|
|||||||
|
|
|
|||||
|
|
|||||||
PLAN OF CA PREREQUISITES ca1 introduction to propositional calculus ca2 introduction to predicate logic caa 2D exact propositional calculus cab ND exact propositional calculus cac implication BACK TO SITE PLAN Site Plan
CA1.INTRODUCTION TO PROPOSITIONAL CALCULUS
Let's start with a short quotation of Boole: -They who are acquainted with the present state of the theory of Symbolic Algebra, are aware that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination.- The common usage forces us to use the term Propositional Calculus (PC). However, we shall use it as a strict synonym of Boolean Algebra (BA). Operands and operators of BA are associated with a variable ranging over binary digits 0-1, which we call "Certainty" for the sake of consistency with Fuzzy Inference (CCA.COGNITIVE NETWORK). Original BA is 2 dimensional: its expressions consist of an operator taking Certainty of two operands as input and determining its own Certainty as output. Operands are determined uniquely as carriers of Certainty and don't admit other interpretation. Operators may become in turn operands of higher level expressions, in which case they are treated as operands, i.e. Certainty carriers and nothing else. 2 dimensional BA admits 16 operators. BA in its improved Polish Notation of Lukasiewicz supports binary gates underlying computing and electronic technology. Its pretended synonym, the pseudo-logical "Propositional Calculus" is a misnomer confusing Operand with "proposition" following naive Logic's traditional striving to be rooted in natural languages and via them in the Naive View. The same tendency confuses the binary, 0-1 range of Certainty with naive noumenalistic meaningless terms "truth-falsity". Throughout its history, naive Logic considered propositions as its unique operands. It never inquired where the "truth" of premises comes from, assumed it arbitrarily and dealt exclusively with its propagation to conclusions. Corrupted Boolean Algebra has been snatched and incorporated into naive Logic as an efficient propagation tool. Rational Logic embodied in the Cognitive Network (CN) (CCA.COGNITIVE NETWORK) may be considered as "Extended BA" (EBA), BA extended over: 1.Network structure with Operands in vertices and Operators in edges. 2.N dimensional vertices (associated with N edges), which involves a very fast increase of operators number (ON) in function of N: ON = 2^(2^N). Original 2D BA has 16 Operators which may be easily learned like the multiplication table and used spontaneously. For 3D we get 256 Operators and for 100 sensors watching malfunctions in a plane, 2^(2^100) Operators, whose enumeration, let alone definition would overflow the Congress Library. And the dimensions of DNA exceed 6000. We discuss practical ways of dealing with this problem in CAB.EXACT ND PROPOSITIONAL CALCULUS. Let's note that dimensionality is local, determined for each vertex by the number of associated edges. 3.continuous Certainty spanning real number range {0-1} supporting Fuzzy Logic and, with open dimensionality, Artificial Intelligence. 4.Operands defined as, or reducible to Phenomena or Entities, which makes CN apt to formulate and to process most complex philosophical, scientific and practical problems. By virtue of these extensions CN accounts for the state of the art of inquiries about human reasoning and thus seems to be the unique contemporary System meriting the name "Logic". One might object to the apparent circularity: Logic being supposed to found Mathematics, cannot be properly founded in Boole Algebra, which is a mathematical construct. However, the circularity is only apparent. CN is derived from inquiry about Mind and Reasoning (BAA.STRUCTURE OF MIND) and our encounter with Boole Algebra during CN derivation is purely coincidental. Yet, while logically immature prior to our extensions, Boole Algebra is an impressing, rigorous theory and we prefer to take over some of its well formulated concepts, than to redundantly redefine them.