Gentzen’s approach to deductive systems, and Carnap’s and Popper’s treatment of probability in logic were two fruitful ideas that appeared in logic of the mid-twentieth century. By combining these two concepts, the notion of sentence probability, and the deduction relation formalized in the sequent calculus, we introduce the notion of ’probabilized sequent’ \ with the intended meaning that “the probability of truthfulness of \ belongs to the interval [a, b]”. This method makes it possible to define a system of derivations (...) based on ’axioms’ of the form \, obtained as a result of empirical research, and then infer conclusions of the form \. We discuss the consistency, define the models, and prove the soundness and completeness for the defined probabilized sequent calculus. (shrink)

A topological class logic is an infinitary logic formed by combining a first-order logic with the quantifier symbols O and C. The meaning of a formula closed by quantifier O is that the set defined by the formula is open. Similarly, a formula closed by quantifier C means that the set is closed. The corresponding models are a topological class spaces introduced by Ćirić and Mijajlović (Math Bakanica 1990). The completeness theorem is proved.