Finite structural axiomatization of every finite-valued propositional calculus

In [2] A. Wroski proved that there is a strongly finite consequence C which is not finitely based i.e. for every consequence C ⁺ determined by a finite set of standard rules C C ⁺. In this paper it will be proved that for every strongly finite consequence C there is a consequence C ⁺ determined by a finite set of structural rules such that C(Ø)=C ⁺(Ø) and = (where , are consequences obtained by adding to the rules of C, C ⁺ respectively the rule of substitution). Moreover it will be shown that under certain assumptions C=C ⁺.     

Strong completeness with respect to finite kripke models

We prove that each intermediate or normal modal logic is strongly complete with respect to a class of finite Kripke frames iff it is tabular, i.e. the respective variety of pseudo-Boolean or modal algebras, corresponding to it, is generated by a finite algebra. The author wishes to thank the Editors for calling his attention to the fact that the result of this paper concerning the intermediate logics was announced earlier by A. Wroński at the conference “Logical calculi”, Wrocław, October 5–7, 1975, though without proof. Wroński's result has not been published.     

On finite approximability of ψ-intermediate logics

The aim of this note is to show (Theorem 1.6) that in each of the cases: = {, }, or {, , }, or {, , } there are uncountably many -intermediate logics which are not finitely approximable. This result together with the results known in literature allow us to conclude (Theorem 2.2) that for each : either all -intermediate logics are finitely approximate or there are uncountably many of them which lack the property.     

Measuring Cultural and Creative Activities in European Cities: Challenges and Practical Problems with Existing Indices

For decades, the focus of cultural, social, and economic policies has been shifting to cities. Cities with the highest concentration of creative employment have demonstrated the greatest resilience in the post-2008 economic recovery. Although, over the last few years, there has been an explosion of cultural or creative related indices for cities, no single reference index is commonly accepted. The main thrust of this article is the provision of a systematic and practical framework for a common point of departure for future index developers. To this end, 38 of the most influential culture and creativity indices were reviewed. By adoption of the indicator-driven approach, 16 dimensions corresponding to aspects of culture and creativity were identified in the city context. These dimensions are presented in a novel taxonomy of 3 clusters that aims to distinguish indicators capturing core cultural and creative activities from those oriented on measuring cities’ organic background or socio-economic outcomes of the aforementioned activities. In addition, the main obstacles most commonly faced by index developers are described and grouped into 4 distinct conceptual challenges, namely (a) precision, (b) diversity, (c) comparability, and (d) spillover. Each challenge is discussed in detail, accompanied by a systematic approach to its solution.