The Logic of Partial Information

This monograph presents the foundations of reasoning with partial information and a theory of common-sense reasoning based on monotonic logic and partial structures. It points out that customary expositions of common-sense reasoning in Artificial Intelligence are based on a flawed non-monotonic reasoning paradigm.

Inhalt
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 AFormal 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.- 5.3.1.1Truth 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 IonicFormulae.- 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 InformationIons.- 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 Connectiv…