Relativistic Dialectics Relativistic Dialectics |
Georges Metanomski Crisis of Logic Implication |
In one of the letters written to the Infeld group in Warsaw Einstein wrote: |
IMPLICATION (Exact 2D Propositional Calculus) Let's consider two statements p: "it has been raining over the street" q: "the street is wet" NOTE: By p and q we mean that: K. The rain was sufficient to wet the street. L. It's been raining recently and the street had no time to dry, M. It's the same street in p and q. Implication in metalanguage: "If it has been raining over the street then the street is wet" or "it has been raining over the street implies that the street is wet" Implication in Calculus: case. imp(pq) 1:....1...11 2:....0...10 3:....1...01 4:....1...00 Let imp(pq) axiom of theory T. We shall discuss: A. Application of T AA.Deductive AB. Inductive B. Research on T A. APPLICATION OF T (Application is based upon belief that imp(pq) holds). ------------------- AA. Deductive ------------- Meteo forcasts rain in concerned area. We deduce from case 1: case. imp(pq) 1:....1...11 that the street will soon be wet and act accordingly, eg. clean the gutters. AB. Inductive ------------- Meteo stated that it has been raining in concerned area, but we state that our street is dry. Upon our belief in imp(pq) holding, we induce from case 4: case. imp(pq) 4:....1...00 that p=0, i. e. that it has not been raining in concerned area and we inform meteo system that it has a bug. On the other hand, if we state that our street is wet, we induce from case 1: case. imp(pq) 1:....1...11 that meteo was right and keep happy and quiet. RESEARCH ON T ------------- We gather FACTUAL INFORMATION, i. e. OBSERVATIONS of p and q to see how they fit pertinent cases of the axiom: "imp(pq)": case. imp(pq) 1:....1...11 2:....0...10 3:....1...01 4:....1...00 Let's note that T CAN BE DISPROVED by a single observation fitting case 2. On the contrary, it cannot be PROVEN. Indeed, no matter how many observations may fit cases 1,3,4, they don't prove that some day we will not observe the case 2. The more observations fit 1,3,4, and the stronger gets our PRAGMATIC BELIEF in imp(pq) holding, but no matter how strong our belief, it is not a PROOF. Billions of observations per minute fit the "gravity theory", making us believe so strongly in gravity, that we take it for obvious and granted. Still, gravity is not PROVEN and while we believe that it will be there to morrow, there is no logical reason to be 100% certain that it will not cease in next second. We encounter here a central premise in the philosophy of science, the Principle of Falsifiability, first formally discussed by Karl Popper. This principle states that in order to be useful (or even scientific at all), a scientific statement ('fact', theory, 'law', 'principle', etc) must be falsifiable, i. e. able to be proven wrong. Without this property, it would be difficult (if not impossible) to test a scientific statement against the evidence. It's surprising to find that it took 2000 years to formulate this Principle, when it is obviously inherent to Implication which has been known to Aristoteles. EXERCISE -------- Let: p: "it has been raining over the street" q: "the street is wet" NOTE: By p we mean additionally that: K. The rain was sufficient to wet the street. L. It's been raining recently and the street had no time to dry, M. The street in p is the same as in q. Let imp(pq) axiom of theory T. case. imp(pq) 1:....1...11 2:....0...10 3:....1...01 4:....1...00 1. Try to find factual examples for case 3. and explain why they do not refute T. 2. Explain why case 1. does not prove T. 3. Try to refute T. |
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