Georges Metanomski LIAR'S "PARADOX"

Relativistic Dialectics Relativistic Dialectics

Georges Metanomski
LIAR'S "PARADOX"

In one of the letters written to the Infeld group in Warsaw Einstein wrote:
"A new manner of thinking is essential if humankind is to survive."

Aristotelian Logic

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 op propositions A, E, I, O: A: a universal affirmative proposition--All S is 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, what we call now "logical operations", but a genial anticipation of naive Set-Theoretical operations as they imply Universal and Existential qualifiers.

Depending on proposition type 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) (A) is true 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) (E) is true 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 our case not_P1 (not(All my_statements are false)) results in "Some of my_statements are false" (I) implying that some are true or: "Any of my_statements is false or true". or "Any of my_statements is logically indeterminate"

The proper deduction becomes: First loop: P1: All my_statements are false P2: P1 is my_statement Thus P3: P1 is false

Second loop: not_P1: Some of my statements are false (or true)(I) P2: P1 is my_statement (A) thus P3': P1 is false or true (I) i. e. indeterminate.

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.

Contemporary Logic
Actually, there ain't no such animal as "Contemporary Logic". As we said in ref. "The Village Barber":

"However, at the end of the 19th century two breaches appeared in the pyramid [of science]:
1. In Physics the Michelson's experiment showing that the speed of light is independent of source and observer,
2. Russell's Paradox calling in question the very foundations of the Set Theory and Logic. Question arose, if single local breaches in the otherwise solid pyramid may be patched locally, or should rather lead to global reconsideration of the whole pyramid. In Physics Einstein has chosen the global approach and revised entire Physics. In Logic and in the Set Theory one has opted for patches and the crisis stays open till today."

As result we faced a multitude of Logical Systems none of them satisfactory even when signed by Russell, Godel or Lukasievicz. Closest to something holding water came Carnap with his concept of "factual truth", but he stopped short of defining it and stayed within the rigid exact, two-valued frame (true/false).

In this situation we had to conceive our own Logic for the needs of AI Systems we had to design and program. This "CN Logic" is described in ref. "Crisis of Logic", mainly in the chapter "Fuzzy Inferencing" and we shall consider the "modern" version of Liar in its light.

However, our arguments hold for any Logic encompassing Propositional Calculus. It's astonishing that Godel could consider seriously the Liar, which simply proves that even greatest thinkers may have moments of weakness.

Now, to the point:

The "modern" version of Liar is the following "statement" L:

L="THIS STATEMENT IS FALSE".

-Which statement?- you may wonder and you would be right: L is NOT a statement. A statement in logical sense is a symbolic construct conveying a meaning usually by assigning some attribute to some entity. Since Carnap this attribution must be reducible via inferencing to verifiable facts: The entity must be (reducible to) observable and so has to be the attribute. Calling observable entity "object" we may say that a valid statement assigns an attribute to an object, that it says something about an object, which may be verified and confirmed (true statement) or disconfirmed (false statement). NOTE: We are close here to Kotarbinski's Reism, which we intend to discuss elsewhere.

Thus when I shout from the cave te sentence S: "The apples are rotten again" my wife understands what I mean, may verify it having herself a look and confirm or refute it. Syntax plays secondary role: if I shouted "hell!" my wife would know from the context that I mean rotten apples.

But "this statement" of L is not an object, nor "false" (or "true") is an attribute. We cannot observe "this statement" as an object, nor "truth"/"falsity" as an observable attribute of anything.

"true"/"false" are values of a (logical) variable associated with a statement, but not elements of the statement itself (object, attribute). My wife looking at the apples may find them rotten indeed and set the associated variable of S to "true". Many other variables may be associated with a statement, for instance "importance": "S" with its rotten apples may be considered as less important than "Tsunami devastated Asia", but neither this "importance", nor the "truth" are intrinsic attributes of involved statements.

In order to be true or false a statement has first to be a statement. Liar in its "modern" version "THIS STATEMENT IS FALSE" is not a statement at all thus it cannot be true, nor false, nor a paradox, but just an empty meaningless noise.

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