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Author: Philip Calabrese
Publisher:
ISBN: 9781848902589
Category : Mathematics
Languages : en
Pages : 286
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Book Description
This monograph develops an algebra of Boolean fractions, (ab) - ordered pairs of propositions or events - "a if b", "event a given event b". In nine chapters, the author shows that these conditional propositions (together with their associated instantiations or models): Provide logical elements that better represent and more faithfully facilitate manipulation of certain and uncertain conditional information Extend the Boole's algebra of 2-valued statements to a 3-valued system that includes "inapplicable statements" - those whose condition may be false in some or all instances (examples, cases, models...) Allow a definition of the probability of an arbitrary Boolean proposition Non-trivially combine Boolean logic with standard conditional probability theory Provide a complete and adequate development of the crucial 4th operation for Boolean logic, namely conditioning, including iterated conditioning Provide an expanded theory of deduction defined in terms of the extended operations on the Boolean fractions Admit a variety of deduction relations, and that the deductively closed sets generated by some initial set of conditionals can be calculated Extend the ordinary function operations of sum, difference, product & quotient to real-valued functions with possibly different or overlapping domains of definition Represent & simplify complex conditional statements in Bayesian expert systems used to calculate maximum information entropy solutions Explicate the logic of quantum measurements by better expressing the changing conditions in quantum mechanics
Author: Philip Calabrese
Publisher:
ISBN: 9781848902589
Category : Mathematics
Languages : en
Pages : 286
Get Book Here
Book Description
This monograph develops an algebra of Boolean fractions, (ab) - ordered pairs of propositions or events - "a if b", "event a given event b". In nine chapters, the author shows that these conditional propositions (together with their associated instantiations or models): Provide logical elements that better represent and more faithfully facilitate manipulation of certain and uncertain conditional information Extend the Boole's algebra of 2-valued statements to a 3-valued system that includes "inapplicable statements" - those whose condition may be false in some or all instances (examples, cases, models...) Allow a definition of the probability of an arbitrary Boolean proposition Non-trivially combine Boolean logic with standard conditional probability theory Provide a complete and adequate development of the crucial 4th operation for Boolean logic, namely conditioning, including iterated conditioning Provide an expanded theory of deduction defined in terms of the extended operations on the Boolean fractions Admit a variety of deduction relations, and that the deductively closed sets generated by some initial set of conditionals can be calculated Extend the ordinary function operations of sum, difference, product & quotient to real-valued functions with possibly different or overlapping domains of definition Represent & simplify complex conditional statements in Bayesian expert systems used to calculate maximum information entropy solutions Explicate the logic of quantum measurements by better expressing the changing conditions in quantum mechanics
Author: E.W. Adams
Publisher: Springer Science & Business Media
ISBN: 940157622X
Category : Philosophy
Languages : en
Pages : 173
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Book Description
Of the four chapters in this book, the first two discuss (albeit in consider ably modified form) matters previously discussed in my papers 'On the Logic of Conditionals' [1] and 'Probability and the Logic of Conditionals' [2], while the last two present essentially new material. Chapter I is relatively informal and roughly parallels the first of the above papers in discussing the basic ideas of a probabilistic approach to the logic of the indicative conditional, according to which these constructions do not have truth values, but they do have probabilities (equal to conditional probabilities), and the appropriate criterion of soundness for inferences involving them is that it should not be possible for all premises of the inference to be probable while the conclusion is improbable. Applying this criterion is shown to have radically different consequences from the orthodox 'material conditional' theory, not only in application to the standard 'fallacies' of the material conditional, but to many forms (e. g. , Contraposition) which have hitherto been regarded as above suspi cion. Many more applications are considered in Chapter I, as well as certain related theoretical matters. The chief of these, which is the most important new topic treated in Chapter I (i. e.
Author: Theodore Hailperin
Publisher: Rowman & Littlefield
ISBN: 1611460107
Category : Mathematics
Languages : en
Pages : 124
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Book Description
The present study is an extension of the topic introduced in Dr. Hailperin's Sentential Probability Logic, where the usual true-false semantics for logic is replaced with one based more on probability, and where values ranging from 0 to 1 are subject to probability axioms. Moreover, as the word "sentential" in the title of that work indicates, the language there under consideration was limited to sentences constructed from atomic (not inner logical components) sentences, by use of sentential connectives ("no," "and," "or," etc.) but not including quantifiers ("for all," "there is"). An initial introduction presents an overview of the book. In chapter one, Halperin presents a summary of results from his earlier book, some of which extends into this work. It also contains a novel treatment of the problem of combining evidence: how does one combine two items of interest for a conclusion-each of which separately impart a probability for the conclusion-so as to have a probability for the conclusion basedon taking both of the two items of interest as evidence? Chapter two enlarges the Probability Logic from the first chapter in two respects: the language now includes quantifiers ("for all," and "there is") whose variables range over atomic sentences, notentities as with standard quantifier logic. (Hence its designation: ontological neutral logic.) A set of axioms for this logic is presented. A new sentential notion-the suppositional-in essence due to Thomas Bayes, is adjoined to this logic that later becomes the basis for creating a conditional probability logic. Chapter three opens with a set of four postulates for probability on ontologically neutral quantifier language. Many properties are derived and a fundamental theorem is proved, namely, for anyprobability model (assignment of probability values to all atomic sentences of the language) there will be a unique extension of the probability values to all closed sentences of the language. The chapter concludes by showing the Borel's early denumerableprobability concept (1909) can be justified by its being, in essence, close to Hailperin's probability result applied to denumerable language. The final chapter introduces the notion of conditional-probability to a language having quantifiers of the kind
Author: Ernest Wilcox Adams
Publisher: Stanford Univ Center for the Study
ISBN: 9781575860664
Category : Mathematics
Languages : en
Pages : 376
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Book Description
This book is meant to be a primer, that is an introduction, to probability logic, a subject that appears to be in its infancy. Probability logic is a subject envisioned by Hans Reichenbach and largely created by Adams. It treats conditionals as bearers of conditional probabilities and discusses an appropriate sense of validity for arguments such conditionals, as well as ordinary statements as premises. This is a clear well written text on the subject of probability logic, suitable for advanced undergraduates or graduates, but also of interest to professional philosophers. There are well thought out exercises, and a number of advanced topics treated in appendices, while some are brought up in exercises and some are alluded to only in footnotes. By this means it is hoped that the reader will at least be made aware of most of the important ramifications of the subject and its tie-ins with current research, and will have some indications concerning recent and relevant literature.
Author: Theodore Hailperin
Publisher: Lehigh University Press
ISBN: 9780934223454
Category : Mathematics
Languages : en
Pages : 316
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Book Description
This study presents a logic in which probability values play a semantic role comparable to that of truth values in conventional logic. The difference comes in with the semantic definition of logical consequence. It will be of interest to logicians, both philosophical and mathematical, and to investigators making use of logical inference under uncertainty, such as in operations research, risk analysis, artificial intelligence, and expert systems.
Author: Gabriele Kern-Isberner
Publisher: Springer Science & Business Media
ISBN: 3540253327
Category : Computers
Languages : en
Pages : 230
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Book Description
This book constitutes the thoroughly refereed postproceedings of the International Workshop on Conditionals, Information, and Inference, WCII 2002, held in Hagen, Germany in May 2002. The 9 revised full papers presented together with 3 invited papers by leading researchers in the area were carefully selected during iterated rounds of reviewing and improvement. The papers address all current issues of research on conditionals, ranging from foundational, theoretical, and methodological aspects to applications in various contexts of knowledge representation.
Author: Donald Nute
Publisher: Springer Science & Business Media
ISBN: 9400989660
Category : Philosophy
Languages : en
Pages : 182
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Book Description
Author: W.L. Harper
Publisher: Springer Science & Business Media
ISBN: 9789027706171
Category : Gardening
Languages : en
Pages : 334
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Book Description
Proceedings of an International Research Colloquium held at the University of Western Ontario, 10-13 May 1973.
Author: Bruno de Finetti
Publisher: Springer Science & Business Media
ISBN: 1402082010
Category : Science
Languages : en
Pages : 239
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Book Description
Bruno de Finetti (1906–1985) is the founder of the subjective interpretation of probability, together with the British philosopher Frank Plumpton Ramsey. His related notion of “exchangeability” revolutionized the statistical methodology. This book (based on a course held in 1979) explains in a language accessible also to non-mathematicians the fundamental tenets and implications of subjectivism, according to which the probability of any well specified fact F refers to the degree of belief actually held by someone, on the ground of her whole knowledge, on the truth of the assertion that F obtains.
Author: Ellery Eells
Publisher: Cambridge University Press
ISBN: 9780521453592
Category : Education
Languages : en
Pages : 224
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Book Description
Essays on the state of research investigating the relationship between conditionals and conditional probabilities.
