This paper examines evidence that the Unified Competency Theory of Knowledge Management (KM) has merit as a precursor to selecting the first formal justification approach. We show that these particular integrals have some similarities with the ‘generalizing deduction’ procedures investigated by Łukasiewicz. Moreover, particular integrals can be constructed based on detachment properties that lead to logical expressions of growing complexity. Within this logical formalism an indefinite integral can always be found for any logical expression. The search for the antiderivatives produces naturally an integral calculus. In addition using the logical derivatives we show that relatively complex formulas can collapse in simple expressions that reveal clearly their hidden logical meaning. We show that some of the basic tautologies (Excluded middle, Modus ponens, Hypothetical syllogism) are members of a kind of hierarchical system linked by the differentiation algorithm. Inside this semantic-algebraic calculus, we obtain expressions for the derivatives of some of the basic logical operations and show the general way to obtain the derivatives of any well-formed formula of propositional calculus. In this way, guided by the laws of matrix algebra, we can construct compact representations for the derivatives and the integrals of logical functions. We show that this extension to logics is strongly helped if we submerge the elementary logical calculus in a matrix-vector formalism that naturally includes a kind of fuzzy-logic. In the present article, we extend this kind of calculus in order to include the semantic of classical logical operations. Mathematics Subject Classification (2000)Primary 00A30–01A55–01A60–01A85–03-03–03-99–Secondary 00A35–00A69–00A71–00A79–03B10–03C55–03G05–30-03–31-03Ī variety of problems emerged investigating electronic circuits, computer devices and cellular automata motivated a number of attempts to create a differential and integral calculus for Boolean functions. Since the article is basically an exercise in historiography, historical examples and case studies are described Finally, the issue of monism versus pluralism in these two disciplines is rehearsed,Īnd some suggestions are made about the special character of mathematical and logical knowledge, and also the differencesīetween them. In algebraic logic and in mathematical logic. Metamathematics, and model theory are assessed, along with the different relationships between the two disciplines adopted The roles in both mathematics and logics of set theory, abstract algebras, The influence of theoriesĪlready in place upon theory-building is emphasised. Theories, such as analogising, revolutions, abstraction, unification, reduction and axiomatisation. With means of developing applied mathematics, and then reconsidering several established ways of elaborating or appraising Various consequences are explored, starting
Primarily to assess the roles of diagrams and notations in these two disciplines. The category of ‘facets’ is also introduced, Some particular period but especially in developments that take place over time: ‘omnipresent’ and ‘multipresent’ theories,Īnd ‘ubiquitous’ notions that form dependent parts, or moments, of theories. The purpose of this article is to presentĪ pair of distinctions that suggest that three kinds of generality are to be found in mathematics and logics, not only at 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.
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