A logical systemBM + is proposed, which, is a prepositional calculus enlarged with prepositional quantifiers and with two modal signs, and These modalities are submitted to a finite number of axioms. is the usual sign of necessity, corresponds to transmutation of a property (to be white) into the abstract property (to be the whiteness). An imbedding of the usual theory of classesM intoBM + is constructed, such that a formulaA is provable inM if and only if(A) is provable inBM +. (...) There is also an inverse imbedding with an analogous property. (shrink)

An attempt is made to include the axioms of Mackey for probabilities of experiments in quantum mechanics into the calculus x0 of ukasiewicz. The obtained calculusQ contains an additional modal signQ and four modal rules of inference. The propositionQx is read x is confirmed. The most specific rule of inference may be read: for comparable observations implication is equivalent to confirmation of material implication.The semantic truth ofQ is established by the interpretation with the help of physical objects obeying to the (...) rules of quantum mechanics. The embedding of the usual quantum propositional logic inQ is accomplished. (shrink)

We say that an n-argument predicate P n is finite, if P is a finite set. Note that the set of individuals is infinite! Finite predicates are useful in data bases and in finite mathematics. The logic DBL proposed here operates on finite predicates only. We construct an imbedding for DBL in a special modal logic MPL. We prove that if a finite predicate is expressible in the classical logic, it is also expressible in DBL. Quantifiers are not necessary in (...) DBL. Some simple algebraic properties of DBL are indicated. (shrink)

I dare say, a set is contranatural if some pair of its elements has a nonempty intersection. So, we consider only collections of disjoint nonempty elements and call them totalities. We propose the propositional logicTT, where a proposition letters some totality. The proposition is true if it letters the greatest totality. There are five connectives inTT: , , , , # and the last is called plexus. The truth of # means that any element of the totality has a nonempty (...) intersection with any element of the totality . An imbeddingG of the classical predicate logicCPL inTT is defined. A formulaf ofCPL is a classical tautology if and only ifG(f) is always true inTT. So, mathematics may be expounded inTT, without quantifiers. (shrink)

Attempts are made to transform the basis of elementary probability theory into the logical calculus.We obtain the propositional calculus NP by a naive approach. As rules of transformation, NP has rules of the classical propositional logic (for events), rules of the ukasiewicz logic 0 (for probabilities) and axioms of probability theory, in the form of rules of inference. We prove equivalence of NP with a fragmentary probability theory, in which one may only add and subtract probabilities.