Ebook: Logical Structures for Representation of Knowledge and Uncertainty

To answer questions concerning previously supplied information the book uses a truth table or 'chain set' logic which combines probabilities with truth values (= possibilities of fuzzy set theory). Answers to questions can be 1 (yes); 0 (no); m (a fraction in the case of uncertain information); 0m, m1 or 0m1 (in the case of 'ignorance' or insufficient information). Ignorance (concerning the values of a probability distribution) is differentiated from uncertainty (concerning the occurrence of an outcome). An IF THEN statement is interpreted as specifying a conditional probability value. No predicate calculus is needed in this probability logic which is built on top of a yes-no logic. Quantification sentences are represented as IF THEN sentences with variables. No 'forall' and 'exist' symbols are needed. This simplifies the processing of information. Strange results of first order logic are more reasonable in the chain set logic. E.g., (p->q) AND (p->NOTq), p->NOT p, (p->q)->(p->NOT q), (p->q)- >NOT(p->q), are contradictory or inconsistent statements only in the chain set logic. Depending on the context, two different rules for the updating of probabilities are shown to exist. The first rule applies to the updating of IF THEN information by new IF THEN information. The second rule applies to other cases, including modus ponens updating. It corresponds to the truth table of the AND connective in propositional calculus. Many examples of inferences are given throughout the book.

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To answer questions concerning previously supplied information the book uses a truth table or 'chain set' logic which combines probabilities with truth values (= possibilities of fuzzy set theory). Answers to questions can be 1 (yes); 0 (no); m (a fraction in the case of uncertain information); 0m, m1 or 0m1 (in the case of 'ignorance' or insufficient information). Ignorance (concerning the values of a probability distribution) is differentiated from uncertainty (concerning the occurrence of an outcome). An IF THEN statement is interpreted as specifying a conditional probability value. No predicate calculus is needed in this probability logic which is built on top of a yes-no logic. Quantification sentences are represented as IF THEN sentences with variables. No 'forall' and 'exist' symbols are needed. This simplifies the processing of information. Strange results of first order logic are more reasonable in the chain set logic. E.g., (p->q) AND (p->NOTq), p->NOT p, (p->q)->(p->NOT q), (p->q)- >NOT(p->q), are contradictory or inconsistent statements only in the chain set logic. Depending on the context, two different rules for the updating of probabilities are shown to exist. The first rule applies to the updating of IF THEN information by new IF THEN information. The second rule applies to other cases, including modus ponens updating. It corresponds to the truth table of the AND connective in propositional calculus. Many examples of inferences are given throughout the book.

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To answer questions concerning previously supplied information the book uses a truth table or 'chain set' logic which combines probabilities with truth values (= possibilities of fuzzy set theory). Answers to questions can be 1 (yes); 0 (no); m (a fraction in the case of uncertain information); 0m, m1 or 0m1 (in the case of 'ignorance' or insufficient information). Ignorance (concerning the values of a probability distribution) is differentiated from uncertainty (concerning the occurrence of an outcome). An IF THEN statement is interpreted as specifying a conditional probability value. No predicate calculus is needed in this probability logic which is built on top of a yes-no logic. Quantification sentences are represented as IF THEN sentences with variables. No 'forall' and 'exist' symbols are needed. This simplifies the processing of information. Strange results of first order logic are more reasonable in the chain set logic. E.g., (p->q) AND (p->NOTq), p->NOT p, (p->q)->(p->NOT q), (p->q)- >NOT(p->q), are contradictory or inconsistent statements only in the chain set logic. Depending on the context, two different rules for the updating of probabilities are shown to exist. The first rule applies to the updating of IF THEN information by new IF THEN information. The second rule applies to other cases, including modus ponens updating. It corresponds to the truth table of the AND connective in propositional calculus. Many examples of inferences are given throughout the book.
Content:
Front Matter....Pages I-XXIII
Introduction....Pages 1-31
Front Matter....Pages 32-32
Chain Set and Probability Overview....Pages 33-51
BP Chain Sets I, Affirmation, Negation, Conjunction, Disjunction....Pages 52-69
BP Chain Sets II, Special Cases of Chain Sets....Pages 70-102
BP Chain Sets III, Precise Formulations....Pages 103-130
Inferences or the Answering of Questions....Pages 131-164
Inferences with Higher Level Chain Sets....Pages 165-181
IF THEN Information....Pages 182-204
Various IF THEN Topics....Pages 205-223
Front Matter....Pages 224-224
The M-Notation and Ignorance vs Uncertainty....Pages 225-242
Two Types of Updating of Probabilities....Pages 243-255
Operations and Ignorance in the M Logic....Pages 256-280
Modus Ponens and Existence Updating....Pages 281-300
IF THEN Information in the M Logic....Pages 301-325
Existence Structures....Pages 326-345
Existence Inferences....Pages 346-355
Conditional and Joint Existence Information and Inferences....Pages 356-379
Front Matter....Pages 380-380
Attributes and the Alex System versus Chain Sets....Pages 381-386
Solutions to Some Exercises....Pages 387-398
Back Matter....Pages 399-419

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To answer questions concerning previously supplied information the book uses a truth table or 'chain set' logic which combines probabilities with truth values (= possibilities of fuzzy set theory). Answers to questions can be 1 (yes); 0 (no); m (a fraction in the case of uncertain information); 0m, m1 or 0m1 (in the case of 'ignorance' or insufficient information). Ignorance (concerning the values of a probability distribution) is differentiated from uncertainty (concerning the occurrence of an outcome). An IF THEN statement is interpreted as specifying a conditional probability value. No predicate calculus is needed in this probability logic which is built on top of a yes-no logic. Quantification sentences are represented as IF THEN sentences with variables. No 'forall' and 'exist' symbols are needed. This simplifies the processing of information. Strange results of first order logic are more reasonable in the chain set logic. E.g., (p->q) AND (p->NOTq), p->NOT p, (p->q)->(p->NOT q), (p->q)- >NOT(p->q), are contradictory or inconsistent statements only in the chain set logic. Depending on the context, two different rules for the updating of probabilities are shown to exist. The first rule applies to the updating of IF THEN information by new IF THEN information. The second rule applies to other cases, including modus ponens updating. It corresponds to the truth table of the AND connective in propositional calculus. Many examples of inferences are given throughout the book.
Content:
Front Matter....Pages I-XXIII
Introduction....Pages 1-31
Front Matter....Pages 32-32
Chain Set and Probability Overview....Pages 33-51
BP Chain Sets I, Affirmation, Negation, Conjunction, Disjunction....Pages 52-69
BP Chain Sets II, Special Cases of Chain Sets....Pages 70-102
BP Chain Sets III, Precise Formulations....Pages 103-130
Inferences or the Answering of Questions....Pages 131-164
Inferences with Higher Level Chain Sets....Pages 165-181
IF THEN Information....Pages 182-204
Various IF THEN Topics....Pages 205-223
Front Matter....Pages 224-224
The M-Notation and Ignorance vs Uncertainty....Pages 225-242
Two Types of Updating of Probabilities....Pages 243-255
Operations and Ignorance in the M Logic....Pages 256-280
Modus Ponens and Existence Updating....Pages 281-300
IF THEN Information in the M Logic....Pages 301-325
Existence Structures....Pages 326-345
Existence Inferences....Pages 346-355
Conditional and Joint Existence Information and Inferences....Pages 356-379
Front Matter....Pages 380-380
Attributes and the Alex System versus Chain Sets....Pages 381-386
Solutions to Some Exercises....Pages 387-398
Back Matter....Pages 399-419
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