Boundary Algebra: A Simpler Approach to Boolean Algebra and the Sentential Connectives

Boundary algebra [BA] is a algebra of type , 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 notion 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 lower 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).Boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; G. Spencer-Brown; C.S. Peirce; existential graphs

Boundary Algebra: A Simple Notation for Boolean Algebra and the Truth Functors

Boundary algebra [BA] is a simpler notation for Spencer-Brown’s (1969) primary algebra [pa], the Boolean algebra 2, and the truth functors. The primary arithmetic [PA] consists of the atoms ‘()’ and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting the presence or absence of () into a PA formula yields a BA formula. The BA axioms are "()()=()" (A1), and "(()) [=?] may be written or erased at will” (A2). Repeated application of these axioms to a PA formula yields a member of B= {(),?} called its simplification. (a) has two intended interpretations: (a) ? a? (Boolean algebra 2), and (a) ? ~a (sentential logic). BA is self-dual: () ? 1 [dually 0] so that B is the carrier for 2, ab ? a?b [a?b], and (a)b [(a(b))] ? a=b, so that ?=() [()=?] follows trivially and B is a poset. The BA basis abc= bca (Dilworth 1938), a(ab)= a(b), and a()=() (Bricken 2002) facilitates clausal reasoning and proof by calculation. BA also simplifies normal forms and Quine’s (1982) truth value analysis. () ? true [false] yields boundary logic.G. Spencer Brown; boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; C.S. Peirce; existential graphs.

forall x (UBC Edition)

This is an open-access introductory logic textbook, prepared by Jonathan Ichikawa, based on P.D. Magnus's forallx. This (v2.0, July 2020) is intended as a stable, ready-for-teaching edition

The Computer Modelling of Mathematical Reasoning

xv, 403 p.; 23 cm

The Logical Writings of Karl Popper

This open access book is the first ever collection of Karl Popper's writings on deductive logic. Karl R. Popper (1902-1994) was one of the most influential philosophers of the 20th century. His philosophy of science ("falsificationism") and his social and political philosophy ("open society") have been widely discussed way beyond academic philosophy. What is not so well known is that Popper also produced a considerable work on the foundations of deductive logic, most of it published at the end of the 1940s as articles at scattered places. This little-known work deserves to be known better, as it is highly significant for modern proof-theoretic semantics. This collection assembles Popper's published writings on deductive logic in a single volume, together with all reviews of these papers. It also contains a large amount of unpublished material from the Popper Archives, including Popper's correspondence related to deductive logic and manuscripts that were (almost) finished, but did not reach the publication stage. All of these items are critically edited with additional comments by the editors. A general introduction puts Popper's work into the context of current discussions on the foundations of logic. This book should be of interest to logicians, philosophers, and anybody concerned with Popper's work

Logic, ontology, and arithmetic : a study of the development of Bertrand Russell’s Mathematical Philosophy from The Principles of Mathematics to Principia Mathematica

O presente trabalho tem por objeto de análise o desenvolvimento da Filosofia Matemática de Bertrand Russell desde os Principles of Mathematics até ­ e inlcuindo ­ a primeira edição de Principia Mathematica, tendo como fio condutor as mudanças no pensamento de Russell com respeito a três tópicos interligados, a saber: (1) a concepção de Russell da Lógica enquanto uma ciência (2) os compromissos ontológicos da Lógica e (3) a tese Russelliana de que a Matemática Pura ­ a Aritmética particular ­ é nada mais do que um ramo da Lógica. Esses três tópicos interligados formam um fio condutor que seguimos na tese para avaliar qual interpretação fornece o melhor relato da evidência textual disponível em Principia Mathematica e nos manuscritos produzidos por Russell no período relevante. A posição geral defendida é que a interpretação de Gregory Landini apresenta argumentos decisivos contra a ortodoxia de comentadores que atribuem à Principia uma hierarquia de tipos ramificada de entidades confusamente formulada, e mostramos que os três pontos apontados acima que formam o fio condutor da tese corroboram fortemente a interpretação de Landini. Os resultados que apontam para a conclusão geral de nossa investigação estão apresentados na tese dividida em duas partes. A primeira parte discute o desenvolvimento da lógica de concepção de Russell e do projeto Logicista desde sua gênese e nos Principles of Mathematics até Principia Mathematica. Esta primeira parte define o contexto para a segunda, que discute a Lógica Russeliana e o Logicismo em sua versão madura apresentada em Principia. Mostramos que, ao fim e ao cabo, o a teoria Lógica e a forma da tese Logicista apresentada em Principia é o resultado do longo processo iniciado com descoberta da Teoria dos Símbolos Incompletos que levou Russell a gradualmente reduzir os compromissos ontológicos de sua concepção da Lógica enquanto uma ciência, culminando na teoria apresentada na Introdução de Principia, na qual ele procura formular uma hierarquia dos tipos que evita o compromisso ontológico com classes, proposições e também com as assim chamadas funções proposicionais e que esse mesmo processo levou Russell a uma concepção da tese de Logicista de acordo com a qual a Matemática é uma ciência cujos compromissos ontológicos não incluem qualquer espécie de objetos (no sentido Fregeano) sejam eles particulares concretos ou abstratos.The present work has as its object of analysis the development of Bertrand Russell’s Mathematical Philosophy from the Principles of Mathematics up to ­ and including ­ the first edition of Principia Mathematica, having as a guiding thread the changes in Russell’s thought with respect to three interconnected topics, namely: (1) Russell’s conception of Logic as a science (2) the ontological commitments of Logic and (3) Russell’s thesis that Pure Mathematics ­ in particular Arithmetic ­ is nothing more than a branch of Logic. These three interconnected topics form a common thread that we follow in the dissertation to assess which interpretation offers the best account of the available textual evidence in Principia Mathematica and in the manuscripts produced by Russell in the relevant period. The general position held is that Gregory Landini’s interpretation presents decisive arguments against the orthodoxy of commentators who attribute to Principia a confusingly formulated hierarchy of ramfified types of entities, and we show that the three points indicated out above that form the main thread of the thesis strongly corroborate Landini’s interpretation. The results that point to the general conclusion of our investigation are stated in the dissertation divided into two parts. The first part discusses the development of Russell’s conception of Logic and the of Logicist project from its genesis and in Principles of Mathematics up to Principia Mathematica. This first part sets the context for a second, which discusses a Russellian Logic and Logicism in its mature version presented in Principia. We show that, in the end, the Logic theory and the form of the Logicist thesis presented in Principia is the result of a long process that started with the discovery of the theory of Incomplete Symbols which led Russell to reduce the ontological commitments of his conception of Logic as a science, culminating in the theory of types presented in Principia’s Introduction, in which Russell seeks to formulate a hierarchy of types that avoids the ontological commitment to classes, propositions and also with so­called propositional functions, and that this same process led Russell to a conception of the Logicist thesis according to Mathematics is a science with no ontological commitments to any kind of objects (in the Fregean sense) whether these are conceived as concrete or abstract particulars