Temporal Logic

This book is an introduction to temporal logic, a now flourishing branch of philosophical logic whose origin is of recent date, its main impetus having been provided by the publication in the late 1950s of A. N. PRIOR'S pioneering book, Time and Modality (Oxford, The Clarendon Press, 1957). Virtually all work in the field to around 1966 is surveyed in PRIOR'S elegant treatise Past, Present and Future (Oxford, The Clarendon Press, 1967). In consequence, it is no simple matter to write a comprehensive book on the subject with­ out merely rehearsing material already dealt with in PRIOR'S works. We believe, however, that the present book succeeds in this difficult endeavor because it approaches established materials from wholly novel points of departure, and is thus able to attain new perspectives and achieve new results. Its introductory character notwithstanding, the present work is consequently in substantial measure devoted to an exposition of new findings and a demonstration of new results. Parts of the book have been published previously. Chapter II is a modified version of an article of the same title by N. RESCHER and JAMES GARSON in The Journal of Symbolic Logic (vol. 33 [1968], pp. 537-548). And Chapter XIII is a modified version of the article "Temporally Conditioned Descriptions" by N. RESCHER and JOHN ROBISON in Ratio, vol. 8 (1966), pp. 46-54. The authors are grateful to Professors GARSON and ROBISON, and to the editors of the jounal involved, for their permission to use this materials here.

I The Background of Temporal Logic.- II Topological Logic.- 1. Introduction.- 2. The P-Operator.- 3. Three Basic Axioms.- 4. The Relation of P-Unqualified to P-Qualified Formulas: The Preferred Position ? : A Fourth Axiom.- 5. The Iteration of P: A Fifth Axiom and the Two Systems PI and PII.- 6. The Possible Worlds Interpretation of Topological Logic.- III Fundamental Distinctions for Temporal Logic.- 1. The Temporal Equivocality of IS.- 2. Translating Temporal to Atemporal IS.- 3. Temporally Definite and Indefinite Statements.- 4. The Implicit Ubiquity of Now in Tensed Statements.- 5. Dates and Pseudo-Dates.- 6. Times of Assertion.- 7. Two Styles of Chronology.- IV The Basic System R of Temporal Logic.- 1. The Concept of Temporal Realization.- 2. The Temporal Transparency of Now .- 3. Temporal Homogeneity.- 4. Axioms for the Logical Theory of Chronological Propositions.- 5. Temporal and Topological Logic.- 6. The Completeness and Decidability of R.- V The Introduction of Tense Operators.- 1. Presentness and Precedence.- 2. Tense.- VI The System Kt of Minimal Tense Logic.- 1. The Problem of a Minimal Tense Logic.- 2. Semantics for Tense Logic.- 3. Semantic Tableaux.- 4. The Completeness of Kt.- 5. Some Corollaries.- 6. Completeness of Kt with Respect to R.- VII Branching Time: The System Kb.- 1. Branching Structures.- 2. The Concept of an Open Future.- 3. The Logic of Branching Time.- 4. Axiomatization of Kb.- 5. Semantic Tableaux.- 6. Systematic Tableaux.- 7. Completeness Proof for Kb.- VIII Linear Time: The System Kl and Its Variants.- 1. The Logic of Linear Time, Kl.- 2. Extensions of Kl.- IX Additive Time: The Systems R? and R?±.- 1. Temporal-Groups and the System R?.- 2. Additive Temporal Logic and the System R?±.- X Metric Time andChronological Logic: The System R+.- 1. The Concept of Metric Time.- 2. Deriving a U-Relation from the Metric.- 3. The System R+.- 4. Distance into Past and Future.- 5. Archimedeanism.- 6. Linear Realizability.- XI Tense Logical Characterizability and Definability.- 1. Expressibility and Characterizability.- 2. Tense-Logical Definability.- XII Temporal Modality.- 1. The Tensed Interpretation of Modality.- 2. Modality in Tense Logic.- 3. Further Definitions of Modality.- XIII Temporally Conditioned Descriptions and the Concept of Temporal Purity.- 1. Temporally Conditioned Descriptions.- 2. Chronological Purity.- 3. The Purely Phenomenological Characterization of the Occurrences of a Moment.- 4. The Absolute vs. the Relative Conception of Time.- XIV The Theory of Processes.- 1. What is a Process?.- 2. The Representation of Processes: Process Implications.- 3. Activities and Processes: Some Applicable Distinctions.- 4. Quasi-Processes: On Coming to Be and Passing Away.- 5. Stochastic vs. Deterministic Processes.- 6. Stochastic Processes and Branching Time.- 7. The Structure of Events.- XV The Logic of World States.- 1. The Concept of a World State.- 2. Some Further Perspectives on Instantaneous World States.- 3. The Concept of a World History.- 4. Development of R-calculi Within Tense Logic.- XVI The Dimensionality of Time.- XVII The Master Argument of Diodorus and Temporal Determinism.- 1. The Master Argument .- 2. Necessity and Determinism in the Context of the Master Argument .- 3. Evading the Deterministic Conclusion of the Master Argument .- 4. The Groundwork of a 3-Valued Conception of Temporal Truth.- 5. Alternative Futures and Future Contingency.- 6. Temporal Determination.- 7. Nomological Necessitation.- XVIII Many-ValuedApproaches to Temporal Logic.- 1. A Mode of Three-Valued Tense Logic.- 2. A Many-Valued Articulation of Temporal Logic.- 3. A Three-Valued Tense Logic: Semantic Considerations.- 4. Generalizing the Preceding Approach.- XIX Propositional Quantification in Tensed Statements.- XX Quantification, Temporal Existence, and Identity.- 1. Individuals and Quantifiers.- 2. Quantification and Temporal Modality.- 3. Quantified Tense Logic.- 4. Temporal Change, Identity, and Leibniz Law.- 5. Alternative Histories in Branching Time.- 6. Quantified Modal Logic in Branching Time.- Appendix I A Summary of Axiom Systems for Topological, Temporal and Modal Logics.- Appendix II The Modal Structure of Tense-Logical Systems.- Bibliography of Temporal Logic.- A. Chronological Listing.- B. Author Listing (Alphabetical).- Index of Names.