The axiom of canonicity was introduced by the famous Polish logician Roman Suszko in 1951 as an explication of Skolem's Paradox and a precise representation of the axiom of restriction in set theory proposed much earlier by Abraham Fraenkel. We discuss the main features of Suszko's contribution and hint at its possible further applications.

We review the four recent monographs by Roman Murawski.

The article presents Konrad Zacharias Lorenz\'s views concerning man\'s axiological space. According to Lorenz the origin of the cognitive apparatus is defined by the need to maintain orientation in the world. The apparatus is determined (ambiguously) by the biological and cultural factors. Also tendencies to accept a given type of values are dependent on inborn instincts. The harmony of mutual coexistence of biological and cultural norms is a manifestation of a \'healthy functioning of the world of values\'. In recognizing values (...) an important role is attributed to emotions. The specificity of the act of valuation results from its subjectivity outside a direct control of acts of consciousness. (shrink)

We investigate computability theoretic and topological properties of spaces of orders on computable orderable groups. A left order on a group G is a linear order of the domain of G, which is left-invariant under the group operation. Right orders and bi-orders are defined similarly. In particular, we study groups for which the spaces of left orders are homeomorphic to the Cantor set, and their Turing degree spectra contain certain upper cones of degrees. Our approach unifies and extends Sikora’s [28] (...) investigation of orders on groups in topology and Solomon’s [31] investigation of these orders in computable algebra. Furthermore, we establish that a computable free group Fn of rank n>1 has a bi-order in every Turing degree. (shrink)

This note discusses some problems concerning intended, standard, and nonstandard models of mathematical theories. We pay attention to the role of extremal axioms in attempts at a unique characterization of the intended models. We recall also Jan Woleński’s views on these issues.

Certain mathematical objects bear the name “pathological”. They either occur as unexpected and unwilling in mathematical research practice, or are constructed deliberately, for instance in order to delimit the scope of application of a theorem. I discuss examples of mathematical pathologies and the circumstances of their emergence. I focus my attention on the creative role of pathologies in the development of mathematics. Finally, I propose a few reflections concerning the degree of cognitive accessibility of mathematical objects. I believe that the (...) problems discussed in the paper may attract the attention of philosophers interested in concept formation and the development of mathematical ideas. (shrink)

In this paper I discuss Ernst Zermelo’s ideas concerning the possibility of developing a system of infinitary logic that, in his opinion, should be suitable for mathematical inferences. The presentation of Zermelo’s ideas is accompanied with some remarks concerning the development of infinitary logic. I also stress the fact that the second axiomatization of set theory provided by Zermelo in 1930 involved the use of extremal axioms of a very specific sort.1.

As early as the beginning of the Second World War J. Maritain in his two articles presented the vision of a United Europe. Our continent in fact was always in the state of war. The 20th century could have become a decline of European history, if the wars had not ended. I. Maritain\'s initiation was therefore to show the ways to peace. One way is unity of Europe, a formation of the federation of European nations. The article shows this way (...) and its theoretical justification, something that can be found in his philosophical conception of politics. (shrink)

Chciałem Go pożegnać. Kilkakrotnie próbowałem napisać choć parę słów. O Nim samym. Okazało się, że nie potrafię. Udało mi się tylko przedstawić Jego poglądy na reizm Tadeusza Kotarbińskiego, ale uczyniłem to gdzie indziej. Muszę tu poprzestać na ofiarowaniu Mu tego, co napisałem na Jego dziewięćdziesiąte urodziny, w osiemdziesięciolecie naszej przyjaźni.

We prove a 0-1 law for the fragment of second order logic SO over parametric classes of finite structures which allow only one unary atomic type. This completes the investigation of 0-1 laws for fragments of second order logic defined in terms of first order quantifier prefixes over, e.g., simple graphs and tournaments. We also prove a low oscillation law, and establish the 0-1 law for Σ14 without any restriction on the number of unary types.