We prove that if f is a partial Borel function from one Polish space to another, then either f can be decomposed into countably many partial continuous functions, or else f contains the countable infinite power of a bijection that maps a convergent sequence together with its limit onto a discrete space. This is a generalization of a dichotomy discovered by Solecki for Baire class 1 functions. As an application, we provide a characterization of functions which are countable unions of (...) continuous functions with domains of type Πn0, for a fixed n<ω. For Baire class 1 functions, this generalizes analogous characterizations proved by Jayne and Rogers for n=1 and Semmes for n=2. (shrink)

An explanation is given of why, after adding to a model M of ZFC first a Solovay real r and next a Cohen real c, in M[ r][ c] a Cohen real over M[ c] is produced. It is also shown that a Solovay algebra iterated with a Cohen algebra can be embedded into a Cohen algebra iterated with a Solovay algebra.

Any finite support iteration of posets with precalibre ℵ 1 which has the length of cofinality greater than ω 1 yields a model for the dual Borel conjecture in which the real line is covered by ℵ 1 strong measure zero sets.

We introduce a new cardinal characteristic r*, related to the reaping number r, and show that posets of size $ r* which add reals add unbounded reals; posets of size $ r which add unbounded reals add Cohen reals. We also show that add(M) ≤ min(r, r*). It follows that posets of size < add(M) which add reals add Cohen reals. This improves results of Roslanowski and Shelah [RS] and of Zapletal [Z].

We construct a G δ set G ⊆ ω ω ×2 ω with null vertical sections such that each perfect set P ⊆2 ω meets almost all vertical sections of G in the following sense: we can define from P subsets S of ω of density zero such that whenever the section determined by x ∈ ω ω does not meet P , then x ∈ S for all but finitely many i . This generalizes theorems of Mokobodzki and Brendle (...) et al. 185–199). (shrink)

We prove that if the Mathias forcing is followed by a forcing with the Laver Property, then any V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {V}$$\end{document}-q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {q}$$\end{document}-point is isomorphic via a ground model bijection to the canonical V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {V}$$\end{document}-Ramsey ultrafilter added by the Mathias real. This improves a result of Shelah and Spinas.

The Covering Property Axiom, which attempts to capture some of the combinatorics of the Sacks model, the model obtained from by countable support iteration of length of the Sacks forcing, seems to miss a Suslin tree. We add a diamond polish to the axiom to remedy this.

The authors investigate an operation * on the subsets of ${\mathcal{P}(\mathbb{R})}$ . It is connected with Borel’s strong measure zero sets as well as strongly meager. The results concern the behaviour of the family of countable sets when * is applied.

Let J be any proper ideal of subsets of the real line R which contains all finite subsets of R. We define an ideal J * ∣B as follows: X ∈ J * ∣B if there exists a Borel set $B \subset R \times R$ such that $X \subset B$ and for any x ∈ R we have $\{y \in R: \langle x,y\rangle \in B\} \in \mathscr{J}$ . We show that there exists a family $\mathscr{A} \subset \mathscr{J}^\ast\mid\mathscr{B}$ of power ω (...) 1 such that $\bigcup\mathscr{A} \not\in \mathscr{J}^\ast\mid\mathscr{B}$ . In the last section we investigate properties of ideals of Lebesgue measure zero sets and meager sets in Cohen extensions of models of set theory. (shrink)

We show that for an uncountable κ in a suitable Cohen real model for any family {A ν } ν<κ of sets of reals there is a σ-homomorphism h from the σ-algebra generated by Borel sets and the sets A ν into the algebra of Baire subsets of 2 κ modulo meager sets such that for all Borel B, B is meager iff h(B) = 0. The proof is uniform, works also for random reals and the Lebesgue measure, and in (...) this way generalizes previous results of Carlson and Solovay for the Lebesgue measure and of Kamburelis and Zakrzewski for the Baire property. (shrink)

We study the degree of (weak) Q-pointness, and that of (weak) P-pointness, of ideals on a regular infinite cardinal.

The class of equivalential logics comprises all implicative logics in the sense of Rasiowa [9], Suszko's logic SCI and many others. Roughly speaking, a logic is equivalential iff the greatest strict congruences in its matrices are determined by polynomials. The present paper is the first part of the survey in which systematic investigations into this class of logics are undertaken. Using results given in [3] and general theorems from the theory of quasi-varieties of models [5] we give a characterization of (...) all simple C-matrices for any equivalential logic C. In corollaries we give necessary and sufficient conditions for the class of all simple models for a given equivalential logic to be closed under free products. (shrink)

In the late forties, Stanisław Jaśkowski published two papers onthe discursive sentential calculus, D2. He provided a definition of it by an interpretation in the language of S5 of Lewis. The knownaxiomatization of D2 with discursive connectives as primitives was introduced by da Costa, Dubikajtis and Kotas in 1977. It turns out, however,that one of the axioms they used is not a thesis of the real Jaśkowski’s calculus. In fact, they built a new system, D∗2 for short, that differs from (...) D2 inmany respects. The aim of this paper is to introduce a direct Kripke-type semantics for the system, axiomatize it in a new way and prove soundness andcompleteness theorems. Additionally, we present labelled tableaux for D∗2.1. (shrink)

We are facing a crisis of trust in expertise today. Fewer and fewer people trust experts, and more and more politicians openly ignore expert consensus. 'Democracy in the Post-Truth Era' asks what might happen to democracy if we reject the fundamental liberal assumption that people are capable of making informed choices. The book explores the potential impact on society if people, including politicians, never appreciate the relevance of expert opinions. What if people cannot choose between supporters and opponents of key (...) issues such as the causes of climate change, effectiveness of vaccines and ways of combating economic and social crises? Applying social epistemology to these questions, the book considers whether democracy can be defended in the post-truth era. (shrink)

The paper is conceived as a first study on the Suszko operator. The purpose of this paper is to indicate the existence of close relations holding between the properties of the Suszko operator and the structural properties of the model class for various sentential logics. The emphasis is put on generality both of the results and methods of tackling the problems that arise in the theory of this operator. The attempt is made here to develop the theory for non-protoalgebraic logics.

Analizy przeprowadzone w artykule pozwalają na wyciągnięcie wniosków, ważnych dla filozofa przyrody. Między filozofią a naukami przyrodniczymi istnieje interakcja, którą można określić jako współprzenikanie się obu tych dziedzin wiedzy. To współoddziaływanie filozofii z nauką podobne jest do niedomykającego się koła. Trudno jest owo współprzenikanie jednoznacznie scharakteryzować, ale można wskazać pewne idee towarzyszące mu. Uświadomienie sobie filozoficznego uwikłania teorii naukowej powoduje często rozszerzenie jej rozumienia oraz ujawnia, jak wiele informacji zawartych jest w teorii naukowej na temat głębokiej struktury świata. Sformułowanie „filozofia (...) w nauce” można by uznać za charakterystyczne dla współczesnego ujmowania związku filozofii z naukami przyrodniczymi. Należy pamiętać, że wspólna droga filozofii i nauk przyrodniczych pozwala osiągać interesujące poznawczo wyniki, ważne dla rozumienia świata zewnętrznego. Współprzenikanie się filozofii z nauk jest procesem dynamicznym i twórczym. Tak filozofia, jak i nauka, ulegają ciągłym zmianom, które w istotny sposób wpływają na rozwijanie się procesu współprzenikania. Trudno jest przewidywać, w którą stronę ten proces będzie ewoluował. Ważne jest jednak, jak twierdzi M. Heller, aby w tym twórczym procesie dostrzec, że największym sukcesem nauk empirycznych, trwającym do dziś, jest coraz lepsze ugruntowanie się przekonania, że Wszechświat stopniowo, choć tylko w przybliżeniu, ujawnia nam tajemnice swojej struktury. -------------- Zgłoszono: 12/09/2020. Zrecenzowano: 10/10/2020. Zaakceptowano do publikacji: 29/10/2020. (shrink)

In this paper, we present a calculus of paraconsistent logic. We propose an axiomatisation and a semantics for the calculus, and prove several important meta-theorems. The calculus, denoted as CB1, is an extension of systems PI, C min and B1, and a proper subsystem of Sette’s calculus P1. We also investigate the generalization of CB1 to the hierarchy of related calculi.

In the paper we study the class of weakly algebraizable logics, characterized by the monotonicity and injectivity of the Leibniz operator on the theories of the logic. This class forms a new level in the non-linear hierarchy of protoalgebraic logics.

This book is both suitable for logically and algebraically minded graduate and advanced graduate students of mathematics, computer science and philosophy, and ...

In 1953, Jerzy Kalinowski published his paper on the logic of normative sentences. The paper is recognized as one of the first publications on the formal system of deontic logic. The aim of this paper is to present a tableau system for Kalinowski’s deontic logic and to discuss some of the topics related to the paradoxes of deontic logic.

This article attempts to answer the question: what has the current migration crisis revealed in relations between Christianity and Islam, two religions with common Abrahamic roots? What conclusions can be drawn about the state of these relations, particularly with regard to ordinary members of the two religions? How should we assess the reaction of Christians in Europe to the inflow of refugees from Muslim countries in the light of the Gospel, the Second Vatican Council and the teaching of Pope Francis?The (...) article consists of four parts. Part I briefly presents Christian-Muslim relationships over the centuries. Part II focuses on the sense of danger occurring in European societies in connection with the influx of Muslims during the migration crisis. Part III presents the teachings of the Second Vatican Council about Islam and Part IV describes Pope Francis’s attitude towards refugees from Muslim countries. (shrink)

This volume explores the range of Alfred North Whitehead's philosophy and his relevance to contemporary philosophical traditions.

This study investigates performance of portfolios composed of British socially responsible investments (SRI) stocks. Using the ‘Global-100 Most Sustainable Corporations in the World’ list (known also as ‘Global-100’) to select the SRI companies, we found that, in the period 2000–2010, the returns of the SRI portfolios were on average higher compared with the corresponding returns of the market indexes. The annual average difference in returns of the SRI portfolios (with dividends) was 5.26 % and 5.69 % relative to the FTSE100 (...) and FTSE4GOOD indexes (the total return versions), respectively, but the differences in returns in the whole period, in individual years and in other sub-periods were in most cases not statistically significant. Positive performance of SRI stocks in the whole sample is, however, evidenced by risk-adjusted measures such as the modified Sharpe ratio (MSR) and certainty equivalent (CEQ) returns, as well as by incorporating various levels of transaction costs. Furthermore, a simple trading strategy relying on selection of SRI stocks from the Global-100 list would beat the market indexes in the whole period 2000–2010, even after inclusion of various levels of transaction costs. We also estimated the Fama–French and Carhart multi-factor models and found that the returns of the SRI portfolios cannot be consistently explained by conventional factors other than the market factor. (shrink)

The class of equivalential logics comprises all implicative logics in the sense of Rasiowa [9], Suszko's logicSCI and many Others. Roughly speaking, a logic is equivalential iff the greatest strict congruences in its matrices (models) are determined by polynomials. The present paper is the first part of the survey in which systematic investigations into this class of logics are undertaken. Using results given in [3] and general theorems from the theory of quasi-varieties of models [5] we give a characterization of (...) all simpleC-matrices for any equivalential logicC (Theorem I.14). In corollaries we give necessary and sufficient conditions for the class of all simple models for a given equivalential logic to be closed under free products (Theorem I.18). (shrink)