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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
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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.
