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Logic Alfred Tarski updated by Karl Frinkle © Draft date April 24, 2019 Contents Preface for New Edition ix Preface xi From the Preface to the Original Edition xvii Compilation of Rules and Laws xix I Elements of Logic1 1 On The Use of Variables3 1.1 Constants and Variables......................3 1.2 Expressions Containing Variables.................4 1.3 Formation of Sentences by Means of Variables..........6 1.4 Universal and Existential Quantifiers, Free and Bound Variables.....................7 1.5 The Importance of Variables in Mathematics.......... 10 Chapter 1 Exercises........................... 11 2 On The Sentential Calculus 15 2.1 The Old Logic and the New Logic................ 15 2.2 Negations, Conjunctions, and Disjunctions............ 16 2.3 Conditional Sentences and Implications in Material Meaning. 19 2.4 The Use of Implication in Mathematics............. 22 2.5 Equivalence of Sentences...................... 25 2.6 Formulations of Definitions and Its Rules............ 26 2.7 The Laws of Sentential Calculus................. 29 2.8 Symbolism and Truth Tables................... 30 2.9 Application of Laws of Sentential Calculus in Inference..... 36 2.10 Rules of Inference, Complete Proofs............... 38 Chapter 2 Exercises........................... 41 v vi Contents 3 On The Theory of Identity 47 3.1 Logical Concepts Outside Sentential Calculus.......... 47 3.2 Fundamental Laws of the Theory of Identity........... 48 3.3 Identity of Objects, Their Designations, and Quotation Marks. 53 3.4 Equality in Arithmetic and Geometry.............. 55 3.5 Numerical Quantifiers....................... 57 Chapter 3 Exercises........................... 59 4 On The Theory of Classes 63 4.1 Classes and Their Elements.................... 63 4.2 Classes and Sentential Functions with One Free Variable.... 64 4.3 Universal and Null Classes..................... 67 4.4 Fundamental Relations Among Classes.............. 68 4.5 Operations on Classes....................... 71 4.6 Cardinal Number of a Class.................... 75 Chapter 4 Exercises........................... 77 5 On The Theory of Relations 81 5.1 Relations and Sentential Functions with Two Free Variables.. 81 5.2 Calculus of Relations........................ 83 5.3 Some Properties of Relations................... 86 5.4 Relations which are Reflexive, Symmetric, and Transitive... 88 5.5 Ordering Relations......................... 90 5.6 One-many Relations or Functions................. 91 5.7 Bijective Functions......................... 94 5.8 Many-Termed Relations...................... 95 5.9 The Importance of Logic for Other Sciences........... 97 Chapter 5 Exercises........................... 97 II The Deductive Method and Formal Systems 103 6 On The Deductive Method 105 6.1 Primitive and Defined Terms, Axioms and Theorems...... 105 6.2 Model and Interpretation of a Deductive Theory........ 107 6.3 Law of Deduction.......................... 113 6.4 Selection of Axioms and Primitive Terms............ 116 6.5 Formalization of Definitions and Proofs............. 117 6.6 Consistency and Completeness of a Deductive Theory..... 119 6.7 Widened Conception of the Methodology of Deductive Sciences........................ 121 Chapter 6 Exercises........................... 122 Contents vii 7 Propositional Languages and the Deductive Method 133 7.1 Formal Languages......................... 133 7.2 Metalogical Theorems About j=P ................. 135 7.3 Formal Systems........................... 136 7.4 Metalogical Theorems About `PS ................ 140 7.5 The Deduction Theorem for PS .................. 141 7.6 Concepts of Semantic Completeness............... 145 Chapter 7 Exercises........................... 155 III Applications of Logic and Methodology: Constructing Mathematical Theories 157 8 Construction of a Mathematical Theory: Laws of Order for Numbers 159 8.1 Primitive Terms and Axioms on Fundamental Relations Among Numbers.................... 159 8.2 Laws of Irreflexivity and Indirect Proofs............. 161 8.3 Further Theorems on the Fundamental Relations........ 162 8.4 Other Relations Among Numbers................. 166 Chapter 8 Exercises........................... 171 9 Construction of a Mathematical Theory: Laws of Addition and Subtraction 173 9.1 Axioms Concerning Addition, Properties of Groups....... 173 9.2 Further Commutative and Associative Laws........... 175 9.3 Laws of Monotony For Addition and Their Converses...... 176 9.4 Closed Systems of Sentences.................... 180 9.5 Consequences of the Laws of Monotony............. 182 9.6 Definition of Subtraction; Inverse Operations.......... 189 9.7 Definitions Containing Identity.................. 191 9.8 Theorems on Subtraction..................... 192 Chapter 9 Exercises........................... 195 10 Methodological Considerations on the Constructed Theory 199 10.1 Elimination of Superfluous Axioms in the Original Axiomatic System.................... 199 10.2 Independence of the Axioms of the Simplified System..... 203 10.3 Elimination of Superfluous Primitive Terms........... 205 10.4 Further Simplifications of the Axiomatic System........ 207 10.5 Problem of the Consistency of the Constructed Theory..... 213 10.6 Problem of the Completeness of the Constructed Theory.... 214 Chapter 10 Exercises........................... 215 viii Contents 11 Foundations of Arithmetic for Real Numbers 219 11.1 First Axiomatic System for the Arithmetic of Real Numbers.......................... 219 11.2 Characterization of the First Axiomatic System......... 220 11.3 Second Axiomatic System for the Arithmetic of Real Numbers.......................... 222 11.4 Characterization of the Second Axiomatic System........ 223 11.5 Equipollence of the Two Systems................. 225 Chapter 11 Exercises........................... 226 Suggested Reading 229 Index 233 Preface for New Edition I should start off by saying that this text is, in this mathematician's opinion, the best, basic introduction to logic textbook available. Many current logical textbooks fail to focus on the most important aspect of logic to the sciences, which is, the deductive method. The rigorous approach to building a deduc- tive theory based on an axiomatic system, and the consequences of selecting axioms and primitive terms is thoroughly, and carefully explained, yet readily accessible to the reader. Many texts spend too much time dealing with logical symbolism, truth ta- bles, and clever word puzzles, while others tend to be only accessible to those studying the fundamental semantics and intricacies of logic. This text is the perfect balance between sufficiently deep logical theory and the application of this theory to algebraic axiomatic systems. One might thus ask: Why would you wish to edit or alter such a text if you praise it so highly? The text itself has been `updated' several times, the last being in 1994. In the last two updates, very little has been done to make the language of the writing or the symbolism more current. Furthermore, af- ter listening to the comments of the students in my Honor's Logic Courses, I realized that the students tolerated the message of the book, just not the style in which that message was presented. Most textbooks today focus heavily on highlighting key items such as definitions, tables, laws, theorems, examples, etc... I have attempted to make the text more easily parsed upon reading. One of the more important aspects of the upgrade to this text is in the rigorous, formal proof system-style approach to proving many of the theorems presented in this text. All previous versions of this text were very quick to wave hands at many of the more subtle steps required to rigorously prove many of the theorems given. In doing so, many logical laws and clever insights were missed, and I have attempted to include them in the details added. I have also taken the liberty of either incorporating optional sections of the text (cordoned off by *'s) or removing them. After using this text numerous ix x Preface for New Edition times, it became clear which material really had an impact on the overall theme of the course, and what material was either not necessary or added confusion to the learning process. In a similar fashion, some of the homework problems have been reworked. Less emphasis has been put on geometrically oriented problems, and most of the questions starting off with \Give examples from..... of such and such a property..." have simply been removed or replaced. I have found Dr. Tarski's writing style to be rather refreshing and con- versational in style, and have therefore attempted to leave most of his words intact. Any additions or further explanations hopefully blend in as seamlessly as I intended. At this point, I need to give a special thanks to two very dedicated stu- dents: Laiken McMurl and Michaela Metts, who were students in my Fall 2015 Logic course. These two students are the definition of ideal students, they read the text multiple times, had intelligent discourse over all of the material, and critiqued every aspect of the book. Many of the modifications done in the text were the direct result of input from these two students. For instance, the com- pilation of rules, laws, and logical identities found after the set of prefaces to this text was one important suggestion they made which I feel will benefit all se- rious students of this text. Included in this list are some clever logical identities that may help solve some of the later exercises and are not discussed in the text. Added in 2019 : After a few years of teaching
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