Russell and His Sources for Non-Classical Logics

Anellis, Irving H.

doi:10.1007/s11787-009-0006-1

Cited by 5 publications

(2 citation statements)

References 91 publications

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“…The last pair also exhibits duality (an important moment in its own right): Hermann Grassmann stressed it in his algebra, including the structurally similar pair of 'analytic' and 'synthetic conjunctions', symbolised respectively '∪' and '∩', with the law (a ∪ b) ∩ b = a [1844, arts. [3][4][5]. Among his followers [156] was to use the pair to denote union and intersection in set theory; and Ernst Schröder [186], who was influenced by both brothers, extensively deployed dual notations and equations in his algebraic logic [157].…”

Section: Analogy and Emulationmentioning

confidence: 99%

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Omnipresence, Multipresence and Ubiquity: Kinds of Generality in and Around Mathematics and Logics

Grattan‐Guinness

2011

Log. Univers.

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A prized property of theories of all kinds is that of generality, of applicability or least relevance to a wide range of circumstances and situations. The purpose of this article is to present a pair of distinctions that suggest that three kinds of generality are to be found in mathematics and logics, not only at some particular period but especially in developments that take place over time: 'omnipresent' and 'multipresent' theories, and 'ubiquitous' notions that form dependent parts, or moments, of theories. The category of 'facets' is also introduced, primarily to assess the roles of diagrams and notations in these two disciplines. Various consequences are explored, starting with means of developing applied mathematics, and then reconsidering several established ways of elaborating or appraising theories, such as analogising, revolutions, abstraction, unification, reduction and axiomatisation. The influence of theories already in place upon theory-building is emphasised. The roles in both mathematics and logics of set theory, abstract algebras, metamathematics, and model theory are assessed, along with the different relationships between the two disciplines adopted in algebraic logic and in mathematical logic. Finally, the issue of monism versus pluralism in these two disciplines is rehearsed, and some suggestions are made about the special character of mathematical and logical knowledge, and also the differences between them. Since the article is basically an exercise in historiography, historical examples and case studies are described or noted throughout. Mathematics Subject Classification (2000). Primary 00A30, 01A55, 01A60, 01A85, 03-03, 03-99; Secondary 00A35, 00A69, 00A71, 00A79, 03B10, 03C55, 03G05, 30-03, 31-03.Keywords. History and philosophy of pure and applied mathematics, history and philosophy of symbolic logics, parts and moments, generality of theories, theory change, monism and pluralism, sets and multisets, abstract algebras, metamathematics, model theory.

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Section: Analogy and Emulationmentioning

confidence: 99%

“…Hugh MacColl advocated modal logics (and some other logics) at that time, but in monistic vein that sought one jumbo logic that also included bivalent logic[91]. On Russell's engagement with modal logics, see[4]. A magisterial historiography of non-bivalent logics is given in[1] 25.…”

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confidence: 99%

Omnipresence, Multipresence and Ubiquity: Kinds of Generality in and Around Mathematics and Logics

Grattan‐Guinness

2011

Log. Univers.

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The Concept of Relevance and the Logic Diagram Tradition

Dejnožka

2010

Log. Univers.

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What is logical relevance? Anderson and Belnap say that the "modern classical tradition [,] stemming from Frege and WhiteheadRussell, gave no consideration whatsoever to the classical notion of relevance." But just what is this classical notion? I argue that the relevance tradition is implicitly most deeply concerned with the containment of truth-grounds, less deeply with the containment of classes, and least of all with variable sharing in the Anderson-Belnap manner. Thus modern classical logicians such as Peirce, Frege, Russell, Wittgenstein, and Quine are implicit relevantists on the deepest level. In showing this, I reunite two fields of logic which, strangely from the traditional point of view, have become basically separated from each other: relevance logic and diagram logic. I argue that there are two main concepts of relevance, intensional and extensional. The first is that of the relevantists, who overlook the presence of the second in modern classical logic. The second is the concept of truth-ground containment as following from in Wittgenstein's Tractatus. I show that this second concept belongs to the diagram tradition of showing that the premisses contain the conclusion by the fact that the conclusion is diagrammed in the very act of diagramming the premisses. I argue that the extensional concept is primary, with at least five usable modern classical filters or constraints and indefinitely many secondary intensional filters or constraints. For the extensional concept is the genus of deductive relevance, and the filters define species. Also following the Tractatus, deductive relevance, or full truth-ground containment, is the limit of inductive relevance, or partial truth-ground containment. Purely extensional inductive or partial relevance has its filters or species too. Thus extensional relevance is more properly a universal concept of relevance or summum genus with modern classical deductive logic, relevantist deductive logic, and inductive logic as its three main domains.

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Jean van Heijenoort’s Conception of Modern Logic, in Historical Perspective

Anellis

2012

Log. Univers.

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Russell and His Sources for Non-Classical Logics

Cited by 5 publications

References 91 publications

Omnipresence, Multipresence and Ubiquity: Kinds of Generality in and Around Mathematics and Logics

Omnipresence, Multipresence and Ubiquity: Kinds of Generality in and Around Mathematics and Logics

The Concept of Relevance and the Logic Diagram Tradition

Jean van Heijenoort’s Conception of Modern Logic, in Historical Perspective

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