Probability not only has a firm mathematical foundation, but also the conditional probability operator captures a form of non-monotonicity of common sense reasoning.Our goal is a more complete and satisfactory theory of "measure-free" conditioning.If the concept of "conditional event" can be formalized and a suitable algebra of operators between such events be developed, then the resulting structure will have use in designing inference rules in expert systems. With probability being the method of choice for handling uncertainty despite the plethora of non-probabilistic procedures such as example, Dubois and Prade (1988)). This development is not to be confused with other "conditional logics", such as that of Nute (1980) and Appiah (1985), which are not j ~ compatible with conditional probability, nor with non-commutative extensions of Boolean logic (Guzman and Squier, 1990). Our approach differs also from that of Adams (1975),[who takes conditionals as primitives in natural language, while ours are mathematical entities.[1 This book is primarily concerned with theory. The reader is expected to be familiar with basic probability theory, elementary logic, and elementary facts from ring theory.However, the text is largely self-contained. The hope is that this book will trigger further JI interest in both the theory and applications of this topic.In c...

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Section: Rb-rirv Ab-a + Rbmentioning

confidence: 92%

Section: !mentioning

confidence: 99%

Section: Three-valued Logicmentioning

confidence: 99%

mentioning

confidence: 91%

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Conditional Inference and Logic for Intelligent Systems: A Theory of Measure-Free Conditioning

Goodman¹,

Nguyen²,

Walker³

1991

128

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“…[15,7,1,14,9,12,6]. In the former, conditionals constitute the core focus of non-monotonic reasoning [10,8,11].…”

Section: Introductionmentioning

confidence: 99%

On Boolean Algebras of Conditionals and Their Logical Counterpart

Flaminio

Godo

Hosni

2017

Lecture Notes in Computer Science

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Abstract. This paper sheds a novel light on the longstanding problem of investigating the logic of conditional events. Building on the framework of Boolean algebras of conditionals previously introduced by the authors, we make two main new contributions. First, we fully characterise the atomic structure of these algebras of conditionals. Second, we introduce the logic of Boolean conditionals (LBC) and prove its completeness with respect to the natural semantics induced by the structural properties of the atoms in a conditional algebra as described in the first part. In addition we outline the close connection of LBC with preferential consequence relations, arguably one of the most appreciated systems of non-monotonic reasoning.

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Probable equivalence, superpower sets, and superconditionals

Kosko

2004

Int. J. Intell. Syst.

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A natural measure of probabilistic equality between sets leads to two measures of probabilistic conditioning that form the endpoints of a conditioning interval. The interval's lower bound is the standard conditional probability or "subconditional" that describes the probability of a subset relation. The upper bound is a new "superconditional" that describes the probability of the corresponding superset relation. These dual conditioning operators correspond to dual set collections and enjoy optimality relations with respect to these set collections. Fuzzy cubes illustrate these set-collection relations in the two-dimensional case. The subconditional operator corresponds to the usual "power set" of a given set. The dual superconditional operator corresponds to what we call the "superpower set" or the set of all supersets of the given set. The two dual conditioning operators can eliminate each other through simple equalities. They obey dual Bayes theorems but differ in how they respond to statistical independence.

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An algebra of conditional events

Cited by 65 publications

References 2 publications

Conditional Inference and Logic for Intelligent Systems: A Theory of Measure-Free Conditioning

Conditional Inference and Logic for Intelligent Systems: A Theory of Measure-Free Conditioning

On Boolean Algebras of Conditionals and Their Logical Counterpart

Probable equivalence, superpower sets, and superconditionals

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