The Logic of Partial Information

ISBN-13: 9783642781629 / Angielski / Miękka / 2011 / 715 str.

Areski Nait Abdallah

Opis

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."

Spis treści

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 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 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 Partial First-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 Partial Information 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 and Semantic 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 for Temporal 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 Behaviour of 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 the Least 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.