W artykule proponuje się aksjomatyzację fragmentu klasycznej logiki zdań, który zostaje nazwany logiką niesymplifikacyjną. Nazwa pochodzi stąd, że unika się w nim, jak przypuszcza pewna ilość badaczy – wskazanych w tekście – paradoksalnych własności klasycznej implikacji i równoważności. Twierdzenie o niewyprowadzalności tez paradoksalnych nie jest jeszcze definitywnie udowodnione. Ale cały bieg myśli wydaje się interesujący z filozoficznego punktu widzenia ze względu na możliwość odróżnienia kilku rodzajów implikacji i równoważności.

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)

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handbooks and various piecemeal contribu tions and also sometimes to largely conceived "mathematical fol klore" of unpublished results.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 aim of this paper is to argue that very often one is not able to distinguish between deterministic processes governed by some dynamical systems and stochastic processes.

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)

Pojęcie kategoryczności zostało wprowadzone w początku obecnego wieku, ażeby scharakteryzować teorie, które w intencji ich twórców mają dokładnie jeden model. Dalszy rozwój badań metalogicznych dotyczących modeli doprowadził do spostrezeżenia, że każda teoria matematyczna opisująca pewien model nieskończony, opisuje równocześnie wiele innych modeli z nim nieizomorficznych. Stąd pierwotne pojęcie kategoryczności straciło na znaczeniu. Pojawiły się natomiast badania mające na celu pewne zmodyfikowanie tego pojęcia. Praca niniejsza omawia różne rodzaje pojęcia kategoryczności znane z literatury logicznej i proponuje pewne nowe.

The contemporary versions of the ontological argument that originated from Charles Hartshorne are formalized proofs based on unique modal theories. The simplest well-known theory of this kind arises from the b system of modal logic by adding two extra-logical axioms: “If the perfect being exists, then it necessarily exists‘ and “It is possible that the perfect being exists‘. In the paper a similar argument is presented, however none of the systems of modal logic is relevant to it. Its only premises (...) are the axiom and, instead of, the new axiom : “If the perfect being doesn’t exist, it necessarily doesn’t‘. The main goal of the work is to prove that is no more controversial than and -- in consequence -- the whole strength of the modal ontological argument lies in the set of its extra-logical premises. In order to do that, three arguments are formulated: ontological, “cosmological‘ and metalogical. (shrink)

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.