PARENTS: LOGIC WORKSHOP WORKSHOPS ARCHIVE =============================================================================== Crisis of Logic =============================================================================== VILLAGE BARBER Some of greatest scientists were at the same time great popularizers and could illustrate most complex issues with perfectly simple examples and allegories accessible to laymen and written with high literary skill. One of them was Russell, who popularized his famous Paradox which shuttered the whole edifice of science with the equally famous puzzle of Village Barber. In order to relax before entering serious considerations I quote it here, wishing you a lot of fun: "A village barber shaves all men of the village who do not shave themselves and only those. Who shaves the barber?" NOTE: Please don't post wisecracks like "Barber is she". The above is the original formulation of Russell, who addressed it to people having some imagination and knowing how to read between lines. If it's not your case, it's your problem. =============================================================================== CANTOR'S PARADISE Around 1873 Cantor formulated the Set Theory which soon became universally accepted as founding discipline of Mathematics. (By 'founding' we understand a discipline whose theorems are accepted as axioms by the 'founded'. We may say that Mathematics is founded since Cantor in the Set Theory). At the same time Physics was mechanistic and deterministic: "give me positions and speeds of all particles and I will tell you the entire history and future of the Universe." The "Cantorian Paradise" may be roughly described by the foundation hierarchy universally accepted at his time: 1.Logic, the universal, absolute, olympic foundation, 2.Set Theory, founded in Logic, 3.Mathematics, founded in Set Theory, 4.Physics, founded in Mathematics, 5.Other "hard" Sciences, founded in Physics, 6."Soft" Sciences, founded in "hard" Sciences. This "Paradise" encompassed the edifice of Science solid and stable as Cheops pyramid. And gave the most comfortable feeling to scientists seeing that Logic, THEIR Logic, explains the Universe and determines its destiny. =============================================================================== BREACHES IN THE PYRAMID However, at the end of the 19th century two breaches appeared in the pyramid: 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. =============================================================================== RUSSEL'S SET-THEORETICAL PARADOX It seems to make perfect sense to inquire, for any given 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 to make perfect sense 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. ------------------------------------------ A glaring contradiction derived from most plausible assumptions by unquestionable inference. =============================================================================== RUSSEL'S LOGICAL PARADOX Russell's Set-Theoretical Paradox shows that a legitimate set may lead via sound inference to a logical contradiction thus proving that something is rotten at the very base of the Set Theory. Actually, the situation was much more serious. Soon after discovering the Set-Theoretical version of the Paradox Russell realized that the Paradox is essentially logical and in no way depends 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 the property of not-applying-to-itself "impredicable" we conclude that IMPREDICABLE IS IMPREDICABLE IFF IMPREDICABLE IS NOT IMPREDICABLE. >>> This logical version of Russell's Paradox shattered the foundations of Logic and of 'Exact Sciences' just as Michelson's Experiment shuttered those of Physics. Unfortunately, unlike Einstein who reconstructed the whole shuttered edifice of Physics, Logicians only tried to patch the breaches locally, thus leaving the crisis of Logic open. =============================================================================== FREGE, THE FIRST VICTIM At the time when Russell discovered the Paradox, Frege was finishing a system of Logic compatible with Cantorian Paradise. Having read the publication of the Paradox he scrapped the work of several years. =============================================================================== CARNAP =============================================================================== Driven out of the paradise of absolute, objective certainty and determination, Carnap jumped to the opposite and became the most extreme thinker of the Vienna Circle, restricting science to empiricism and empiricism to introspection. Thus objective science appeared to be impossible and, on the other hand, introspection being unable to produce ultimate general premises all deductions became logically indeterminate. This extreme position is surprising, because Carnap subsequently moved to the least extreme position of the Vienna Circle and went further than anybody towards solving the crisis of Logic. He evolved in this direction under the influence of Popper, who was the fist in Vienna Circle to secure a basis of objective science, swampy and relative, but no more impossible. Under this influence Carnap conceived his duality of "truth": "logical truth" resulting from logical inferencing and "factual truth". However, he searched the definition of "factual truth" within Logic: <<< factually true is what is true without being logically true. >>> This definition resembles a trial to lift oneself by pulling on one's own shoes and Carnap's tentative finished as a failure. Nevertheless, it introduced the duality of "truth" and indicated the direction of searching the "factual truth" in semantic analysis, thus opening a way for post-Carnapian logicians. =============================================================================== POST-CARNAPIAN SITUATION After Carnap, Logic has to satisfy following requirements: -Fuzzy, "swampy", gray "probability" or "certainty" replacing the traditional black and white "truth/falsity", -Axiomatic ultimate premises similar to Popper's "piers" driven into "swamps", -Factual ultimate conclusions similar to physical observations, -Unlimited dimensionality allowing to deduce from any number of premises and to induce from any number of conclusions, which implies network as fundamental logical structure. -Customized logic: algorithms of fuzzy operators vary from application to application. We show a small example of a logical system satisfying all those requirements in FUZZY LOGIC EXAMPLE . =============================================================================== This logical system is a generalization of Propositional Calculus, which we describe shortly in: EXACT 2-DIM PROPOSITIONAL CALCULUS and EXACT N-DIM PROPOSITIONAL CALCULUS . These steps are indispensable for understanding of the FUZZY LOGIC EXAMPLE =============================================================================== PARENTS: LOGIC WORKSHOP WORKSHOPS ARCHIVE