The author reflects on the moral attitudes displayed by Poles fighting in the Warsaw Uprising. He believes that the sacrifice and selflessness with which Varsovians battled for their city had its roots in the general mentality of the Poles, who for generations had been raised in the spirit of “mutual and willing endowment”. He also notes that the noble ideals of the wartime generations have today been largely replaced by mercenary selfishness.

Let us consider an arbitrary semantical relation. It holds between some linguistic entities and pieces of reality referred to. We may call it meaning. The controversy between psychologism and antipsychologism therefore may be exhibited as an ontological dilemma:Antipsychologism PsychologismThe relation of meaning is independent of human beings The relation of meaning is established by human beingsWhen we describe the meaning of words we do not need to refer to human behavior When we describe the meaning of words we need to (...) refer to human behaviorIt seems to be safe to assume that reality is not contradictory and a good description of reality cannot lead to the contradiction. All presentations of semantical antinomies rest on antipsychologistic description of semantical relations. Thus we can blame the contradictions called semantical antinomies on the antipsychologistic description. (shrink)

In the present paper the well-known Gödels – Churchs argument concerning the undecidability of logic (of the first order functional calculus) is exhibited in a way which seems to be philosophically interestingfi The natural numbers are not used. (Neither Chinese Theorem nor other specifically mathematical tricks are applied.) Only elementary logic and very simple set-theoretical constructions are put into the proof. Instead of the arithmetization I use the theory of concatenation (formalized by Alfred Tarski). This theory proves to be an (...) appropriate tool. The decidability is defined directly as the property of graphical discernibility of formulas. (shrink)

The author reflects on the moral attitudes displayed by Poles fighting in the Warsaw Uprising. He believes that the sacrifice and selflessness with which Varsovians battled for their city had its roots in the general mentality of the Poles, who for generations had been raised in the spirit of “mutual and willing endowment”. He also notes that the noble ideals of the wartime generations have today been largely replaced by mercenary selfishness.

Recent years have seen the appearance of many English-language hand books of logic and numerous monographs on topical discoveries in the foundations of mathematics. These publications on the foundations of mathematics as a whole are rather difficult for the beginners or refer the reader to other handbooks and various piecemeal contribu tions and also sometimes to largely conceived "mathematical fol klore" of unpublished results. As distinct from these, the present book is as easy as possible systematic exposition of the now (...) classical results in the foundations of mathematics. Hence the book may be useful especially for those readers who want to have all the proofs carried out in full and all the concepts explained in detail. In this sense the book is self-contained. The reader's ability to guess is not assumed, and the author's ambition was to reduce the use of such words as evident and obvious in proofs to a minimum. This is why the book, it is believed, may be helpful in teaching or learning the foundation of mathematics in those situations in which the student cannot refer to a parallel lecture on the subject. This is also the reason that I do not insert in the book the last results and the most modem and fashionable approaches to the subject, which does not enrich the essential knowledge in founda tions but can discourage the beginner by their abstract form. A. G. (shrink)

The paper proposes a new definition of the conception of effectiveness . A good name for this version of effectiveness is discernibility. The definition is based on the fact that every computation may be reduced to the operation of discerning the fundamental symbols and concatenation of formulas. This approach to effectiveness allows us to formulate the proof of undecidability in such a way that arithmetization of the syntax may be replaced by the use of concatenation in metalogic.

Some people approve of certain general rules of behavior, or some concrete cases. The others disapprove of or are indifferent to them. In this paper I suggest an axiom system which formalizes the use of these utterances. It may be considered as a special (individualistic) approach to deontic logic.

Modern methodology furnishes two partly competitive and partly complementary views on structure of the development of scientific investigation. According to the first view the development of science consists in enlargement of the set of empirical theorems; according to the other it consists, rather, in the narrowing of the set of possible theoretical hypotheses. A particular kind of assertion is associated with each of these views. The first is associated with the relation of assertion expressed in the statement: “the state α (...) of scientific investigationB forces us to accept the statement Τ”. The other view is associated with the relation of assertion expressed in the statement: “the state α of scientific investigationB admits the acceptance of the statement Τ”. Under a suitable definition of forcing we obtain the conclusion that intuitionistic logic is the class of only those formulas which we are forced to accept by every state of every scientific investigation. The notion of admission under a certain definition analogically leads to intuitionistic logic; on the other hand in the case of another, more natural definition, it leads to justification of a certain system of strict implication which is stronger than Lewis’ system S4 and which overlaps system S5.It can be said that assuming quite a natural definition of admission this system contains only those formulas the acceptance of which is admitted by every state of every scientific investigation. (shrink)

In the second installment to Gruszczyński and Pietruszczak we carry out an analysis of spaces of points of Grzegorczyk structures. At the outset we introduce notions of a concentric and \-concentric topological space and we recollect some facts proven in the first part which are important for the sequel. Theorem 2.9 is a strengthening of Theorem 5.13, as we obtain stronger conclusion weakening Tychonoff separation axiom to mere regularity. This leads to a stronger version of Theorem 6.10. Further, we (...) show that Grzegorczyk points are maximal contracting filters in the sense of De Vries, but the converse inclusion is not necessarily true. We also compare the notions of a Grzegorczyk point and an ultrafilter, and establish several properties of topological spaces based on Grzegorczyk structures. The main results of the paper are representation and completion theorems for G-structures. We prove both set-theoretical and topological representation theorems for various classes of G-structures. We also present topological object duality theorem for the class of complete G-structures and the class of concentric spaces, both restricted to structures which satisfy countable chain condition. We conclude the paper with proving equivalence of the original Grzegorczyk axiom with the one accepted by us as axiom. (shrink)

This is the first, out of two papers, devoted to Andrzej Grzegorczyk’s point-free system of topology from Grzegorczyk :228–235, 1960. https://doi.org/10.1007/BF00485101). His system was one of the very first fully fledged axiomatizations of topology based on the notions of region, parthood and separation. Its peculiar and interesting feature is the definition of point, whose intention is to grasp our geometrical intuitions of points as systems of shrinking regions of space. In this part we analyze separation structures and (...) class='Hi'>Grzegorczyk structures, and establish their properties which will be useful in the sequel. We prove that in the class of Urysohn spaces with countable chain condition, to every topologically interpreted representative of a point in the sense of Grzegorczyk’s corresponds exactly one point of a space. We also demonstrate that Tychonoff first-countable spaces give rise to complete Grzegorczyk structures. The results established below will be used in the second part devoted to points and topological spaces. (shrink)

Grzegorczyk's modal logic corresponds to the class of upwards well-founded partially ordered Kripke frames, however all known proofs of this fact utilize some form of the Axiom of Choice; G. Boolos asked in [1], whether it is provable in plain ZF. We answer his question negatively: Grz corresponds to a class of frames, which does not provably coincide with upwards well-founded posets in ZF alone.