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

Genuine Process Logic

The Genuine Process Logic described here (abbreviation: GPL) places the object-bound process itself at the center of formalism. It should be suitable for everyday use, i.e. it is not primarily intended for the formalization of computer programs, but instead, as a counter-conception to the classical state logics. The new and central operator of the GPL is an action symbol replacing the classical state symbols, e.g. of equivalence or identity. The complete renunciation of object-language state expressions also results in a completely new metalinguistic framework, both regarding the axioms and the expressive possibilities of this system. A mixture with state logical terms is readily possible

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.

From Many-Valued Consequence to Many-Valued Connectives

Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but on what we call Gentzen-regular connectives (including negation, conjunction, and disjunction). For arbitrary N-valued logics, we state necessary and sufficient conditions for the existence of such connectives in a multi-premise multi-conclusion setting. The results show that mixed consequence relations admit all classical connectives, and among them pure consequence relations are those that admit no other Gentzen-regular connectives. Conditionals can also be found for a broader class of intersective mixed consequence relations, but with the exclusion of order-theoretic consequence relations.Comment: Updated version [corrections of an incorrect claim in first version; two bib entries added

Forall x: Introduction to Formal Logic, version 1.29

In formal logic, sentences and arguments in English are translated into mathematical languages with well-defined properties. If all goes well, properties of the argument that were hard to discern become clearer. This book covers translation, formal semantics, and proof theory for both sentential logic and quantified logic. Each chapter contains practice exercises; solutions to selected exercises appear in an appendixhttps://scholarsarchive.library.albany.edu/cas_philosophy_scholar_books/1000/thumbnail.jp

Forall x: Introduction to Formal Logic, version 1.40

In formal logic, sentences and arguments in English are translated into mathematical languages with well-defined properties. If all goes well, properties of the argument that were hard to discern become clearer. This book covers translation, formal semantics, and proof theory for both sentential logic and quantified logic. Each chapter contains practice exercises; solutions to selected exercises appear in an appendixhttps://scholarsarchive.library.albany.edu/cas_philosophy_scholar_books/1004/thumbnail.jp

Forall x: An introduction to formal logic 1.30

In formal logic, sentences and arguments in English are translated into mathematical languages with well-defined properties. If all goes well, properties of the argument that were hard to discern become clearer. This book covers translation, formal semantics, and proof theory for both sentential logic and quantified logic. Each chapter contains practice exercises; solutions to selected exercises appear in an appendix.https://scholarsarchive.library.albany.edu/cas_philosophy_scholar_books/1002/thumbnail.jp

Forall x: Introduction to Formal Logic, version 1.29

In formal logic, sentences and arguments in English are translated into mathematical languages with well-defined properties. If all goes well, properties of the argument that were hard to discern become clearer. This book covers translation, formal semantics, and proof theory for both sentential logic and quantified logic. Each chapter contains practice exercises; solutions to selected exercises appear in an appendixhttps://scholarsarchive.library.albany.edu/cas_philosophy_scholar_books/1000/thumbnail.jp

Forall X: An Introduction to Formal Logic, version 1.28

This is a textbook covering translation, formal semantics, and proof theory for both sentential logic and quantified logic. Each chapter contains practice exercises; solutions to selected exercises appear in an appendixhttps://scholarsarchive.library.albany.edu/cas_philosophy_scholar_books/1001/thumbnail.jp

forall x: Calgary. An Introduction to Formal Logic

forall x: Calgary is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity of arguments, the syntax of truth-functional propositional logic TFL and truth-table semantics, the syntax of first-order (predicate) logic FOL with identity (first-order interpretations), translating (formalizing) English in TFL and FOL, and Fitch-style natural deduction proof systems for both TFL and FOL. It also deals with some advanced topics such as truth-functional completeness and modal logic. Exercises with solutions are available. It is provided in PDF (for screen reading, printing, and a special version for dyslexics) and in LaTeX source code