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NOTE: neologies and ambiguous terms clarified in GLOSSARY
are marked "[G]".
CBB.ESTABLISHED LOGIC
1.Overview
Strangely enough, Logic was seldom, if ever, rigorously
defined and founded. True, one would expect only a vague
intensional definition for a concept of that generality, but
solid, stable foundations and an extensional definition
enumerating its objectives, procedures, structures and rules
seem possible and advisable. One would expect Logic to be
founded in Ontology and Epistemology and to found in turn
Mathematics, Linguistics and other disciplines.
In this chapter we recapitulate established Logic showing
that its hardly enumerable competing systems share the
foundation in Mathematics and in Naive View via naively
interpreted constructs of natural languages. They all turn
in idle circularity of Naive View founding Naive View and
Mathematics founding Mathematics. Dressed up in highbrow
pseudo-scientific verbiage they are all empty metaphysical
speculations.
Incidentally, numbers speak for themselves: while the totally
renovated post-Crisis Physics covers just two basic Models,
Extended Relativity and Quantum Field Theory, the post-Crisis
established Logic counts over 150 "Systems", all equally
naive and empty.
2.Marginalia.
2.1.Propositional Calculus (PC)
In the chapter CA1.INTRODUCTION TO PROPOSITIONAL CALCULUS
we have shown that the Propositional Calculus (PC) is in
fact corrupted Boole Algebra (BA). BA does not interpret
in any way its symbolic Operands, but depends solely upon
the laws of their combination. Operands are associated with
a variable ranging over binary digits 0-1, which we call
"Certainty" for the sake of consistency with Fuzzy Inference.
The laws are embodied in Operators, which get the resulting
Certainty of a particular combination of 2 Operands.
CAA.EXACT TWO-DIMENSIONAL PROPOSITIONAL CALCULUS
PC 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 replaces the binary 0-1 range of BA's Certainty
with naive undefined and meaningless terms "truth-falsity".
While BA is a useful mathematical support of binary gates
underlying computing and electronic technology, its corrupted
version "PC" is an empty, naive procedure illegally
pretending to be a "logical system".
2.2.Modal Systems
PC forks into 70(!) "Modal Logics". Modality denotes a
linguistic concept of subparts of sentences modifying their
semantics. For example, "We go to the games" can be modified
to give "We should go to the games", "We can go to the games"
or "We will go to the games". More generally, we might say
that modality affects the circumstances in which we take an
assertion to be satisfied. Whatever its merits for Semantics
of natural languages, when dragged into PC, modality only
aggravates the corruption of Boole Algebra and stresses
the pseudo-foundations in Naive View.
2.3.Extended Boole Algebra (EBA)
Vacuity of 2 dimensional (2 operands), 2 valued ("true"/
"false") PC motivated some Logicians to develop Multivalued
Systems, although nobody considered the 2D restriction.
Lukasiewicz toyed with 3 valued PC, replacing 2 valued
certainty range [0/1] with 3 valued [0/0.5/1], followed with
5 valued [0/0.25/0.5/0.75/1]. Finally, Logicians realized
that fixed-valued systems were simple useless toys and moved
to timid trials of continuous distribution of values, or
"Fuzzy" Logic. As EBA they produced results in technical
control systems, but in spite of being misnamed "Artificial
Intelligence", or "Fuzzy Logical Systems" thay can only be
considered as technological applications of Mathematics and
not as Logic supporting human reflection by embodying the
dynamic function of Mind processing its Symbolic domain.
BAA.STRUCTURE OF MIND
The typical EBA Systems have the folloving restrictions:
-Two Dimensions, ie. Fuzzy Operators defined 2 Operands,
-Just three Extended Operators: OR, AND, NOT.
-Fixed Operator Algorithms, as for instance:
"crt" = certainty)
crt(A(x) OR B(x)) = MAX(crt(A(x)),crt(B(x)))
crt(A(x) AND B(x)) = MIN(crt(A(x)),crt(B(x)))
crt(NOT A(x)) = 1 - crt(A(x))
The "anti-logical" sense of these restrictions may best be
understood by comparison with the Phenomenal Cognitive
Network (CN) Logic. ( CCA.COGNITIVE NETWORK )
We shall just mention here three features of CN:
-Illimited dimensionality (number of Operands per Operator),
implying multi-operand algorithms for Operators.
-Comprehensive set of Operators including a multidimensional
fuzzy extension of Exclusive Or, viz. "One Of".
-Customizable operator algorithms supporting particular
experience in form of Expert Systems.
3. Mainstream, the Predicate Logic (PL).
In CA2.INTRODUCTION TO PREDICATE LOGIC
we have described in detail the naive, noumenal "foundations"
of PL and the useless fatuity of innumerable "logical"
systems based upon them. Here we shall copy its final
paragraph:
Strangely enough, Logic was seldom, if ever, rigorously
defined and founded. True, one would expect only a vague
intensional definition for a concept of that generality, but
solid, stable foundations and an extensional definition
enumerating its objectives, procedures, structures and rules
seem indispensable to talk about Logic. We have seen that
extensional definition of PL presents it vaguely as a branch
of Mathematics without anything properly "logical" and that
its "foundations" in naive structures of common languages
hardly merit this name.
One would expect Logic to be founded in Ontology and
Epistemology and to found in turn Mathematics, Linguistics
and other disciplines, as schematically shown in figure 1.
---------------------------------------------
| Fig.1 |
| Ontology |
| Epistemology |
| Logic Languages |
| | Natural Formal |
| | | | |
| ----------------------- -------- |
| | | | | | |
| Mathematics ... ... Linguistics |
---------------------------------------------
PL's foundations schema looks instead as shown in figure 2.
-----------------------------------
| Fig.2 |
| Natural Languages Mathematics |
| | | |
| Naive View | |
| | | |
| ------------------ |
| | |
| Predicate Logic |
| | |
| ------------------ |
| | | |
| Naive View Mathematics |
-----------------------------------
We arrive at the source of Paradoxes, Antinomies and other
"reflexive fallacies" constitutional in PL, viz. fallacious
pseudo-foundations in the Natural Languages via Naive View
and the circularity NV - PL - NV (Pl being founded in NV and
founding it in turn).
The other circularity, Mathematics - PL - Mathematics
accounts for the success of PL as universally accepted
foundation of Mathematics, but in fact boils down to the
conjurer trick of Mathematics founding Mathematics.
Our critique of PL and its authors and adherents, to mention
Frege, Russell, Wittgenstein, Quine, Tarski, Goedel, may seem
excessively severe. Yet, we believe to have shown, that it
is a naive, dogmatic phantasm which has resulted, like Aether
in Physics, in wasted efforts of generations of scientists
and in wrecked established Logic. Actually, PL turns out more
noxious than Aether, whose destructive effects were after all
restricted to Physics. PL, contrariwise, impacts the totality
of human praxis, leaving the mankind in want of means to
formulate, let alone to solve current problems critical for
its very survival.
We believe that when a Russell or a Hegel, no matter how
famous and celebrated, makes a blunder, it stays a blunder.
And, when a Smith or a Dupont says something reasonable, it
stays reasonable.
For the sake of Smiths and Duponts somebody's got to say
that the Emperor is naked.
