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.

Rationalism is the ontological and epistemological view to the effect that there exist abstract entities—as classes, numbers, points, etc.—being cognitively available to our minds. Thus, the axiom of abstraction can be interpreted as the most concise statement of the rationalist position in its ontological facet. The acknowledging of abstracts may be motivated either with their direct intellectual vision, as claimed by Plato, or with their ‘indispensability’ for scientific progress, as claimed by Quine and a host of other famous thinkers. For such a practical approach the term ‘Pragmatic Rationalism’ should be in order (more frequently one speaks of ‘pragmatic platonism’ but in the present context this term would be misleading). Among its followers there are some eminent members of the Lvov-Warsaw School. Predominantly, Kazimierz Ajdukiewicz for his program of empirical methodology of sciences, and Alfred Tarski for the abundant use of highly abstract concepts in his mathematical research. Radical nominalism pioneered by Stanisław Leśniewski and Tadeusz Kotarbiński, denying the existence of classes, did not comply with methods and results of sciences. However, there is a moderate version of nominalism which pragmatically does admit classes, for the sake of scientific progress. This version has been suggested by Andrzej Grzegorczyk under the name of ‘liberal reism’. In such a pragmatic form, nominalism can comply with pragmatic rationalism, and the other way around. The latter is not bound to deny what nominalism says about ontological and epistemological priority of physical individuals. Even such a convinced rationalist as Gödel shared with nominalists (unlike Plato) that observational statements concerning physical individuals are most basic in the structure of our knowledge. A nice example of such reconciliation is found in Tarski’s work ‘Foundations of the geometry of solids’. He recedes from Euclid’s idea of point as a primitive concept, in accordance with the claim of reistic nominalism that only solids can play such a role. Instead, he starts from the concept of sphere, defining ‘the point as the class of all spheres which are concentric with a given sphere’. Thus, the typically abstract concept of class (attacked by radical nominalism) proves indispensable to do justice to the ontological priority of physical individuals—a hallmark of nominalism. In such a way, Tarski, when doing justice to nominalism, at the same time gains considerable scores for pragmatic rationalism as well. Such a constructive collaboration of the two opposing camps within the School sheds a new light on the School’s ability to solve hard problems with a vigorous and ‘penetrative discussion’ between opposite standpoints.

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 paper contains a rew postulates concerning the method of philosophy. In the beginning there are presented some fundamental views on the method of philosophical investigation. An approach which can be named deductive program of philosophy is the oldest one. It is represented among others by Aristotle, St. Thomas Aquinas and great rationalists of the XVII century. In the XIX century a new claim arose to assimilate the structure of philosophical theory with the theories of natural sciences; the theorems of thus constructed philosophy should be founded also on the results of sciences. This was the so called „inductive philosophy“. At last in phenomenology and other schools the rights of intuition were defended and this view can be named an intuitionistic one.In neothomism all of these methods are „de facto“ applied, though the methodological program formulated by St. Thomas was only a deductive one. The author tries to demonstrate upon the example of the problem of substance that the deductive approach is not sufficient to construct a whole philosophical theory. What is meant as deductive proof is often a pseudo-proof which has at most a value of „philosophical agitation“. Hence other methods of arguing are necessary. One of them is treating our proposition as a hypothesis which is an explanation of empirical data. If the proposition well explains an experience, then it is justified in our theory these data can be taken from empirical sciences or from everyday experience). This procedure resembles the explanation in natural sciences, though the explanation in philosophy shows some specific characteristics. It is showed also upon the example of substandertheory.Another method of philosophical demonstration is that of „reductio ad absurdum“: we argue that all opposite views have consequences which are absurd i.e. self-contradictory or incompatible with empirical data er — at least — leading to some complications in theory. As theories opposite to the thomistic theory of substance are considerjed: the thesis that no properties of substance exist (held by prof. T. Kotarbiński) and the thesis that there are not substances but every object is a bundle of properties (this view known as „positivistic“ was held among others by Locke and Hume). By a critical examination of these theories the thomi-.stie theory of substance is grounded.

The paper begins with the presentation of the views of Hume and logical positivism, that statements about facts (the so called real statements) cannot be necessary (if „necessary' is understood as „the one which is known true independently of experience“).This point of view ought to be carefully analysed by the adherents of Aristotle and St. Thomas Aquinas who to the statements of their philosophy attributed two simultaneous properties of logical necessity and i eality.The method here applied is a sort of conceptual analysis. This method consists in defining concepts in terms of relevant metalinguistic predicates (such as „true“, „atomic“ and similar other ones) and of logical constans. Owing to this method the application becomes possible of logical calculus to the comparison of the extensions of the predicates (we confirm, for instance, the equivalence or lack of equivalence between analysed definitions). In this way we can formulate the following definition of necessary statement:(N) The statement p is necessary in language L if and only if no true atomic sentence implies here the negation of p.The definition (N) allows also to establish the relation of inclusion between the class of analytic and the class of necessary sentences, where the former are meant as statements reducible to logical truths by means of mere definitions.Next a satisfactory condition is formulated, which, when fulfilled by a sentence, makes this sentence real:(R) If from a sentence p and some true atomic sentence q follows an other true atomic (sentence, then p is real.It is not difficult to realize that (N) and (R) do not bring forward sufficient grounds to establish an exclusion between the classes of necessary and real sentences. Morever, one may easily show the instances satisfying simultaneously (N) and (R), what is a quite satisfactory argument, for the thesis that necessity does not exclude reality.In the last part of the article two statements of thomistic mataphysics ere discussed. These statements enjoy logical necessity as being a substitution of logical truths; on the other hand the same propositions satisfy the condition of reality (R). One may object to a thus formulated condition on the ground that it is too weak and, therefore, it cannot secure the knowledge against the metaphysics. But one can point that there are many theorems of science acknowledged as real by virtue of the proposed test of reality, i. e. the same test, which is satisfied by discussed metaphysical statements.

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.

The first good message is to the effect that people possess reason as a source of intellectual insights, not available to the senses, as e.g. axioms of arithmetic. The awareness of this fact is called rationalism. Another good message is that reason can daringly quest for and gain new plausible insights. Those, if suitably checked and confirmed, can entail a revision of former results, also in mathematics, and - due to the greater efficiency of new ideas - accelerate science’s progress. The awareness that no insight is secured against revision, is called fallibilism. This modern fallibilistic rationalism, i.e. the belief in the attainability of inviolable truths of reason which would forever constitute the foundations of knowledge. Fallibilistic rationalism is based on the idea that any problem-solving consists in processing information. Its results vary with respect to informativeness and its reverse - certainty. It is up to science to look for highly informative solutions, in spite of their uncertainty, and then to make them more certain through testing against suitable evidence. To account for such cognitive processes, one resorts to the conceptual apparatus of logic, informatics, and cognitive science.