Would you like to make this site your homepage? It's fast and easy...
Yes, Please make this my home page!
CBD.GOEDEL AND LIAR'S PARADOX
Liar's "Paradox" has been formulated in Greece
about 600 BC so it's reasonable to push it first through the
filter of Aristotelian Logic. As we shall see below, it's no
paradox at all, but simple error in negating a proposition.
Aristoteles would get rid of it in under a minute if he
considered it worth bothering.
Let's recall some elementary concepts of Aristotelian Logic.
Four types of propositions A, E, I, O:
A: a universal affirmative proposition All S are P
E: a universal negative proposition No S is P
I: a particular affirmative proposition Some S is P
O: a particular negative proposition Some S is not P
SYLLOGISM: a logical deductive structure consisting of three
propositions: First and Second Premises and Conclusion.
NOTE: Strictly speaking Syllogisms were not "logical
operations", but a genial anticipation of naive
Set-Theoretical operations as they use Universal and
Existential quantifiers and anticipated Contain operator
"is/are".
Depending on proposition types one may distinguish:
AAA or Barbara (see below) for instance:
P1: All humans are mortal (A)
P2: All Greeks are humans (A)
P3: All Greeks are mortal (A)
EAE or Celarent (see below) for instance:
P1; No human is immortal (E)
P2: All Greeks are humans (A)
P3: No Greek is immortal (E)
etc.
For mnemotechnic reasons all Syllogism types ordered in four
groups have been enumerated in form of hexameter:
1. Barbara, Celarent, Darii, Ferioque, prioris:
2. Cesare, Camestres, Festino, Baroco, secundae:
3. Tertia, Darapti, Disamis, Datisi, Felapton, Bocardo,
Ferison, habet:
4. Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo,
Fresison.
With this light baggage we may easily find that Liar is no
paradox, but a simple deduction error.
In syllogistic form the Liar becomes:
Let
P1="All my_statements are false" (A)
giving
not_P1: "None of my_statements is false" (E)
(under the ASSUMPTION: not_A --> E)
Then starting with assuming P1 to be true:
First loop (Barbara):
P1: (All my_statements are false) is true (A)
P2: P1 is my_statement (A)
Thus
P3: P1 is false (A)
(thus not_P1 is true (E))
Second loop (Celarent):
not_P1: (None of my_statements is false) is true (E)
P2: P1 is my_statement (A)
thus
P3': P1 is not false (E)
thus
P1 is true (A) and etc. da capo al fine.
We negated First Premise of Barbara getting Celarent,
returned to Barbara, etc.
However, our ASSUMPTION was wrong: negation of A results not
in E but in I: not_A --> I.
Indeed,
"All cars are red" (A)
negated does not give
"No car is red" (E)
but
"Some cars are not-red" (I)
In Liar's case
not_P1 (not(All my_statements are false)) not_(A)
results in
"At least one of my_statements is true" (I)
implying that other ones are either true or false
i.e. logically indeterminate.
The proper deduction becomes: First loop:
P1: All my statements are false (A)
P2: P1 is my_statement (A)
Thus
P3: P1 is false (A)
Second loop:
not_P1: Some of my statements are true (or false) (I)
P2: P1 is my_statement (A)
thus
P3': P1 is true or false i.e. indeterminate. (I)
So, by proper negation of Barbara's First Premise we don't
get Celarent, but Disamis which says that we cannot conclude
anything about P1.
In other terms we don't know anything about P1 and "Liar"
does not loop infinitely but peters out after the second
correct loop.
Liar and Contemporary Established Logic
In CBA.LOST PARADISE we have said that 19th Century Logic
was dominated by the Predicate Logic (PL) and shown that PL
is a noumenal, ill-founded, naive 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 remedes 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 (CA1.INTRODUCTION TO
PROPOSITIONAL CALCULUS) as meaningless and useless as PL
remedes, 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 remedes. We don't intend to discuss all 100+
of them and shall have a short look on two leading and
typical ones, 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 noumenal Ontology underlying these assertions
is discussed in CA1.INTRODUCTION TO PROPOSITIONAL CALCULUS .
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 Disamis with Celarent, which would have saved
him from writing thick treaties about it and us from being
muddled.
But that's not the worst. 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". Recalling it one feels embarrassed like
listening to a friend 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. "S" (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, nothing that
may possibly be true or false.
Let's now pass to Goedel.
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.
Shalving the rather clear upshot that it's probably the most
inane pile of garbage ever dumped in "Logic", let's go to
brass tacks, to the famous G: "This sentence is not provable"
and to 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.
