The Logic of Partial Information

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 Connectives, Ionic Operators and Sets of 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.- 9.2.6Case 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 Partial Information Ionic Logic.- 10.1 Axiomatics and Proof Theory of Propositional Partial Information Ionic Logic.- 10.1.1 Axioms and Proof Rules for Propositional Partial Information Ionic Logic (PIL).- 10.1.1.1 Axioms Inherited from Propositional Logic.- 10.1.1.2 The Propositional Logic of Partial Information Ions: P*-logic.- 10.1.1.3 Axioms that Are Specific to Partial Information Ions.- 10.1.1.4 Proof Rules.- 10.1.2 Lakatosian Logics: IC-logic and J-logic.- 10.1.3 Non-Lakatosian Logics: E-logic and N-logic.- 10.1.3.1 The Logic of Elementary Justifications E.- 10.1.3.2 Truth Maintenance Logic N.- 10.1.3.3 Modal Properties of N-logic.- 10.2 Application of Conjugated Pairs: a Semantic Definition of Possibility and Necessity.- 10.3 Weak Implication in Partial Information Ionic Logic; Proof Theory.- 10.3.1 Proof-Theoretic Properties of Weak Implication.- 10.3.1.1 Transitivity and Modus Tollens.- 10.3.1.2 "a Implies Weakly b" : a ? [b].-10.3.1.3 Weakly "a Implies b" : [a ? b].- 10.3.2 Applications of Weak Implication.- 10.3.2.1 Is Tweety a Bird?.- 10.3.2.2 Is John a Person?.- 10.3.3 Lottery Paradox.- 10.3.4 Use of Disjunctive Information.- 10.3.4.1 Case Analysis Using Two Hard Statements.- 10.3.4.2 Case Analysis Using Two Weak Statements.- 10.3.5 Lukaszewicz Rules as Metatheorems in the IC-logic and the J-logic.- 10.3.6 An Example of Reiter, Criscuolo and Lukaszewicz Revisited in the J-logic.- 10.3.6.1 Model-Theoretic Analysis of the Example.- 10.3.6.2 Proof-Theoretic Analysis of the Example.- 11 Soundness of Propositional Partial Information Ionic Logic.- 11.1 Soundness of Propositional Partial Information Ionic Logic.- 11.1.1 Potential Validity of the Axioms of Propositional IC-logic.- 11.1.2 Potential Validity of the Axioms of Propositional E-logic.- 11.1.3 Potential Validity of Propositional N-logic Axioms.- 11.1.4 Truth Versus Potential Truth of Theorems.- 12 Formal Axiomatics of Propositional Partial Information Ionic Logic.- 12.1 Strengthening the Axioms of Partial Information Ionic Logic.- 12.2 Formal Axiomatics of Ionic Logic.- 12.2.0.1 Strong Axiomatics of Propositional Partial Information Ionic Logic.- 12.2.0.2 Inference Rules.- 12.2.0.3 Weak Axiomatics of Propositional Partial Information Ionic Logic.- 13 Extension and Justification Closure Approach to Partial Information Ionic Logic.- 13.1 Justification Closure and Extensions.- 13.1.1 Justification Closures.- 13.1.2 Extensions in the Sense of a Given Justification Closure.- 13.1.3 Ionic Extensions.- 13.1.4 Examples of Extensions in the Sense of Reiter.- 13.1.5 Comparing Reiter's and Lukaszewicz' Logics.- 13.2 Ionic Models and Extensions.- 13.2.1 A Heuristic for Building Ionic Extensions of Default Theories.- 14 PartialFirst-Order Logic.- 14.1 Partial First-Order Logic.- 14.1.1 The Language of Partial First-Order Logic (FOL).- 14.1.1.1 Alphabet, Terms and Formulae.- 14.1.2 Semantics of Partial First-Order Logic.- 14.1.2.1 The Set of Interpretations for Partial First-Order Logic.- 14.1.2.2 Truth Versus Potential Truth in Partial First-Order Logic.- 14.1.2.3 Truth and Potential Truth Under Some First-Order Valuation.- 14.1.3 Algebraic Properties of Partial First-Order Logic.- 14.1.4 The Generalized Gylindric Algebra of Partial First-Order Logic.- 14.1.5 Beth Tableaux Rules and Entailment in Partial First-Order Logic.- 14.1.5.1 Smullyan's Classification of Signed Quantified Formulae of Partial FOL.- 14.1.5.2 Existential Type Rules for ? Type Formulae.- 14.1.5.3 Universal Type Rules for ? Type Formulae.- 14.1.6 Naive Axiomatics and Proof Theory for Partial First-Order Logic.- 14.1.6.1 Axioms of Partial First-Order Logic.- 14.1.6.2 Soundness of Partial First-Order Logic.- 14.2 Partial First-Order Logic with Equality.- 14.2.1 Objects and Fictions.- 14.2.2 Designating and Potentially Designating Terms.- 14.2.3 Partial FOL with Equality.- 14.2.3.1 Quantifying Over Actual Objects Versus Quantifying Over Potential Objects.- 14.2.3.2 The Language of Partial FOL with Equality.- 14.2.3.3 Truth Versus Potential Truth in Partial First-Order Logic with Equality.- 14.2.3.4 Truth and Potential Truth Under Some First-Order Valuation.- 14.2.3.5 Existence Issues in Partial First-Order Logic with Equality.- 14.2.4 The Generalized Cylindric Algebra of Partial First-Order Logic with Equality.- 14.2.5 Beth Tableaux for Partial FOL with Equality.- 14.2.5.1 Tableaux for Equality.- 14.2.5.2 Tableaux for Quantified Formulae in Partial FOL with Equality.- 15 Syntax and Semantics of First-Order PartialInformation Ions.- 15.1 Syntax of the Language of First-Order Partial Information Ions (FIL).- 15.1.1 Alphabet.- 15.1.2 Terms.- 15.1.3 Formulae of First-Order Partial Information Ionic Logic (FIL).- 15.1.4 Occurrences and Their Justification Prefixes.- 15.2 Towards a Model Theory for First-Order Partial Information Ionic Logic.- 15.3 Interpretation of FIL Formulae.- 15.3.1 Domain ?1 of (First-Order) Interpretations of Rank 1.- 15.3.2 Interpretation of First-Order Ionic Formulae of Rank 1.- 15.3.2.1 Truth.- 15.3.2.2 Soft Truth.- 15.3.3 Example: Sorites Paradox.- 15.3.4 Continuous Bundle ?? for FIL.- 15.4 Algebraic Properties of First-Order Partial Information Ionic Logic.- 15.4.1 The Generalized Cylindric Algebra of First-Order Partial Information Ionic Logic.- 16 Beth Tableaux for First-Order Partial Information Ions.- 16.1 Beth Tableaux for FIL of Rank 1.- 16.1.1 Tableaux Rules for Equality.- 16.1.2 Tableau Rules for Quantification.- 16.1.2.1 Existential Type Rules (Actual and Potential Quantification).- 16.1.2.2 Universal Type Rules.- 16.2 Applications to Reasoning with Partial Information.- 16.2.1 Counter-Example Axioms.- 16.2.2 Separating "Optimism" from Universal Quantification.- 16.2.3 Basic Default Reasoning.- 16.2.4 Default Reasoning with Irrelevant Information.- 16.2.5 Default Reasoning with Incomplete Information.- 16.2.6 Default Reasoning in an Open Domain.- 16.2.7 Default Reasoning with Incomplete Information in an Open Domain.- 16.2.8 Default Reasoning with a Disabled Default.- 16.3 Deriving Presuppositions in Natural Language (First-Order Case).- 16.3.1 Computing Presuppositions of Complex Sentences (First-Order Case).- 16.3.2 Presuppositions of Propositional Logic Structures: the Projection Problem.- 16.3.2.1 Possibly.- 16.3.2.2 Conditional.-16.3.3 Computing Presuppositions of Quantification Logic Structures: the Existential Presupposition Problem.- 17 Axiomatics and Proof Theory of First-Order Partial Information Ionic Logic.- 17.1 Definition of a Formal Deductive System of First-Order Partial Information Ionic Logic (FIL).- 17.1.1 Naive Axiomatics and Proof Theory of First-Order Partial Information Ionic Logic.- 17.1.1.1 Axioms Inherited From Propositional Partial Information Ionic Logic.- 17.1.1.2 Quantification Logic Axioms Inherited From First-Order Logic.- 17.1.1.3 Axioms That Are Specific to First-Order Partial Information Ions.- 17.1.1.4 Proof Rules.- 17.1.1.5 Lakatosian Versus Non-Lakatosian First-Order Logics.- 17.2 Weak Implication in First-Order Partial Information Ionic Logic.- 17.2.1 Sorites Paradox.- 17.2.2 The Yale Shooting Problem Revisited.- 17.3 Potential Validity.- 18 Partial Information Ionic Logic Programming.- 18.1 Propositional Partial Information Logic Programming.- 18.1.1 Syntax of Propositional Partial Information Logic Programs.- 18.1.2 Derivation Steps.- 18.1.3 Least Fixpoint Semantics of Propositional Partial Information Logic Programs in Terms of the T Operator.- 18.2 First-Order Partial Information Logic Programming.- 18.2.1 Syntax of First-Order Partial Information Logic Programs.- 18.2.2 Derivation Steps.- 18.2.3 Least Fixpoint Semantics of First-Order Partial Information Logic Programs in Terms of the T Operator.- 18.3 Applications of First-Order Logic Programs.- 18.3.1 Undesirable Properties of Skolemization in Reiter's Default Logic.- 18.3.2 Poole's Logical Framework for Default Reasoning.- 18.3.2.1 Poole's Programs As Logic Fields.- 18.3.2.2 Poole's Programs As Partial Information Logic Programs.- 18.3.3 Reasoning About Unknown Actions.- 19 Syntactic andSemantic Paths; Application to Defeasible Inheritance.- 19.1 Syntactic Paths and Semantic Paths.- 19.1.1 Regular Models and Continuous Models of Propositional Partial Information Logic Programs.- 19.1.1.1 Syntactic Paths, Semantic Paths.- 19.1.1.2 Semantic Paths in Sets of Interpretation Schemes.- 19.1.1.3 Constrained Regular Models.- 19.1.2 Regular Models and Continuous Models of First-Order Partial Information Logic Programs.- 19.2 Application: the Axiomatization of Multiple Defeasible Inheritance.- 19.2.1 Sandewall's Primitive Structures.- 19.2.2 Sandewall's Structures As a Path Rule.- 20 The Frame Problem: The Dynamics of Logic Systems.- 20.1 The Dynamics of Logic Systems.- 20.1.1 Towards a Least Action Principle for the Dynamics of Logic Systems.- 20.1.2 The Oceania Problem.- 20.1.2.1 The Global Approach to the Oceania Problem.- 20.1.2.2 Deductive Sequence Approach to the Oceania Problem.- 20.1.2.3 Dynamic Approach to the Oceania Problem.- 20.1.3 The Characteristic Surface of a Dynamic Logic System.- 20.1.3.1 Syntactic Paths.- 20.1.3.2 Variety Defined by a Syntactic Path.- 20.1.3.3 Characteristic Surface of a Syntactic Path.- 20.1.3.4 Semantic Paths in a Variety.- 20.1.3.5 Semantic Paths on a Charateristic Surface.- 20.1.3.6 Galois Connection Between the Variety of a Syntactic Path and Its Characteristic Surface.- 20.1.4 The Least Action Principle of the Dynamics of Logic Systems.- 20.2 The Marathon Problem.- 20.2.1 Operational Semantics of the Marathon Problem.- 20.2.2 Least Fixpoint of the Marathon Problem.- 20.2.3 Deductive Sequence Approach to the Marathon Problem.- 20.2.4 Dynamic Sequence Approach to the Marathon Problem.- 20.2.5 Practical Meaning of the Models Obtained.- 20.3 The Vanishing Car Problem.- 20.4 The Yale Shooting Problem, Frame Problem forTemporal Projection.- 20.4.1 The Global Approach to the Yale Shooting Problem, and Its Weaknesses.- 20.4.1.1 Hanks and McDermott's "Construction".- 20.4.1.2 Making Sure Fred Dies: Morris' Formalization of the Yale Shooting Problem.- 20.4.1.3 Extensional Supports in the YSP.- 20.4.1.4 Should Fred Actually Die?.- 20.4.2 Operational Semantics of the Yale Shooting Problem.- 20.4.3 Least Fixpoint of the Yale Shooting Problem.- 20.4.3.1 Minimal Models of the Yale Shooting Problem.- 20.4.3.2 Deductive Sequence Approach to the Yale Shooting Problem.- 20.4.3.3 Phase Diagram of the Yale Shooting Problem.- 20.4.4 Dynamic Approach to the Yale Shooting Problem.- 20.5 Reasoning About Actions Within the Framework of Extensions and Justification Closures.- 20.5.1 The Extension and Justification Closure Approach to the YSP.- 20.5.2 The Extension and Justification Closure Approach to the Marathon Problem.- 21 Reasoning About Actions: Projection Problem.- 21.1 Modified Frame Problem for Temporal Projection.- 21.2 The Assassin Problem.- 21.2.1 Least Fixpoint and Minimal Models of the Assassin Problem.- 21.2.2 Deductive Sequence Approach to the Assassin Problem.- 21.3 Forcing Discontinuity Into the Yale Shooting Problem.- 21.3.1 First Discontinuous YSP.- 21.3.2 Second Discontinuous YSP: Separating the Deductive Approach from the Dynamic Approach.- 21.4 The Spectre Problem (Temporal Explanation).- 21.4.1 Least Fixpoint of the Spectre Problem.- 21.4.2 Minimal Models of the Spectre Problem.- 21.4.2.1 Dynamic Sequence of the System of the Spectre Problem.- 21.5 The Robot Problem.- 21.5.1 Least Fixpoint of the Robot Problem.- 21.5.2 Minimal Models of the Robot Problem.- 21.5.2.1 Deductive Sequence Approach.- 21.5.2.2 Dynamic Sequence Approach.- 21.5.3 Diagnosing the Anomalous Behaviourof the Robot.- 21.5.3.1 The Robot is Observed Moving Forward.- 21.5.3.2 The Robot is Observed Moving Backward.- 21.5.3.3 The Robot is Observed Moving.- 21.6 The Yale Shooting Problem, Temporal Projection.- 21.6.1 Least Fixpoint of the Temporal Projection Problem.- 21.6.2 Minimal Models of the Temporal Projection Problem.- 21.6.3 Continuous Model of the Temporal Projection Problem.- 21.7 Reasoning About the Unknown Order of Actions.- 22 Reasoning About Actions: Explanation Problem.- 22.1 The Explanation Problem.- 22.2 The Abductive Variational Principle for Reasoning About Actions.- 22.2.1 The Generation of the Variation by Means of Abduction.- 22.2.2 The Abduction Principle in Partial Information Logic.- 22.2.2.1 The Abduction Inference Rule.- 22.2.2.2 The Abduction Principle.- 22.2.3 Algorithmic Description of the Abductive Variational Principle.- 22.2.3.1 The Generation of the Variation and of the "Nearby" Logic Program.- 22.2.3.2 Application of the Least Action Principle to the Variational Dynamic Sequence.- 22.2.3.3 Fixing the Ending Point of the Variational Syntactic Path.- 22.2.3.4 Applying the Abductive Variational Principle.- 22.3 Application of the Abductive Variational Principle.- 22.3.1 Generating the "Nearby" Program P'.- 22.3.2 Application of the Least Action Principle to the Variational Dynamic Sequence.- 22.3.3 Application of the Abductive Variational Principle for Reasoning About Actions.- 22.4 The Murder Mystery Problem (Temporal Explanation Problem).- 22.4.1 Operational Semantics of the Murder Mystery Problem.- 22.4.2 Least Fixpoint of the Murder Mystery Problem (Temporal Explanation).- 22.4.3 Application of the Variational Principle to the Murder Mystery Problem.- 22.4.4 Generation of the "Nearby" Program.- 22.4.5 Application of theLeast Action Principle to the Variational Dynamic Sequence.- 22.4.6 Fixing the Ending Point of the Variational Logic System.- 22.5 Dynamics of Logic Systems and Psychological Processes.