Formal Logic, Models, Reality

(5-2) |— A Ais classically provable. Intuitionistic Logic is the logic of constructive mathematics.The Law of Excluded Middle fails in intuitionistic logic. This shows that a classicaldeductive system can lead from premises which can be jointly true in intuitionisticmathematics to a conclusion which is false in intuitionistic mathematics. Thereforethe Soundness Theorem of classical logic fails in constructive mathematics.(II) The Distributive Law(5-3) A (B C) |— (A B) (A C)is classically valid. But it fails in Quantum Logic based on the Hilbert space model ofquantum mechanics. When the Distributive Law is combined with premises whichare true in the Hilbert space model of quantum mechanics, it can lead to conclusionswhich are false in the same model. This shows that there are Hilbert space models ofquantum mechanics which cannot be adequately represented in set-theoretical semanticmodels. Classical formal logic cannot be applied directly to such Hilbert spacemodels and their corresponding quantum systems.(III) The Law of Conjunction Introduction(5-4) A, B |— A Bis provable in classical logic, intuitionistic logic, and in the standard quantum logic.It is generally believed that this logical law cannot possibly fail. Nevertheless I showin my book Logical Physics: Quantum Reality Theory that if there is a local quantumreality, then the law (5-4) must fail there. Moreover, it can be shown that the failureof (5-4) is not as absurd as it first appears since the invalidity of (5-4) in the quantumworld is a consequence of an ontic interpretation of the indeterminacy relations. Ifthe logical law (5-4) is applied to premises which are true in a local quantum reality,this can lead to false conclusions like for instance Bell's inequality. Therefore classicalformal logic is not sound when it is applied to a local quantum reality, and classicalformal logic cannot be applied directly to a local quantum reality. It can only beapplied to set-theoretical semantic models of such a kind of reality which can be adequatelyrepresented in such models.1.6 ANALYSIS. We consider logics of truth, that is, such logics which apply tofields of sentences which can be assigned exactly one of the truth values True andFalse. Such a logic is formal iff every logically valid inference(6-1) A 1 , …, A n |= Bis valid due only to the form of the sentences A 1 , …, A n , B and not dependend ontheir contents. As examples of such inferences, we consider the Modus Ponens bothin formal logic and in the nonformal logic used in ordinary language:(6-2) A, A B |= B4