NOTES ON THE SCIENCE OF LOGIC Nuel Belnap University of Pittsburgh 2009 Copyright c 2009 by Nuel Belnap February 25, 2009 Draft for use by friends Permission is hereby granted until the end of December, 2009 to make single copies of this document as desired, and to make multiple copies for use by teach- ers or students in any course offered by any school. Individual parts of this document may also be copied subject to the same con- straints, provided the title page and this page are included. Contents List of exercises vii 1 Preliminaries 1 1A Introduction . 1 1A.1 Aim of the course . 1 1A.2 Absolute prerequisites and essential background . 3 1A.3 About exercises . 4 1A.4 Some conventions . 5 2 The logic of truth functional connectives 9 2A Truth values and functions . 10 2A.1 Truth values . 11 2A.2 Truth functions . 13 2A.3 Truth functionality of connectives . 16 2B Grammar of truth functional logic . 19 2B.1 Grammatical choices for truth functional logic . 19 2B.2 Basic grammar of TF . 21 2B.3 Set theory: finite and infinite . 24 2B.4 More basic grammar . 31 2B.5 More grammar: subformulas . 36 iii iv Contents 2C Yet more grammar: substitution . 38 2C.1 Arithmetic: N and N+ ................... 39 2C.2 More grammar—countable sentences . 43 2D Elementary semantics of TF . 45 2D.1 Valuations and interpretations for TF . 46 2D.2 Truth and models for TF . 51 2D.3 Two two-interpretation theorems and one one-interpretation theorem . 54 2D.4 Quantifying over TF-interpretations . 61 2D.5 Expressive powers of TF . 69 2E Elementary proof theory of STF ................... 76 2E.1 Proof theoretical choices . 78 2E.2 Basic proof-theoretical definitions for STF . 80 2E.3 Structural properties of derivability . 85 2E.4 Derivability and connectives . 88 2F Consistency and completeness of STF . 95 2F.1 Consistency of STF ..................... 96 2F.2 Completeness of STF .................... 97 2F.3 Quick proof of Lindenbaum’s lemma . 103 2F.4 Set theory: chains, unions, and finite subsets . 104 2F.5 Slow proof of Lindenbaum’s lemma . 106 2F.6 Lindenbaum and Zorn . 109 2G Consistency/completeness of STF and its corollaries . 111 2G.1 Finiteness and compactness . 112 2G.2 Transfer to subproofs . 113 2G.3 Set theory: sequences, stacks, and powers . 116 2G.4 Back to subproofs . 120 Contents v 3 The first order logic of extensional predicates, operators, and quanti- fiers 124 3A Grammatical choices for first order logic . 124 3A.1 Variables and constants . 124 3A.2 Predicates and operators . 126 3A.3 Sentences, connectives, and quantifiers . 129 3A.4 Other variable-binding functors . 131 3A.5 Summary . 134 3B Grammar of Q . 134 3B.1 Fundamental grammatical concepts of Q . 134 3B.2 Substitution in Q . 143 3B.3 Occurrence, freedom, closed, and open . 144 3C Elementary semantics of Q . 151 3C.1 Semantic choices . 151 3C.2 Basic semantic definitions for Q . 158 3C.3 Fundamental properties of V alj . 164 3C.4 Semantic consequence for Q . 169 3D Elementary proof theory of SQ . 172 3D.1 Proof theoretical choices for SQ . 173 3D.2 Basic proof theoretic definitions for SQ . 174 3D.3 Universal generalization in SQ . 178 3E Consistency and completeness of SQ . 185 3E.1 Consistency of SQ . 185 3E.2 Completeness of SQ . 186 3E.3 Set theory: axiom of choice . 193 3E.4 Slow proof of the Lindenbaum lemma for SQ . 196 3F Wholesale substitution and economical sets . 199 vi Contents 3F.1 Grammar of wholesale substitution . 200 3F.2 Semantics of wholesale substitution . 204 3F.3 Proof theory of wholesale substitution . 206 3F.4 Reducing consequence questions to economical sets . 208 3G Identity . 209 3G.1 Grammar of Q= . 210 3G.2 Semantics of Q= . 211 3G.3 Proof theory of SQ= . 211 3G.4 Consistency of SQ= . 212 3G.5 Completeness of SQ= . 213 Bibliography 217 vii List of exercises Exercise 1 p. 11 Exercise 25 p. 85 Exercise 49 p. 147 Exercise 2 p. 12 Exercise 26 p. 86 Exercise 50 p. 149 Exercise 3 p. 14 Exercise 27 p. 87 Exercise 51 p. 150 Exercise 4 p. 15 Exercise 28 p. 88 Exercise 52 p. 160 Exercise 5 p. 16 Exercise 29 p. 90 Exercise 53 p. 161 Exercise 6 p. 18 Exercise 30 p. 94 Exercise 54 p. 163 Exercise 7 p. 18 Exercise 31 p. 96 Exercise 55 p. 165 Exercise 8 p. 24 Exercise 32 p. 101 Exercise 56 p. 168 Exercise 9 p. 28 Exercise 33 p. 102 Exercise 57 p. 172 Exercise 10 p. 29 Exercise 34 p. 105 Exercise 58 p. 175 Exercise 11 p. 31 Exercise 35 p. 107 Exercise 59 p. 177 Exercise 12 p. 37 Exercise 36 p. 107 Exercise 60 p. 184 Exercise 13 p. 39 Exercise 37 p. 108 Exercise 61 p. 191 Exercise 14 p. 40 Exercise 38 p. 109 Exercise 62 p. 193 Exercise 15 p. 45 Exercise 39 p. 111 Exercise 63 p. 197 Exercise 16 p. 53 Exercise 40 p. 113 Exercise 64 p. 198 Exercise 17 p. 58 Exercise 41 p. 113 Exercise 65 p. 198 Exercise 18 p. 59 Exercise 42 p. 114 Exercise 66 p. 203 Exercise 19 p. 63 Exercise 43 p. 115 Exercise 67 p. 203 Exercise 20 p. 64 Exercise 44 p. 119 Exercise 68 p. 209 Exercise 21 p. 67 Exercise 45 p. 122 Exercise 69 p. 212 Exercise 22 p. 68 Exercise 46 p. 123 Exercise 70 p. 216 Exercise 23 p. 73 Exercise 47 p. 132 Exercise 24 p. 75 Exercise 48 p. 138 Source: /Users/nuelbelnap/Documents/texcurr/nsl/2009-nsl/nsl-preface-2009.tex. Printed February 25, 2009 Preface These notes began many years ago as an auxiliary to the fine text Hunter 1971, which he updated in 2001 (University of California Press). Please bear with inad- equacies. P. Woodruff taught us several of the key ideas around which these notes are orga- nized. Thanks to J. M. Dunn for suggestions, and to W. O’Donahue and B. Basara for collecting errors. G. A. Antonelli and P. Bartha contributed greatly, as did a variety of necessarily anonymous students. viii Chapter 1 Preliminaries 1A Introduction This course assumes you know how to use truth functions and quantifiers as tools; such is part of the art of logic. Our principal task here will be to study these very tools; we shall be engaged in part of the science of logic. 1A.1 Aim of the course Our aim is not principally mathematical, but foundational; undoubtedly “logical” is the best word. We are not aiming to hone your mathematical intuitions, and this is not a “mathematical” study. When we prove a theorem, we shall therefore not try to do it as swiftly and intuitively as possible, which is the aim of the mathematician, but instead we shall try to keep track of the principles involved. Principles, however, come at many levels, so we must be more specific. In contrast to much of mathematical logic, proofs in these notes will use what most logicians consider only the most elementary of steps. In fact, this presentation of logic is unlike any other of which we know in that (unless clearly labeled) there is no armwaving in the following perfectly good sense: Whenever we give a proof, it is possible for you to fully formalize this proof using only techniques you learned in a one-term course in logic. In particular, (1) no proof relies on geometrical or arith- metical intuition to gain your assent to its conclusion, since we do not presuppose your mastery of either of these topics, and (2) no proof is so complicated or so long that you cannot see how it goes. With regard to (1), if our aim were the mathemat- ical one of helping you to acquire the ability to see that our conclusions were true, 1 2 Notes on the science of logic then reliance on arithmetic or geometrical intuitions would be reasonable; but in- stead our principal aim is the logical one of seeing how, using only quantifier and truth-functional principles, our conclusions follow from the fundamental axioms and definitions that characterize our subject matter. For example, it is “obvious” that every formula contains at most a finite number of atomic symbols—you can “see” that this must be so. These notes will rely on no such “vision,” but will instead analyze the principles lying behind such intuitions so that for justifying proofs the intuitions themselves can be dispensed with. With regard to (2), it is equally important that we organize the work in such a way that you can see how all the quantifier and truth-functional steps are put together— so that you can “survey” our proofs; for unless you can do that, you cannot after all see how our conclusions follow. In short, we bring to proofs in the subject the same standards of rigor and comprehensibility that are enshrined in the logic of which we are speaking. You must be aware, however, that to be rigorous is an ideal, and one that we have not always attained. Our more realistic hope is that we have been sufficiently clear so that with diligence you can see where we have fallen short—as successive groups of students have done. On the other hand, without intuition conception is blind, and we shall make every effort to give you “pictures” of theorems and proofs to help you see what is going on. Attaining a how-to-carry-out-the proof understanding of these matters is not enough—you also need to acquire an arithmetic and geometrical and philosophical intuitive understanding.