عنوان

The Logic of Partial Information

(Number (ISBN

3642781608

(Number (ISBN

9783642781605

Number

b573780

Title Proper

The Logic of Partial Information

General Material Designation

[Book]

First Statement of Responsibility

by Areski Nait Abdallah.

Place of Publication, Distribution, etc.

Berlin, Heidelberg

Name of Publisher, Distributor, etc.

Springer Berlin Heidelberg

Date of Publication, Distribution, etc.

1995

Specific Material Designation and Extent of Item

(XXV, 715 p.).

Series Title

Monographs in Theoretical Computer Science An EATCS Series.

Text of Note

Justifications --; 8.3.3 Derived Beth Tableaux Rules for Canonical Justification Formulae of Rank 1 --; 8.3.4 Closure Conditions for Beth Tableaux in Partial Information Ionic Logic --; 8.3.5 Closure Properties of Beth Tableaux --; 8.3.5.1 Closure Properties Inherited from Partial Propositional Logic --; 8.3.5.2 Closure Properties "Soft. Knowledge Extends Hard Knowledge" --; 8.3.5.3 Closure Properties "Justification Knowledge Extends Hard Knowledge" --; 8.3.5.4 General Closure Rules for Justifications --; 8.3.5.5 Closure Properties for Connectives in Elementary Canonical Justifications --; 8.3.6 Syntactic Entailment, Soundness of the Tableau Method for Ionic Logic --; 8.3.7 Sorted Patterns of Rank 1, and Their Satisfaction --; 8.3.7.1 Simple Patterns --; 8.3.8 The Continuity of the Beth Tableau Technique for Partial Information Ionic Logic --; 9 Applications; the Statics of Logic Systems --; 9.1 The Statics of Logic Systems --; 9.2 Weak Implication in Partial Information Ionic Logic; Tableaux and Model Theory --; 9.2.1 Introduction to Weak Implication --; 9.2.2 Formal Properties of Weak Implication --; 9.2.3 Applications of Weak Implication --; 9.2.3.1 Example 1: Is Tweety a Bird? --; 9.2.3.2 Example 2: Is John a Person? --; 9.2.4 Contraposition --; 9.2.5 Lottery Paradox: Models --; 9.2.6 Case Analysis Using Two Strong Statements: Tableaux and Models --; 9.2.7 Case Analysis Using Two Weak Statements --; 9.3 Truth Maintenance --; 9.4 Expressing Partialness of Information Using Partial Information Ions --; 9.5 The Heisenberg Principle and Quantum Mechanics --; 9.5.1 Heisenberg's Principle and Quantum Mechanics --; 9.5.2 Specializing the Value of the Conditional Ionic Operator Into * = ? --; 9.5.3 General Structure of the Electron Interference Problem --; 9.6 Alexinus and Menedemus Problem --; 9.7 Deriving Presuppositions in Natural Language --; 9.7.1 Presuppositions and Partial Information Logic --; 9.7.2 Defining a Formal Notion of Presupposition in Partial Information Logic --; 9.7.3 A Semantic Definition of Presuppositions --; 9.7.4 Computing Presuppositions of Complex Sentences --; 10 Naive Axiomatics and Proof Theory of Propositional Pa.

Text of Note

1 Introduction --; 1.1 Introduction --; 1.1.1 The Logic of Non-monotonie Reasoning --; 1.1.1.1 Practical Problems --; 1.1.1.2 Theoretical Problems --; 1.1.2 Changing Paradigms: The Logic of Reasoning with Partial Information --; 1.2 Principles of Our Approach --; 1.2.1 The Separation Between Hard Knowledge, Justification Knowledge and Tentative Knowledge --; 1.2.2 Partial Information and Partial Models --; 1.3 Conclusion --; 2 Partial Propositional Logic --; 2.1 Syntax and Semantics of Partial Propositional Logic --; 2.1.1 Syntax of (Partial) Propositional Logic --; 2.1.2 Semantics of Partial Propositional Logic --; 2.1.2.1 Partial Interpretations for Propositional Logic --; 2.1.2.2 The Set of Interpretations for Partial Propositional Logic --; 2.1.2.3 Truth Versus Potential Truth in Partial Propositional Logic --; 2.1.2.4 Truth of Propositional Formulae Under Some Valuation --; 2.1.2.5 Potential Truth Under Some Valuation --; 2.1.3 Algebraic Properties of Partial Propositional Logic --; 2.1.3.1 Semantic Scope in Partial Propositional Logic --; 2.1.3.2 The Generalized Boolean Algebra of Partial Propositional Logic --; 2.1.3.3 Saturated Pairs of Sets --; 2.1.4 Semantic Entailment --; 2.2 Beth Tableau Method for Partial Propositional Logic --; 2.2.1 Beth Tableau Rules for Partial Propositional Logic; Syntactic Entailment --; 2.2.1.1 Beth Tableaux for Negation --; 2.2.1.2 Beth Tableaux for the Bottom Function --; 2.2.1.3 Beth Tableaux for Conjunction --; 2.2.1.4 Beth Tableaux for Disjunction --; 2.2.1.5 Beth Tableaux for Implication --; 2.2.1.6 Beth Tableaux for Interjunction --; 2.2.1.7 Closure Conditions for Partial Propositional Logic Formulae --; 2.2.1.8 Linear Representation of Beth Tableaux --; 2.2.1.9 Syntactic Entailment, Soundness and Completeness of the Tableau Method --; 2.3 Axiomatization of Partial Propositional Logic --; 2.3.1 A Formal Deductive System with Axioms and Proof Rules for Partial Propositional Logic --; 2.3.1.1 Generalizing Classical Propositional Logic --; 2.3.2 Strong Theorems Versus Weak Theorems --; 2.3.2.1 Strong Axiomatics of Partial Propositional Logic --; 2.3.2.2 Weak Axiomatics for Partial Propositional Logic --; 2.3.3 Monotonicity Issues in Partial Propositional Logic --; 3 Syntax of the Language of Partial Information Ions --; 3.1 The Language of Partial Information Ions --; 3.1.1 Partial Information Ions --; 3.1.2 Alphabet --; 3.1.3 Formulae of Propositional Partial Information Ionic Logic --; 3.1.4 Occurrences and Their Justification Prefixes --; 3.1.4.1 Occurrences --; 3.1.4.2 Justification-bound and Justification-free Occurrences --; 3.1.4.3 Prefix and Justification Prefix of a Formula --; 3.1.4.4 Rank of a Formula --; 4 Reasoning with Partial Information Ions: An Overview --; 4.1 From Reasoning with Total Information to Reasoning with Partial Information --; 4.2 Reasoning with Partial Information in Propositional Logic --; 4.3 Global Approach to Reasoning with Partial Information Ions --; 4.4 Reasoning with Partial Information in First-order Logic --; 4.5 The Dynamics of Logic Systems: Is There a Logical Physics of the World? --; 4.5.1 Using the Least Action Principle --; 4.5.2 Combining the Least Action Principle with Abduction: An Abductive Variational Principle for Reasoning About Actions --; 4.6 A Geometric View of Reasoning with Partial Information --; 4.6.1 Static Logic Systems --; 4.6.2 Dynamic Logic Systems --; 4.7 Conclusion --; 5 Semantics of Partial Information Logic of Rank 1 --; 5.1 Towards a Model Theory for Partial Information Ionic Logic --; 5.2 The Domain ?1 of Ionic Interpretations of Rank 1 --; 5.3 The Semantics of Partial Information Ions of Rank 1 --; 5.3.1 The Semantics of Ionic Formulae of Rank 1 --; 5.3.1.1 Truth of Formulae with Respect to Sets of Valuations --; 5.3.2 Canonical Justifications and Conditional Partial Information Ions --; 5.3.2.1 Acceptability, Conceivabihty of Propositional Formulae --; 5.3.2.2 Canonical Justification Formulae and Their Interpretation --; 5.3.2.3 Acceptability and Conceivabihty as Levels of Truth --; 5.3.2.4 Acceptable and Unacceptable Elementary Canonical Justification Formulae; Semantics of Partial Information Ions --; 5.3.2.5 Semantics of Conditional Ions --; 5.3.3 Canonical Justification Declarations and Coercion Ions --; 5.4 Interpretation of Propositional Ionic Formulae of Rank 1 --; 5.4.1 Acceptance, Rejection of a Justification by a Conditional Ion --; 5.4.2 Truth Versus Potential Truth in Partial Information Ionic Logic --; 5.4.3 Truth of Ionic Formulae of Rank 1 --; 5.4.3.1 Plain Truth: ?? --; 5.4.3.2 Plain Potential Truth: ?? --; 5.4.4 Soft Truth of Ionic Formulae of Rank 1 --; 5.4.4.1 Soft Truth: ?soft? --; 5.4.4.2 Soft Potential Truth: ?soft? --; 5.4.5 Semantic Entailments and Equivalence --; 5.4.6 Decomposition of Conditional Partial Information Ions into Elementary Justifications and Soft Formulae --; 5.4.7 Truth and the Information Ordering --; 5.4.7.1 Acceptable Versus Unacceptable Justifications --; 5.4.8 Elementary Justifications Versus Canonical Justifications of Rank 1 --; 5.4.8.1 The Semantics of Elementary Justifications (Universal Ions Case) --; 5.4.8.2 The Semantics of Elementary Justifications (Existential-Universal Ions Case) --; 6 Semantics of Partial Information Logic of Infinite Rank --; 6.1 The Continuous Bundle ?? of Ionic Interpretations --; 6.1.1 The Category of Continuous Bundles --; 6.1.2 Ionic Interpretations and Continuous Bundles --; 6.1.3 The Projective/Injective System --; 6.2 Interpretation of Propositional Partial Information Ionic Formulae --; 7 Algebraic Properties of Partial Information Ionic Logic --; 7.1 Scopes and Boolean Algebra --; 7.1.1 Semantic Scopes --; 7.1.1.1 Semantic Scope --; 7.1.1.2 Potential Semantic Scope --; 7.1.1.3 Semantic Scope Ordering Between Formulae --; 7.1.2 Justifiability Scope --; 7.1.3 The Generalized Boolean Algebra of Propositional Partial Information Ionic Logic --; 7.1.4 Warrant Scope --; 7.1.4.1 The Semantics of Elementary Justifications (Existential-Universal Ions Case) --; 7.2 Orderings on Ionic Interpretations; Interpretation Schemes --; 7.2.1 Quasi-Orderings and Partial Orderings --; 7.2.2 Justification Orderings --; 7.2.2.1 Justification Ordering --; 7.2.2.2 Justification Ordering with Respect to a Given Set of Formulae, Single Operator Case --; 7.2.2.3 Justification Ordering with Respect to a Given Set of Formulae, General Case --; 7.2.3 Warrant Orderings --; 7.2.3.1 Warrant Ordering, Interpretation Schemes and Model Schemes --; 7.2.3.2 Warrant Equivalence with Respect to a Given Set of Formulae, Single Operator Case --; 7.2.3.3 On the Non-Monotonicity of Truth with Respect to the Warrant Ordering --; 7.2.4 Default Orderings on Ionic Interpretations --; 7.2.4.1 Default Ordering --; 7.3 Semiotic Orderings and Galois Connection --; 7.3.1 Semiotic Ordering on Justification Equivalence Classes --; 7.3.2 Semiotic Ordering on Warrant Equivalence Classes; Galois Connection --; 7.3.3 Semiotic Ordering with Respect to a Given Set of Justifications --; 8 Beth Tableaux for Propositional Partial Information Ionic Logic --; 8.1 Semantic Entailment in Propositional Ionic Logic --; 8.1.1 Satisfaction of General Signed Formulae --; 8.1.2 Semantic Entailment in Propositional Partial Information Ionic Logic --; 8.2 Beth Tableaux in Propositional Partial Information Ionic Logic --; 8.2.1 Tableau Rules for Conditional Partial Information Ions --; 8.2.1.1 Beth Tableaux for Universal Ions --; 8.2.1.2 Beth Tableaux for Existential-Universal Ions --; 8.2.1.3 Beth Tableaux for Universal-Existential Ions --; 8.2.1.4 Beth Tableaux for Canonical Justification Formulae with Sets --; 8.2.2 Beth Tableaux for Coercion Partial Information Ions --; 8.3 The General Tableau Method for Propositional Ionic Logic --; 8.3.1 General Tableau Rules for Quantification in Canonical Justifications --; 8.3.2 General Tableau Rules for Propositional Logic Connectives, Ionic Operators and Sets of

Text of Note

One must be able to say at all times - in­ stead of points, straight lines, and planes - tables, chairs and beer mugs. (David Hilbert) One service mathematics has rendered the human race. It has put common sense back where it belongs, on the topmost shelf next to the dusty canister labelled "discarded nonsense. " (Eric T. Bell) This book discusses reasoning with partial information. We investigate the proof theory, the model theory and some applications of reasoning with par­ tial information. We have as a goal a general theory for combining, in a principled way, logic formulae expressing partial information, and a logical tool for choosing among them for application and implementation purposes. We also would like to have a model theory for reasoning with partial infor­ mation that is a simple generalization of the usual Tarskian semantics for classical logic. We show the need to go beyond the view of logic as a geometry of static truths, and to see logic, both at the proof-theoretic and at the model-theoretic level, as a dynamics of processes. We see the dynamics of logic processes bear with classical logic, the same relation as the one existing between classical mechanics and Euclidean geometry.

Parallel Title

Logic of Partial Information

Computer algorithms.

Computer science.

Programming languages (Electronic computers) -- Semantics.

Class number

Book number

by Areski Nait Abdallah.

Areski Nait Abdallah

Electronic name

[Book]

Y