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LIAR, RUSSELL AND GOEDEL
Liar's Paradox has been formulated in numerous forms, the
simplest being "I always lie", supposed to loop infinitely:
"I always lie", thus I lied, thus "I always tell truth",
thus "I always lie" was true, thus "I always lie", etc.
Actually, it was no paradox at all but a sophism, coined
at the time when Greeks discovered the delights of ingenious,
unbridled reasoning, before conceiving ways to keep it
under control.
In order to demystify the alleged paradox, we reformulate
"I always lie" as equivalent, but more rigorous:
P="all propositions I utter are false" and reconstruct
the pretended loop:
A. I utter as true: P="all propositions I utter are false".
thus
B. P is false
thus
C. not-P="no propositions I utter is false" is true
thus
D. P which I uttered is not-false, i.e. true and we are
back at A.
However, since Liar's inception about 600 BC many ways to keep
sophisms under control have been defined, starting with the
Aristotelian Logic, which would clearly falsify C.
Indeed, negation of the universal affirmative proposition
P="all propositions I utter are false" does not give the
universal negative proposition not-P="no propositions I
utter is false", but the particular affirmative proposition
not-P="some propositions I utter are false" and D. becomes
"P is either false or true", i.e. logically indeterminate.
Reconstructing correctly the chain of reasoning we get:
A. I utter as true: P="all propositions I utter are false".
thus
B. P is false
thus
C. not-P="some propositions I utter are false" is true
thus
D. P is either false or true, i.e. logically indeterminate.
In other terms we don't know anything about P and Liar
does not loop infinitely but peters out after the first
correct conclusion.
Liar and the Established Logic
In "PREDICATE LOGIC" we have shown that the currently
dominating Predicate Logic (PL) is a noumenal, ill-founded,
instance of Naive View. Russell's paradox shuttered PL with
its feeble foundations, but logicians did not dare to abandon
the Naive View and tried vainly to save and rebuild the
collapsed structure upon its rotten base rather than construct
a new edifice upon new rational foundations.
As result, the established Logic became an extraordinary
proliferation of competitive remedies of PL, all noumenal,
ill-founded and meaningless. We have counted about 100 of
them. There are also about as many non-PL systems, mainly
based upon corrupted Boolean Algebra ("BOOLEAN SUPPORT OF
ERN LOGIC") as meaningless and useless as PL remedies, but
we shall consider here only the latter, as on the one hand
they constitute the mainstream founding the official Set Theory
and, on the other hand, they reserve an eminent place for Liar's
Paradox. Indeed, after 2000 years of quiet rest at the cemetery
of deduction errors it resurrected, donned the old moth eaten
dress of Paradox and made a triumphal come-back as a principal
hinge of most, if not all PL remedies. We don't intend to discuss
all 100+ of them, but shall have a short look on two leading and
typical "applications" of Liar, those of Russell and Goedel.
Russell
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.
The super-naive Ontology underlying these assertions is
discussed in "PREDICATE LOGIC". Here we are mainly interested
in surprising resurgence of Liar's "Paradox" in the monumental
Principia Mathematica often considered as the principal
contribution to foundations of Logic and Mathematics:
-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.-
If Russell studied Aristoteles, he would have noticed that
the "contradiction" disappeared 2000 years ago, reduced to
confusion of a particular affirmative proposition with
a universal negative one, which would dispense him from
writing thick fallacious treaties and us from being muddled.
But things went still much deeper. Liar had many shades,
one of them, the Eubulides "paradox": "This statement is false,
thus it's true, thus it's false", etc. Russell treated it
seriously and even went to the trouble of creating its two
statement version: "The following statement is true. The
preceding statement is false".
Talking about it one feels embarrassed like listening to a
fellow who had one over the eight and laughs heartily at his
own silly jokes.
Symbolizing "This statement is false" with "R", R is neither
false, nor true for the simple reason that it is NO statement
at all. By the standards of Russell's own Predicate Logic
a statement is a predication, an assignment of a property to
a subject. "Truth/Falsity" qualifies the predication itself
and not the subject of predication. A statement "all cars are
red" is a valid predication or statement which may be true or
false and by virtue of observations turns out to be false.
Now, R does not assign any property to any subject, thus is
not a predication, not a statement at all, a meaningless chain
of characters that may not be true or false.
As meaningless as the built upon it Types Theory.
Goedel
-For any consistent formal theory that proves basic
arithmetical truths, it is possible to construct an
arithmetical statement that is true but not provable in the
theory. That is, any theory capable of expressing elementary
arithmetic cannot be both consistent and complete.-
Whatever the meaning, if any, of "basic arithmetical truths"
may possibly be, we shall recall how Goedel describes his
"theory" and the famous "G" (Goedel Sentence) true but not
provable in the theory.
"Theory" refers to an (infinite) set of statements, some of
which are taken as true without proof (these are called
axioms), and others (the theorems) are taken as true because
they are provable from the axioms. "Provable in the theory"
means "derivable from the axioms and primitive notions(?) of
the theory, using standard first order logic." A theory is
"consistent" if it never proves a contradiction. "It is
possible to construct" means that there exists some
mechanical(?) procedure which can construct the statement,
given the axioms, primitives, and first order logic. The
resulting true but unprovable statement is often referred to
as "the Goedel sentence" for that theory. In fact, there are
infinitely many statements in the theory that share with the
Goedel sentence the property of being true but not provable
from the theory. "Elementary arithmetic" consists merely(?)
of addition and multiplication over the natural numbers.
We have seen in "PREDICATE_LOGIC" and "SET_THEORY" the inanity
of founding mathematics and mathematical proofs in "standard
first order logic" and more generally, in discreteness. For an
arithmetic theory properly founded in continuous geometry, the
question of completeness could not arise. If consistently
reducible to geometric continuum, it would discretize only a
partial area thereof and thus be incomplete; otherwise, it would
be inconsistent.
But let's consider for oddity's sake the famous Goedel sentence G:
"This sentence is not provable" and the Olympian theorem:
"G is true but not provable in the theory".
(G cannot be false, because then it would be provable and
all provable sentences are true; so it's necessarily true
and unprovable).
Now, as Russell's "R", Goedel's "G" is a shade of the
exhumated Liar's "Paradox" and no "sentence" at all.
Why did Goedel play amateur gravedigger and disinter the
poor Liar, when he had any number of true and unprovable
axioms at hand, will stay for ever a closed book.
Unless, as some people say, he considered axioms as provable
from the theory. But it must be a malignant calumny. Even
Goedel could not be that inane.
At least...
In "Goedel's Proof" by Peter Suber, Philosophy Department,
Earlham College,we read:
***
Suppose we added G to the axiom set of S. Then G would become
provable, since all axioms are provable by definition.
***
The mind boggles. Since 2300 years all scientific theories
are axiomatic and their founding axioms are unanimously
considered as "by definition" true and unprovable. So Suber's
"definition" barring the whole scientific history seems
a bit exotic and one would associate it rather with some
exclusive loony bin than with science, mathematics, logic
or anything rational.
And Mr Suber continues:
***
Goedel proved that if we do this (add G to the axiom set of S),
creating S', then we could always construct another undecidable
wff, G', which asserted that it could not be proved in S'.
Of course we could then add G' to S', creating S'', but then
we could construct G'', and so on ad infinitum.
***
Very exclusive must indeed be the loony bin where Peter Suber
may venerate his co-certified master Goedel confusing mathematics
with russian dolls.
This Goedel's betise may be added to all those shown in the
chapter "PREDICATE LOGIC".
