Witold Marciszewski

Building on our diverse research traditions in the study of reasoning, language and communication, the Polish School of Argumentation integrates various disciplines and institutions across Poland in which scholars are dedicated to understanding the phenomenon of the force of argument. Our primary goal is to craft a methodological programme and establish organisational infrastructure: this is the first key step in facilitating and fostering our research movement, which joins people with a common research focus, complementary skills and an enthusiasm to work together. This statement—the Manifesto—lays the foundations for the research programme of the Polish School of Argumentation

For a free filter F on $$\omega $$ ω, endow the space $$N_F=\omega \cup \{p_F\}$$ N F = ω ∪ { p F }, where $$p_F\not \in \omega $$ p F ∉ ω, with the topology in which every element of $$\omega $$ ω is isolated whereas all open neighborhoods of $$p_F$$ p F are of the form $$A\cup \{p_F\}$$ A ∪ { p F } for $$A\in F$$ A ∈ F. Spaces of the form $$N_F$$ N F constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson–Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter F, the space $$N_F$$ N F carries a sequence $$\langle \mu _n:n\in \omega \rangle $$ ⟨ μ n : n ∈ ω ⟩ of normalized finitely supported signed measures such that $$\mu _n(f)\rightarrow 0$$ μ n ( f ) → 0 for every bounded continuous real-valued function f on $$N_F$$ N F if and only if $$F^*\le _K{\mathcal {Z}}$$ F ∗ ≤ K Z, that is, the dual ideal $$F^*$$ F ∗ is Katětov below the asymptotic density ideal $${\mathcal {Z}}$$ Z. Consequently, we get that if $$F^*\le _K{\mathcal {Z}}$$ F ∗ ≤ K Z, then: (1) if X is a Tychonoff space and $$N_F$$ N F is homeomorphic to a subspace of X, then the space $$C_p^*(X)$$ C p ∗ ( X ) of bounded continuous real-valued functions on X contains a complemented copy of the space $$c_0$$ c 0 endowed with the pointwise topology, (2) if K is a compact Hausdorff space and $$N_F$$ N F is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck.

This volume is written jointly by Witold Marciszewski, who contributed the introductory and the three subsequent chapters, and Roman Murawski who is the author of the next ones - those concerned with the 19th century and the modern inquiries into formalization, algebraization and mechanization of reasonings. Besides the authors there are other persons, as well as institutions, to whom the book owes its coming into being. The study which resulted in this volume was carried out in the Historical Section of the research project _Logical Systems and Algorithms for Automatic Testing of Reasoning,_ 1986-1990, in which participated nine Polish universities; the project was coordinated by the Department of Logic, Methodology and Philosophy of Science of the Bia??l??ystok Branch of the University of Warsaw, and supported by the Ministry of Education 1987). The major part of the project was focussed on the software for computer-aided theorem proving called Mizar MSE ) due to Dr. Andrzej Trybulec. He and other colleagues deserve a grateful mention for a hands-on experience and theoretical stimulants owed to their collaboration

The affirmative answer to the title question is justified in two ways: logical and empirical. The logical justification is due to Gödel’s discovery that in any axiomatic formalized theory, having at least the expressive power of PA, at any stage of development there must appear unsolvable problems. However, some of them become solvable in a further development of the theory in question, owing to subsequent investigations. These lead to new concepts, expressed with additional axioms or rules. Owing to the so-amplified axiomatic basis, new routine procedures like algorithms, can be reached. Those, in turn, help to gain new insights which lead to still more powerful axioms, and consequently again to ampler algorithmic resources. Thus scientific progress proceeds to an ever higher scope of solvability. The existence of such feedback cycles –in a formal way rendered with Turing’s systems of logic based on ordinal – gets empirically supported by the history of mathematics and other exact sciences. An instructive instance of such a process is found in the history of the number zero. Without that insight due to some ancient Hindu mathematicians there could not arise such an axiomatic theory as PA. It defines the algorithms of arithmetical operations, which in turn help intuitions; those, inturn, give rise to algorithmic routines, not available in any of the previous phases of the process in question. While the logical substantiation of the point of this essay is a well-established result of logico-semantic inquiries, its empiricalclaim, based on historical evidences, remains open for discussion. Hence the author’s intention to address philosophers and historians of science, and to encourage their critical responses.