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CBA.LOST PARADISE
NOTE: neologies and ambiguous terms clarified in GLOSSARY are marked "[G]".
1.Cantorian Paradise
19th Century Physics and Logic were dominated by reaction of Dogmatism against Rationality of the First Enlightenment. Physics was founded in the dogmatic delusion of Aether. Logical systems culminated in Predicate Logic (PL) founded in linguistic structures reducible to "[subject] is [property]", e.g. "my-car is green" or "my-car has (the property of being) green" or "my-car has greenness" or, generally "[entity] has [attribute]" ("attribute" being synonym of "property"). This, rather surprising, ontological foundation in wrongly interpreted structures of some selected natural languages (BAC.LINGUISTIC INVERSION) sees Entity as "container of Attributes", which clearly defines PL as a noumenal system. Indeed, Noumenalism considers Entity and Attribute as ontologically distinct constructs, Entity being a "Noumenon", a "Container", which may receive and "have" Attributes but exists as such, even empty and prior to receiving them. Kant called Noumenon "Ding an sich" (thing in itself) and banned it from cognition and rationality. Now, Noumenon is neither observable nor factually verifiable, so that whatever we assert about it is clearly a Dogma[G]. Thus, Noumenalism is synonymous with Dogmatism and like the latter is nothing else than Naive View[G] (also called "Naive Realism") dressed up in highbrow verbiage. Logic founded thus in Naive View may be called naive. This noumenal, naive Logic was assumed to be the universal, absolute foundation of transcendental "Reality" and of human knowledge thereof. All areas of Science were assumed to be recursively founded in Logic, although Mathematics stayed for some time unfounded. Finally, in 1873 Cantor conceived the naive Set Theory which supplied the missing link and completed the hierarchy of "Cantorian Paradise": 1.(naive) Logic, the universal, absolute, olympian foundation, 2.(naive) Set Theory, founded in (naive) Logic, 3.Mathematics, founded in (naive) Set Theory, 4.Physics, founded in Mathematics and in the mechanistic Dogma, itself founded in (naive) Logic. 5.Other "hard" Sciences, founded in Physics, 6."Soft" Sciences, founded in "hard" Sciences. This "Paradise" represented the edifice of Science solid and stable as Cheops pyramid, bestowing upon scientists the gratifying conviction that Logic, THEIR Logic, explains and determines the "Real Universe".
2.Breaches in the pyramid
At the end of the 19th century two breaches appeared in the pyramid: 1.In Physics the Michelson-Morley experiment showing that the speed of light is invariant with respect to the speed of source and observer and thus falsifying the mechanistic, Dogma. 2.Russell's Paradoxes calling in question the foundations and sense of naive PL and founded in it naive Set Theory. Question arose, if apparently local breaches in otherwise solid pyramid may be patched locally, or should lead to global reconsideration of the whole structure. In Physics Einstein has chosen the global approach, revised entire Physics and laid the cornerstone of the Second Enlightenment with its new Rationality. Logicians did not notice that Russell's Paradoxes wrecked only the naive PL and try in vain, till our days, to patch it rather than to scrap the wreck and conceive in its place a new rational Logic.
2.1.Russell's Set-Theoretical Paradox
It seems to make sense to inquire, for any set, whether it is member of itself or not. The set of horses, e. g., is certainly not a horse. The set of all sets, on the contrary, being a set, is clearly member of itself. Therefore it seems reasonable to ask the same question with regard to "S" the 'set of all sets that are not members of themselves'. The answer is alarming: S IS MEMBER OF S IFF S IS NOT MEMBER OF S. (IFF="if end only if"). A contradiction derived from apparently plausible assumptions.
2.2.Russell's Logical Paradox
Russell's Set-Theoretical Paradox shows that an apparently legitimate set may lead via sound inference to a logical contradiction thus proving that something is rotten at the very base of the naive Set Theory. Actually, the situation was much more serious. The Paradox turned out to be essentially logical and in no way depending on some perhaps out-of-the-way peculiarities of sets: We may inquire whether a property applies to itself or not. Property of being red is not red while property of being abstract is abstract. Calling "impredicable" the property of not-applying-to-itself we conclude that IMPREDICABLE IS IMPREDICABLE IFF IT IS NOT IMPREDICABLE.
3.Aftermath
It has been said that Russell's "Logical Paradox" shattered Logic, just like the MM experiment shuttered Physics. However, Logic should not be confused with PL. The Paradox indeed (and not surprisingly) wrecked the noumenal naive PL with its feeble foundation in misinterpreted linguistic expressions. One would expect PL, like mechanistic Physics, to be dropped and replaced by some redefined and refounded rational Logic. Yet, such is the sticking power of Dogma, such is the dread of quitting Naive View and of following the "Sapere Aude" (Dare to reason) of Horace and Kant, that all Logicians (some considered as leading thinkers of 20th Century) 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 and stays till our days an extraordinary proliferation of competitive remedial systems all noumenal, ill-founded and meaningless. This critic which may appear severe is justified in more detail in CA2.INTRODUCTION TO PREDICATE LOGIC. Somebody's got to say at last that the Emperor is naked.