Generalization of Scott's formula for retractions from generalized Alexandroff's cube

In the paper [2] the following theorem is shown: Theorem (Th. 3,5, [2]), If α=0 or δ=∞ or α?δ, then a closure space X is an absolute extensor for the category of 〈α, δ〉 -closure spaces iff a contraction of X is the closure space of all 〈α, δ〉-filters in an 〈α, δ〉-semidistributive lattice. In the case when α=ω and δ=∞, this theorem becomes Scott's theorem: Theorem ([7]). A topological space X is an absolute extensor for the category of all topological spaces iff a contraction of X is a topological space of “Scott's open sets” in a continuous lattice. On the other hand, when α=0 and δ=ω, this theorem becomes Jankowski's theorem: Theorem ([4]). A closure space X is an absolute extensor for the category of all closure spaces satisfying the compactness theorem iff a contraction of X is a closure space of all filters in a complete Heyting lattice. But for separate cases of α and δ, the Theorem 3.5 from [2] is proved using essentialy different methods. In this paper it is shown that this theorem can be proved using, for retraction, one uniform formula. Namely it is proved that if α= 0 or δ= ∞ or α ? δ and \(F_{\alpha ,\delta } \left( L \right) \subseteq B_{\alpha ,\delta }^\mathfrak{n} \) and if L is an 〈α, δ〉-semidistributive lattice, then the function $$r:{\text{ }}B_{\alpha ,\delta }^\mathfrak{n} \to F_{\alpha ,\delta } \left( L \right)$$ such that for x ε ? ( \(\mathfrak{n}\) ): (*) $$r\left( x \right) = inf_L \left\{ {l \in L|\left( {\forall A \subseteq L} \right)x \in C\left( A \right) \Rightarrow l \in C\left( A \right)} \right\}$$ defines retraction, where C is a proper closure operator for \(B_{\alpha ,\delta }^\mathfrak{n} \) . It is also proved that the formula (*) defines retraction for all 〈α, δ〉, whenever L is an 〈α, δ〉 -pseudodistributive lattice. Moreover it is proved that when α=ω and δ=∞, the formula (*) defines identical retraction to the formula given in [7], and when α = 0 and δ=ω, the formula (*) defines identical retraction to the formula given in [4].     

Q-ultrafilters and normal ultrafilters in B-algebras

The first part of the paper deals with some subclasses of B-algebras and their applications to the semantics of SCI _B , the Boolean strengthening of the sentential calculus with identity (SCI). In the second part a generalization of the McKinsey-Tarski construction of well-connected topological Boolean, algebras to the class of B-algebras is given.     

Ideal objects as models in science

Three main concepts of model in science are distinguished: (1) semantical model of a theory; (2) real model of another real thing; (3) mathematical model of a real thing. The last concept is the most important for the empirical sciences. The mathematical model is not identical with a theory: it is an ideal object which is directly described by the theory. We have here an intermediate level between reality and theory.     

From the Profession

Studies in East European Thought -     

VEP latency and some properties of simple motor reaction-time distribution

Summary The median of simple motor reaction (RT) to a flash parallels the latencies of visually evoked potential (VEP) deflections if flash intensity is varied. However, mean and median reaction times are not equal because of the skewness of RT distribution. It therefore seemed plausible that the mean reaction time — intensity relationship would be steeper than that for VEP latency. Such divergence would account for the intensity-dependent motor component of RT revealed by other psychophysical studies. The latencies of VEP deflections were measured as a function of intensity and the results were compared with mean and median RTs. The difference between mean and median RT is constant and independent of flash intensity. Moreover, both values are parallel to VEP latency. The general pattern of results remains the same after a change in the distribution from which the foreperiod is sampled.     

Three contributions to the two-valued propositional calculus

Summary Three chapters contain the results independent of each other. In the first chapter I present a set of axioms for the propositional calculus which are shorter than the ones known so far, in the second one I give a method of defining all ternary connectives, in the third one, I prove that the probability of propositional functions is preserved under reversible substitutions. This paper appeard orginally under the title “Trois contributions an calcul des propositions bivalent” inStudia Societatis Scientiarum Torunensis, Toruń, Polonia, Sectio A, vol. I (1948), pp. 3–15.     

Concerning axiomatizability of the quasivariety generated by a finite Heyting or topological Boolean algebra

In classes of algebras such as lattices, groups, and rings, there are finite algebras which individually generate quasivarieties which are not finitely axiomatizable (see [2], [3], [8]). We show here that this kind of algebras also exist in Heyting algebras as well as in topological Boolean algebras. Moreover, we show that the lattice join of two finitely axiomatizable quasivarieties, each generated by a finite Heyting or topological Boolean algebra, respectively, need not be finitely axiomatizable. Finally, we solve problem 4 asked in Rautenberg [10].     

O związkach pomiędzy wynikaniem,sprzecznością i zupeŀnością

Allatum est die 6Aprilis 1965     

O klasach normalnych i nienormalnych na terenie języka potocznego

Problematyka i glówne tezy tej pracy pochodz od pierwszego autora. Udzial drugiego autora sprowadza sie do propozycji utosamienia jeezyka potocznego z formalizmem logiki zda i logiki nazw oraz do pewnych sugestii odnonie systematyzacji materialuAllatum est die 2 Octobris 1965