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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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Site Plan
NOTE: neologies and ambiguous terms clarified in GLOSSARY
are marked "[G]".
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 its meaning and the world.
It's a literary version of the Axiom of Abstractions
which underlies science and epistemology since the
Second Scientific Revolution of the Extended Relativity:
**ABSTRACT CONSTRUCTS may only be justified by their
capacity to coordinate events, which represent their
unique justification and meaning.**
CA2.INTRODUCTION TO PREDICATE LOGIC
An unfortunate homonymy collects under the term "Philosophy"
two strictly disjoined domains and conflicting attitudes:
1.Dogmatism, which is nothing else than Naive View (NV)[G],
also called Naive Realism or Naive Common Sense. NV presumes
a directly cognizable "Reality", consisting of "material"
"Objects" occupying space and having properties such as size,
shape, smell, taste and colour exactly as we perceive them.
NV is Dogmatic: it reposes in unshakable faith in "reality"
and in its "objects" being directly cognizable a priori.
NV is Noumenal: it takes its "objects" as Noumena "having"
Properties (or Attributes), as autonomous "containers" of
Attributes, existing as such (an sich), independently from
them, also in their total absence.
Reciprocally, Dogmatism and Noumenalism are synonyms of NV
often dressed-up in formidable highbrow verbiage.
2.Rationality, which, unlike Dogmatism reposing in faith,
reposes in doubt. It considers naive "reality" as uncertain
appearance, endeavors to interpret it and subjects the
interpretations to empiric verification.
It considers its Entities[G] or Phenomena as structures of
Attributes (or Aspects[G]) and nothing else: a single
Attribute is already a full-fledged Entity and Entity void
of Attributes is a nonsense.
Rationality or the empirically controllable Phenomenology
starts by trying to overcome the naive View.
Naive Common Sense guided original Humans in their struggle
for survival. It was assisted by natural languages which
came forth and developed to express NV and to support human
groups in their struggle against the hostile "Reality".
The principal linguistic construct supporting the NV is the
Predicative Assignment Expression "[E] & [A]", where "E"
stands for "Entity", "A" - for "Attribute" and "&", the
symbolic Assign Operator, replaces the copula "is/has" (e.g.
"my-car"[E] is "green"[A] or "my-car"[E] has "greenness"[A]).
Symbolic Operator "&" takes in natural languages the form of
respective copulas or, like in semitic languages, is implied.
It protects us also from the usual noumenalistic jumble
elevating illegal verbal inflections of the copula "be" to
the Olympian heights of ontological "Beings".
The Assignment Expression "[E] & [A]" is glaringly a naive,
noumenal, "container/contained" structure.
And this naive 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 naive, "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 naive
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, exhumate 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 naive
"Logic" under the misnomer "Propositional Calculus".
(CA1.INTRODUCTION TO PROPOSITIONAL CALCULUS).
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 B. Russell as an "apparent variable" and
"generalizing" him denotes a total lack of respect.
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.
Let's mention by anticipation that in network-structured
phenomenal Logic (CCA.COGNITIVE NETWORK) it suffices to avoid
upwards directed arrows to eliminate antinomies, paradoxes
and other"reflexive fallacies".
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
LIAR'S PARADOX 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 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.
