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Formal Semantics and Logic Bas C. van Fraassen Copyright c 1971, Bas C. van Fraassen Originally published by The Macmillan Company, New York This eBook was published by Nousoul Digital Publishers. Its formatting is optimized for eReaders and other electronic reading devices. For information, the publisher may be con- tacted by email at: [email protected] 2 To my parents 3 Preface to the .PDF Edition With a view to the increasing academic importance of digital media this electronic edition was created by Nousoul Digital Publishers. Thanks to the diligent work and expertise of Brandon P. Hopkins this edition has fea- tures that no book could have in the year of its original publication: searchable text and hyperlinked notes. The text itself remains essentially unchanged, but in addition to typographical corrections there are also some substantive corrections. Apart from the change in the solution to exercise 5.4 of Chapter 3, none comprise more than a few words or symbols. However, as different as digital media are from print media, so too is digital for- matting different from print formatting. Thus there are significant formatting differences from the earlier edition. The font and page dimensions differ, as well as the page numbering, which is made to accord with the pagina- tion automatically assigned to multi-paged documents by most standard document-readers. Bas van Fraassen 2016 4 Contents Preface (1971)9 Introduction: Aim and Structure of Logical Theory 12 1 Mathematical Preliminaries 19 1.1 Intuitive Logic and Set Theory....... 19 1.2 Mathematical Structures.......... 23 1.3 Partial Order and Trees.......... 29 1.4 Mathematical Induction.......... 35 1.5 Algorithms.................. 38 2 Structure of Formal Languages 45 2.1 Logical Grammar.............. 45 2.2 Syntactic Systems.............. 48 2.3 Semantic Concepts............. 53 2.4 Valuation Space of a Language....... 58 5 2.5 Semantic Entailment and Axiomatizability 68 2.6 Theory of Deductive Systems....... 72 2.7 System Complementation and Axiomatizability............ 77 2.8 Filters and the Compactness Problem... 85 2.9 Ultraproducts and the Compactness Problem........... 98 2.10 Partial Valuations and the Compactness Problem......... 107 3 Appraisal of Logical Systems 115 3.1 Logical Systems............... 116 3.2 Classical Propositional Logic: Axiomatics. 125 3.3 Classical Propositional Logic: Soundness and Completeness........ 132 3.4 Interpretations of Logical Systems..... 139 3.5 Interpretation Through Matrices...... 141 3.6 Interpretation Through Supervaluations. 150 4 Classical Quantification and Identity Theory 156 4.1 Syntax of Quantifier Languages...... 157 4.2 Axiomatics of Quantificational Logic... 163 4.3 Referential Interpretation: Models..... 168 4.4 Soundness and Completeness Theorems.. 174 4.5 Compactness and Countable Models... 180 4.6 Elementary Relations Among Models... 188 6 4.7 L¨owenheim{Skolem Theorem....... 191 4.8 Deductive Theories............. 195 4.9 Substitution Interpretation......... 199 4.10 Extensions of Quantificational Logic.... 206 5 Nonclassical Logics 212 5.1 Many-Valued Logics............ 213 5.1.1 Substitution and Lindenbaum Algebras............... 213 5.1.2 Compactness and Finite Matrices. 219 5.2 Modal Logics................ 222 5.2.1 Normal Propositional Modal Logics................ 223 5.2.2 Transformation Semantics for Modal Logics............ 232 5.3 Logic of Presuppositions.......... 237 5.3.1 Presupposition and Semantic Entailment............. 238 5.3.2 Policies on Presupposition..... 242 5.3.3 Epitheoretic Arguments...... 247 5.4 Concept of Truth.............. 252 5.4.1 Truth and Bivalence........ 252 5.4.2 Designation of True Sentences... 257 5.4.3 Truth Assertions in General.... 260 Appendix A: Completeness of the Calculus of Systems 270 7 Appendix B: Topological Matrices 276 Appendix C: Satisfiability and Semantic Paradoxes 285 Problems 289 Solutions to Selected Problems 312 Indices 334 Index of Searchable Terms............ 334 Index of Unsearchable Terms.......... 342 Index of Symbols................. 343 8 Preface This book is based on my lectures in advanced and in- termediate logic courses at Yale University 1966{1968, Indiana University 1969{1970. These courses were in- tended specifically for philosophy students with one pre- vious course in formal logic. The general aim of this book is to provide a broad framework in which both classical and nonclassical logics may be studied and appraised. The semantic approach adopted here was first systematically developed by Alfred Tarski; its development over the past forty years can only be very partially documented in our bibliographi- cal notes. After a preliminary chapter presenting the fairly ele- mentary techniques to be utilized, Chapters2 and3 pro- vide the general concepts and methods of formal seman- tics of logic. These chapters are illustrated throughout by the propositional calculus, the most familiar logical system we have. Chapters4 and5 are devoted to appli- cations to quantificational logic and to various nonclassi- cal logics, respectively. In Chapter 4 we develop first the usual semantics for quantificational logic. We then add a brief introduction to model theory, and a discussion of several forms of the L¨owenheim-Skolem theorem. More than half of this chapter is devoted to standard material: for example, Lindenbaum's theorem concerning charac- 9 teristic matrices, and the usual Kripke-style semantics for modal logic. But in view of the increasing influence of formal se- mantics on contemporary philosophical discussion, the emphasis is everywhere on applications to nonclassical logics and nonclassical interpretations of classical logic. In the Introduction I sketch a view of the nature of logic that is meant to to accommodate the existence and im- portance of nonclassical logics. In Chapter 2 the syntac- tic and semantic concepts used are purposely presented in a manner so general that they apply to languages of arbitrary structure. The calculus of systems as developed there and the problems of axiomatizability that are con- sidered pertain to deductive systems formulated in any kind of formal language. The concept of compactness is dissected into a family of concepts, equivalent in the classical case but not elsewhere. Proof-theoretic methods are entirely avoided in this chapter, and the methods of proving compactness studied do not involve reference to any logical system. Chapter 3 is devoted to the semantic appraisal of logical systems. As in the case for Chapter 2, the con- cepts and methods introduced are illustrated throughout with reference to the most familiar logical system, clas- sical propositional logic. Three kinds of interpretations of this system are considered: the usual one, interpreta- tions through matrices, and interpretations through su- 10 pervaluations. In Chapter 4, concerned with applications to quantificational logic, we only consider one unusual interpretation of the familiar logic: what Ruth Barcan Marcus calls the substitution interpretation. In Chapter 5, devoted to nonclassical logics, there are also sections dealing with subjects that have so far been discussed only in the journals: transformation semantics for modal logic, supervaluations, and presuppositions. This chapter ends with an analysis of the concept of truth, in which it is argued that Tarski's theory of truth does not carry over unchanged to nonclassical cases. In conclusion I would like to acknowledge gratefully my many debts to teachers, colleagues, and students: especially my teachers Karel Lambert and Nuel D. Belnap, Jr., but also Alan Anderson, Nino Cocchiarella, J. Michael Dunn, Frederic Fitch, Hugues Leblanc, Robert Meyer (who read an earlier draft of this book and made many valuable suggestions), Nicholas Rescher, Richmond H. Thomason, and many others. Whatever shortcomings this work has, it has in spite of what I learned from them. Bas C. van Fraassen 1971 11 Introduction: Aim and Structure of Logical Theory Logical studies comprise today both logic proper and metalogic. We distinguish these subjects by their aims: the aim of logic proper is to develop methods for the logi- cal appraisal of reasoning,1 and the aim of metalogic is to develop methods for the appraisal of logical methods. In pursuing the aims of logic, it has been fruitful to proceed systematically, that is, to construct formal axiomatic sys- tems of various kinds. These logical systems provide the immediate subject matter for metalogical investigation. Metalogic can in turn be roughly divided into two parts: proof theory and formal semantics.2 In proof the- ory, the logical systems are treated as abstract math- ematical systems, and the questions dealt with relate 12 directly to the specific set of axioms and rules used to formulate the system.3 In formal semantics, the logical systems are studied from the point of view of their pos- sible interpretations|with special reference to their in- tended interpretation, if such there be. This has led to a profound analysis of the structure of language, which has proved to be of importance for many philosophical discussions. While it is not possible to proceed with the semantic analysis of a logical system without due atten- tion to some proof-theoretical results, it is important to emphasize their relative independence. This is nowhere clearer than with respect to the compactness problem, a central problem studied in this book. For the usual procedure in logic texts is to use proof-theoretic results concerning a system to establish a certain semantic
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