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Working Paper DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND Boundary Algebra: A Simpler Approach to Boolean Algebra and the Sentential Connectives Philip Meguire WORKING PAPER No. 50/2010 Department of Economics and Finance College of Business and Economics University of Canterbury Private Bag 4800, Christchurch New Zealand Boundary Algebra: A Simpler Approach to Boolean Algebra and the Sentential Connectives. Philip Meguire Department of Economics University of Canterbury Christchurch, New Zealand [email protected] August 2010. Abstract Boundary algebra [BA] is a 〈--,(-),()〉 algebra of type 〈2,1,0〉, and a simplified notation for Spencer- Brown’s (1969) primary algebra. The syntax of the primary arithmetic [PA] consists of two atoms, () and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive no- tion of distinction. Inserting letters denoting, indifferently, the presence or absence of () into a PA formula yields a BA formula. The BA axioms are A1: ()()= (), and A2: “(()) [abbreviated ‘⊥’] may be written or erased at will,” implying (⊥)=(). The repeated application of A1 and A2 simplifies any PA formula to either () or ⊥. The basis for BA is B1: abc=bca (concatenation commutes & associates); B2, ⊥a=a (BA has a low- er bound, ⊥); B3, (a)a=() (BA is a complemented lattice); and B4, (ba)a=(b)a (implies that BA is a distributive lattice). BA has two intended models: (1) the Boolean algebra 2 with base set B={(),⊥}, such that () ⇔ 1 [dually 0], (a) ⇔ a′, and ab ⇔ a∪b [a∩b]; and (2) sentential logic, such that () ⇔ true [false], (a) ⇔ ~a, and ab ⇔ a∨b [a∧b]. BA is a self-dual notation, facilitates a calculational style of proof, and simplifies clausal reasoning and Quine’s truth value analysis. BA resembles C.S. Peirce’s graphical logic, the symbolic logics of Leibniz and W.E. Johnson, the 2 notation of Byrne (1946), and the Boolean term schemata of Quine (1982). Keywords: Boundary algebra, boundary logic, primary algebra, primary arithmetic, Boolean algebra, calculation proof, G. Spencer-Brown, C.S. Peirce, existential graphs. Dedication. Historical: C. S. Peirce, the first boundary logician. Past: To the memories of my father and of my mentor R.C.K. Present: To George Spencer-Brown, to my spouse and helpmeet Ruth, and to my mother. Future: To my daughter. Table of Contents Preface Acknowledgements Boundary Algebra and Theoretical Physics 1. Introduction 1 2. The Primary Arithmetic, PA 4 2.1. Syntax 5 2.2. PA: Axiomatics, Simplification, Semantics 8 2.3. PA: Canons and (Meta)theorems 12 3. The Primary Algebra, pa: Syntax and Algebra 16 3.1. Consequences, Canons, Theorems 17 3.2. Order Irrelevance, Tacit and Explicit 22 3.3. The Lattice Road from Antisymmetry to BA 24 3.4. Boundary Notations for Other Algebraic Structures 29 4. pa Semantics: From BA to Boundary Logic 34 4.1. Duality 35 4.2. Boundary Logic 37 4.3. The Enigmatic Degeneracy of BA 40 4.4. pa: Metatheory 41 5. pa: Proof 45 5.1. A Decision Procedure 46 5.2. More on the Initials B1-B4 47 5.3. The Usual Inference Rules of Logic 49 5.4. Some Worked Examples from Logic Texts 51 5.5. Syllogisms as Clauses 55 5.6. Segue to First Order Logic or How to Quantify sans Quantifiers 57 6. Historical Antecedents and More Axiomatics 61 6.1. Peirce’s Alpha Existential Graphs 61 6.2. Some Ba Postulate Sets 64 6.3. Other Historical Systems Related to the pa 69 7. pa: Why the Indifference? 72 8. Conclusion 75 Bibliographic Postscript 77 Appendices: The Null Individual and Its Controverted Ontology 78 Demonstrations, Proofs, Etc.: A.1 − 10 80 Systems Going Beyond LoF: A.11 − 17 91 A Précis of Mathematical Logic 98 Table of Cross-References Between LoF and This Book 99 References 100 Subject Index 106 Name Index 109 Preface “Logic is better presented as algebra.” Hehner (2004) This book sets out a simplified variant of the classic two-element Boolean algebra, and argues that this variant is an easy way to carry out the calculations elementary logic requires. In 1969, a British free-lance intellectual named George Spencer-Brown published a curious short book called Laws of Form (LoF). It was based on a university extension course in elementary logic he had taught for some years. In 1974, I chanced on the American mass market paperback edition of LoF, and over the next quarter century, intermittently struggled to grasp its content. In 2001, despite having no training in algebra or logic, I decided to make explicating LoF the primary focus of my academic research; thus began a decade’s reflection that culminated in this book. I began this research because I was especially intrigued by Spencer-Brown’s provocative claim that “…the cal- culus published in this text renders [standard university logic problems] so easy that we need not trouble ourselves further with them...” (LoF, p. viii). LoF sets out a simplified approach, called the primary algebra (pa), to that hoary mathematical chestnut, the two-element Boolean algebra to which Paul Halmos gave the name 2. At the outset, I thought I was primarily exploring elementary logic, but soon saw that the logic was nothing more than an interpretation of 2. However, LoF confused this picture by claiming to do much more, phil- osophically as well as mathematically. This book adopts a notation due to Croskin, one I call boundary notation, that is not only more keyboard-friendly than the notation of LoF, but is also more in the spirit of C.S. Peirce’s graphical logic which the pa startlingly parallels (Kauffman 2001). In the pa, the concatenation of subformu- lae may be interpreted either as Boolean sum or as Boolean product. Because both interpretations are equally valid, the pa is self-dual. Because Boolean sum and product both associate, there is no need to indicate grouping, freeing up parentheses to denote Boolean complementation. LoF un- wittingly rediscovered an enigmatic fact that Peirce discovered in the late 19th century: comple- mentation with an empty scope (hence “()”) can be interpreted as a lattice bound and primitive value. Boundary algebra (BA) combines the pa with LoF’s primary arithmetic, also notated using bound- ary notation. Employing conventional Boolean notation for the nonce, the primary arithmetic is grounded in two facts from Boolean arithmetic: 1∪1= 1, and “1′ may be written or erased at will.” The first fact is very well known; the second is is much less so. LoF then invoked two algebraic postulates: the distributive law and (a′∪a)′ = 0. That ‘∪’ commutes and associates was asserted true by default. This book sets out a new postulate set for the pa that first makes the commutativity and associativity of ∪ explicit, then invokes two very familiar laws, 0∪a = a and a′∪a = 1, and the less familiar a∪(a∪b)′ = a∪b′. These postulates often simplify proofs (here called “demonstrations,” as per LoF) of identities. This book also often draws on the fact that proving that α′∪β = β′∪α = 1 amounts to a proof of α=β. While the pa is the focus of this book, I gradually came to appreciate that boundary notation can be applied to other algebraic structures. Thus chapter 3 also speaks to lattices and groups, and men- tions other algebraic structures. The pa postulates I now prefer highlight how near the BA is to an abelian group. Chapter 5 shows by example how the methods of this book greatly facilitate the sort of exercises one does in undergraduate logic courses. It also shows how boundary methods can be employed in first order logic, and sheds light on the hoary syllogism. Chapter 6 lays out the close connection between BA and Peirce’s graphical logic. Chapter 7 proposes an explanation for why the pa has had little impact even though LoF has never gone out of print. Spencer-Brown intended that LoF be a contribution to philosophy, especially to the philosophy of mathematics and logic. A glance at the reference section for this book reveals that my intentions parallel his. This book is silent about my professional discipline, economics. More generally, it appears that economic reasoning never invokes Boolean algebra (Ba) in any way (although Boolean logic is fundamental to the computers economists use daily in their professional and personal activities). But starting around 1990, work in economic theory began to appear that drew on the generalisation of Ba called lattice theory. Topkis (1998) shows how the theories of the consumer, firm, general equilibrium, and non-cooperative games can be re-exposited using lattice theory. This is one reason why §3.3 introduces a boundary approach to lattice theory. This book’s preferred pa basis, B1-B4, and its numbering system for the derived consequences differ significantly from those in Meguire (2003). §§3.4, 4.3, 5.6, and 6.3 are all new, as are all appendices except A.6-9 and A.17. A new section on the syllogism, §5.5, replaces my earlier dis- cussion of monadic predicate logic. §§2.3, 3.3, and the balance of §5 and §6 are revised and ex- panded. The former §3.4 and §6.0 are now §4.1 and §6.1, respectively. I have moved material from §§3.1, 5.0, 5.2, and 6.1 in the earlier version, to §§4.4, 5.4, and 6.2-3 here. There are, of course, re- visions of detail everywhere. Acknowledgements. I thank Taylor & Francis for kindly permitting me to draw freely on Meguire (2003) in writing this book. I am also very grateful to George Klir, the editor of the International Journal of General Systems, for not asking me to shorten that paper, even though it filled 63 pages of his journal.
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