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PREDICATE LOGIC
Sartre said:
"Having discovered the world through language,
I have over long time taken language for the world".
We say with him:
He who starts with the world will find in it
the meaning of the language.
But he who starts with the language will fail
to find both, its meaning and the world.
It's a literary version of the Abstractions Postulate
which underlies science and epistemology since the
Second Scientific Revolution of the Extended Relativity
(see "STRUCTURES OF MIND"):
**
Abstract, symbolic constructs may be justified solely by
their capacity to coordinate events which represent their
unique meaning and justification, where coordination of
events implies considering them in their context, i.e.
upon their background of continuum.
**
In the "Natural Model" we saw that extrinsic logical systems
may only be justified by extending and simulating the ER
(Expression/Relation) structures of the natural inferring
faculty. None of the established logical systems satisfies
this requirement, all being based on reifying, noumenal
natural languages. In the present chapter we present their
flagship, the Predicate Logic (PL).
The crucial for PL noumenal "container" structure of natural
languages is represented by the Predicative Assignment
Expression symbolized by "[E][&][A]", "E"(entity) standing
for "object", "&"(assign symbol) for "contains" and "A"
(attribute(s)) for contained sense impressions.
The symbolic expression [E][&][A] has the syntactic form
[Subject][copula][Property], e.g. "my-car is green", or
"my-car has greenness", where the copula "is/has" embodies
the assign symbol "&" and can be implied, like in Semitic
languages or in Russian. It may also have the apparently
recursive form [E][&][e], "e" standing for a more general
entity than "E", e.g. "my car is a vehicle", but it is just
a shortcut for [E][&][all attributes of (e)].
And this noumenal predicative expression is the only foundation
of PL. One would expect Logic to be founded in Epistemology
and Ontology, one would hope to see it founded in rational
study of basic functions of human Mind. It's with astoundment
that one discovers a noumenal, "kitchen" language expression as
cornerstone of "Logic", which pretends to be the absolute
foundation of Mathematics and Science.
At best, one could understand it in the pre-Crisis context
of the flat out reaction of Dogmatism against the First
Enlightenment. But one can think only with disappointment
about Russell. His own Paradox wrecked exclusively the
noumenal PL and in no way Logic in general. Yet, he tried
vainly to save and rebuild the collapsed structure upon its
rotten base rather than to construct a new edifice upon solid
rational foundations. And, what's worse, he roped in all his
followers, the entire established Logic for this enormous
wasted effort.
As result, our epoch stays without any well defined and well
founded Logic. Facing global problems calling in question
mankind's survival and impossible to be even formulated, let
alone solved, without pertinent logical structures, it needs
it more than any other one in history.
A randomly chosen example from 1 160 000 Web articles on PL
listed by Google tells us what PL can do:
### First-order logic permits reasoning about the
propositional connectives (as in propositional logic) and
also about quantification ("all" or "some"). A classic, if
elementary, example of what can be done with the predicate
logic is the inference from the premises:
* All men are mortal.
* Socrates is a man.
to the conclusion
* Socrates is mortal ###
Yes, disinter syllogisms after 2000 years and unnecessarily
muddle them, that's all what it can do.
One may object that First Order PL does much more, namely
founds Mathematics. But does it?
It IMPORTS from Mathematics several concepts (marked "#*):
-relations# (also called "predicate variables")
-constants#
-variables#
-valence# (of relations# and variables#) greater#/equal# 1#
(numbers and 1 in particular are not yet defined)
-equality# and its symbol "="
-"logical" boolean# operators#, or, better said, the whole
Boolean Algebra# corrupted and incorporated into noumenal
"Logic" under the misnomer "Propositional Calculus" (see
"BOOLEAN SUPPORT OF ERN LOGIC").
So, PL imports from Mathematics enough to found Mathematics
in Mathematics. This may look to any normal Human as a
vicious circle, but who would insult established Logicians
by calling them normal Humans?
PL does not restrict its imports to Mathematics, but rubbing
shoulders with Gods upon the vertiginous heights of the
Olympus a priori, juggles joyously with metaphysical Beings,
Objects, Things, Individuals, Properties, Existences, Truths,
Falsities, Falsehoods.
In his Types Theory Russell introduces the notion of
first-order, second-order and higher order logics in this
way:
-...We may define an individual as something destitute of
complexity; it is then obviously not a proposition, since
propositions are essentially complex. Hence in applying the
process of generalization to individuals we run no risk of
incurring reflexive fallacies.
Elementary propositions together with such as contain only
individuals as apparent variables we will call first-order
propositions. We can thus form new propositions in which
first-order propositions occur as apparent variables. These
we will call second-order propositions; these form the third
logical type.
-Thus, for example, if Epimenides asserts "all first-order
propositions affirmed by me are false," he asserts a
second-order proposition; he may assert this truly, without
asserting truly any first-order proposition, and thus no
contradiction arises.-
One seems to be daydreaming. Facing an "individual", let's
say, B. Russell himself, we are impressed by his biological,
genetical, social and spiritual complexity, the latter so
clearly manifested by his writings. Compared with intricacy
of the "individual" B. Russell, the proposition "B. Russell
is dead" appears to be totally "destitute of complexity".
Algebraizing a human being, or for that matter any perceivable
object, as a "variable" void of complexity would indicate the
incapacity of elementary reflection, let alone of conceiving
logical systems.
The obsession of "reflexive fallacies" pertains to mental
disorder and not to Logic. It's true that Russell has the
merit of having discovered and honestly published Paradoxes,
whose "reflexive fallacy" shuttered his dear PL. It's a pity
that he did not take the hint and scrap the wrecked phantasm
rather than spend the rest of his life hunting "reflexive
fallacies" as if they were rats and by virtue of his
celebrity transforming established Logic into rat hunting.
Rats hunt started in Russell's 1908 paper, "Mathematical
Logic as Based on the Theory of Types", where he enumerated
seven paradoxes, starting with that of Epimenides' Liar
followed by his own Paradox and other "reflexive fallacies".
The choice of "Liar" as leader speaks by itself. In the
chapter "LIAR, RUSSELL AND GOEDEL" we show that "Liar" is not
a paradox, but a simple sophism, a Syllogism error dead and
buried at Aristoteles time. And, its Eubulides' version
"This statement is false", a starting block of Russell's Types,
is by his own standards not a proposition at all. Indeed,
it does not assign any property to anything: "falsity" is not
a property of "things", but of Assignment Expression.
Statement "Joe is tall" is an assignment of tallness to Joe,
a proposition which may be true or false. But "This statement
is false" does not assign anything to anything, does not
assert anything about anything, so it's no proposition,
but just empty, meaningless noise, which cannot be true,
nor false.
Definition and Foundations
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 noumenal structures of common languages
are no foundations at all.
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 |
| | | |
| Reifications | |
| | | |
| ------------------ |
| | |
| Predicate Logic |
| | |
| ------------------ |
| | | |
| Reifications Mathematics |
-----------------------------------
We arrive at the source of Paradoxes, Antinomies and other
"reflexive fallacies" constitutive of PL, viz. fallacious
pseudo-foundations in the Natural Languages via Reifications
and the circularity Reifications - PL - Reifications.
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 noumenal, 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.
