Abstracts of papers published in 1980

A. Pelc, An example of formalizing some intensional expressions, Studia Filozoficzne 5 (1980), pp. 105-109 (in Polish).


One of the main difficulties encountered while using intensional sentences is that applying usual logical operations to them, such as substitution or changing a sub-sentence into an equivalent one, can turn the original true sentence into a false one. We give an example of formalizing this problem which shows that a suitable understanding of the above operations permits to apply them to intensional sentences as well, without risk of changing their logical value. As a tool we use Kripke's semantics.

A. Pelc, On degrees of uncertainty in mathematics, Studia Filozoficzne 3 (1980), pp. 91-94 (in Polish).


Given a formal theory T, additional hypotheses that do not follow from axioms of T may contradict T even when T itself is consistent. We compare this risk of contradiction for various theories and various additional hypotheses, indicating that there are many possible degrees of this risk and consequently a gradation of uncertainty of such hypotheses.

A. Krawczyk, A. Pelc, On families of sigma-complete ideals, Fundamenta Mathematicae 109 (1980), pp. 155-161.


Our main results are the following: Assume martin's Axiom. Then

1. For every family of fewer than continuum two-valued uniform measures on the reals, there exists a set of reals non-measurable with respect to any of them.

2. For every cardinal k larger than continuum but smaller than the first cardinal carrying a continuum-complete continuum-saturated ideal, the following holds: For every family of fewer than continuum two-valued continuum-additive measures on k, there exists a subset of k non-measurable with respect to any of them.