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

Church's Thesis (CT) was first published by Alonzo Church in 1935. CT is a proposition that identifies two notions: an intuitive notion of an effectively computable function defined in natural numbers with the notion of a recursive function. Despite the many efforts of prominent scientists, Church's Thesis has never been disproven. There exists a vast literature concerning the thesis. The aim of this book is to provide a one volume summary of the state of research on Church's Thesis. These include (...) the following: different formulations of CT, CT and intuitionism, CT and intensional mathematics, CT and physics, the epistemic status of CT, CT and philosophy of mind, provability of CT and CT and functional programming. Adam Olszewski, is assistant professor of the philosophy of mathematics and mathematical logic at the Department of Philosophy of Pontifical Academy of Theology, Cracow (Poland). Jan Wolenski is professor of philosophy, Institute of Philosphy, Jagiellonian University, Cracow (Poland). He is one of the most distinguished logicians in Poland. Robert Janusz is assistant professor in the Department of Philosophy of Ignatius College, Cracow (Poland). (shrink)

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.

The thought of Lev Karsavin—like all representatives of the Russian religious and philosophical renaissance—is deeply rooted in the work and ideas of Fyodor Dostoyevsky. However, instead of focusing on the broad relationship between the two thinkers, we focus here on a specific aspect of their connection: two significant articles by Karsavin on Dostoevsky. These are: Dostoevsky and Catholicism (1922) and—or rather primarily—Fyodor Pavlovich Karamazov as an ideologist of love (1921). After a short presentation of the first text, I will try (...) to show the debatable, controversial, but also inspiring and even daring nature of the second text. (shrink)

The article reviews the book Jak to jest być świadomym? Analityczne teorie umysłu a problem świadomości [What Is It Like to be Conscious? Analytical Theories of Mind and the Problem of Consciousness], by Józef Bremer.

The aim of the article is to outline the historical background and the present state of the methodology of deductive systems invented by Alfred Tarski in the thirties. Key notions of Tarski's methodology are presented and discussed through, the recent development of the original concepts and ideas.

The principle of explosion asserts that any formula can be derived from any pair of other contradictory formulas. Paraconsistent logic is typically regarded as a logic in which the universal validity of this principle is questioned. Therefore, a key point is determining when the validity can be considered universal to classify a logic as paraconsistent. A pertinent example to illustrate this point is the calculus CB1 that admits the principle but only for negated formulas, i.e., from any set {α, ∼α} (...) any other formula follows if and only if α is of the form ∼γ. Another example is Sette’s calculus P1, which is paraconsistent at the level of variables but not complex formulas. Both serve as compelling examples of the so-called borderline cases. In this paper, we examine several calculi expected to be paraconsistent at the level of literals. It means that a pair of formulas, α and ∼α, can yield any β if, and only if α is neither a propositional variable nor is its iterated negation. Furthermore, it is assumed that in some calculi presented here, β must adhere to specific restrictions. Once these conditions are satisfied, we refer to calculus as paraconsistent in a “controllable manner”. (shrink)

Paul Edwards' October 1989 essay, ‘Heidegger's Quest for Being’ illustrates the wisdom of the dictum that the only legitimate criticism of a philosopher's position is a constructive criticism. His negative critique amounts to no more than expressions of distaste, dislike, and displeasure, and nothing of philosophical interest follows from his assertions, as the discourse is primarily an argument ad hominem . He grossly misinterprets Heidegger on some elemental and fundamental matters as well. In particular, he sorely misinterprets what he refers (...) to as the ‘Quest for Being’, which only further contributes to the objectionable character of the essay. (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.

The philosophical challenge that religious diversity poses for religious belief has become in recent years the focal point of a very engaging theological and philosophical debate. The debate began in the Christian context and it would be fair to say that its main issue remains the relationship of Christianity to other major religions. Traditionally Christian thinkers faced with the fact of religious plurality have assumed that Christianity is the only way to salvation, and the truth-claims of other religions can be (...) refuted by way of argument. This position is described today as 'exclusivist'. John Hick's name has become synonymous with a radically different approach to the whole issue. Hick argues that all religious traditions make contact with the same Ultimate Reality, each encountering it through a variety of culturally shaped forms of thought and experience, but all offering equally effective paths to 'salvation/liberation'. Hick's pluralistic hypothesis, although very popular in some quarters, appears to many Christian and non-Christian thinkers as highly controversial. (shrink)

This book makes a significant contribution to the standard decision theory, that is, the theory of choice built around the principle of maximizing expected utility, both to its causal version and to the more traditional noncausal approach. The author’s success in clarifying the foundations of the standard decision theory in general, and causal decision theory in particular, also makes the book uniquely suitable for a person whose research in philosophy has led her to want to learn about contemporary decision theory. (...) The book presupposes some mathematical sophistication, and it contains more philosophical and mathematical argumentation per page than may be suitable for an undergraduate level course; but it stands out as an ideal textbook for a graduate-level course focused on the foundations of the standard decision theory. (shrink)

From the perspective of Marek J. Siemek’s theory of modernity, one of the most important problem is to include conflicts into institutional framework of the modern society. He reinterprets Hegel’s dialectics of the struggle for recognition by conceptual tools of Hobbes and Marx in order to uncover hidden assumptions and conditions of possibility of the social rationality. For Siemek, law as purely formal, autopoetic social system or social subject, which produces individual subjects, is the first of the conditions of possibility (...) of modernity. The second one is the convergence of formal and material presuppositions of law—or, speaking generally—the convergence of form and content of the social reason. Form and content, facticity and normativity, instrumentality and communicativity are aspects of the process of rationalization andof the only one reason, self-generating in the history. So for Siemek, the Hegelian model of the struggle for recognition gains its theoretical power only when it is interpreted from the perspective of economical, technical and legal rationalization of modernity. Only such perspective is able to construct “the transcendental social philosophy” which starts from critique of “the non-instrumental reason”. (shrink)

The paper is concerned with the old conjecture that there are infinitely many twin primes. In the paper we show that this conjecture is true, that is, it is true in the standard model of arithmetic. The proof is based on Rasiowa–Sikorski Lemma. The key role are played by the derived notion of a Rasiowa–Sikorski set and the method of forcing adjusted to arbitrary first–order languages. This approach was developed in the papers Czelakowski [ 4, 5 ]. The central idea (...) consists in constructing an appropriate countable model \(\textbf{A}\) of Peano arithmetic by means of a Rasiowa–Sikorski set. This model validates the twin prime conjecture. Since \(\textbf{A}\) is elementarily equivalent to the standard model, the conjecture follows. Thus the standard model validates the twin primes conjecture. More generally, it is shown that de Polignac’s conjecture has a positive solution. The paper employs methods borrowed from the contemporary mathematical logic. Such a ’logical’ approach may be viewed as a useful addition to the dominant methodology in number theory based on mathematical analysis. (shrink)

A self-conscious antagonism to Newtonian science is widely seen as characteristic of the Romantic movement, and Coleridge is routinely portrayed as one of the major representatives of this anti-Newtonian sentiment. Although such a view of Coleridge is correct, his hostility to Newton is puzzling. The attitudes that Coleridge objected to are often expressly denied in Newton's published writings, and Coleridge's own ‘dynamic’ philosophy was, in fact, remarkably like the conception of nature personally favoured by Newton. Coleridge, then, must have been (...) objecting to something other than Newton's beliefs—to beliefs not in fact held by Newton, but by others. Commentators have noted Coleridge's distortion of Newton's philosophies, but have not explored this matter in any detail. The present paper attempts to redress this situation. As much of the historiography of Western philosophy presumes an unequivocal opposition between Romanticism and the Enlightenment, it is of importance to find that Coleridge and Newton held strikingly similar views about the nature of the universe. (shrink)

The article outlines Vladimir Soloviev’s views at the time of his fascination by the theocracy, Christian policy and United Church concepts. His standpoint then was to place the “Godmanhood” idea underlying his philosophy in a realistic, historically and socially factual—hence universalistic—context. This led him to confer a special role in the historical process to the Christian church, which he saw as a dynamic institution adding energy to history. Soloviev considered this energy crucial in the rebirth of Christian unity around the (...) Holy See and the fulfillment of the “social trinity” reflecting the structure of the Divine Absolute and harmoniously uniting three relatively independent seats of social power: clerical, state and prophetic. For Soloviev the fulfillment of this project consisted in a lasting alliance between the papacy and the Tsar’s court, a concept which sounds very eccentric today. (shrink)

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)

By a sentential logic we understand a pair, where S is a sentential language, i.e. an absolutely free algebra freely generated by an infinite set p, q, r,... of sentential variables and endowed with countably many finitary connectives §1, §2,... and C is a consequence operation on S, the underlying set of S, satisfying the condition of structurality: eC ⊆ C, for every endomorphism e of S and for every X ⊆ S. If no confusion is likely we shall identify (...) a logic with its consequence operation C. A logic C is standard if C = [ {C : Y ⊆ X & Y is finite}, for all X ⊆ S. (shrink)

In the article some philosophical issues regarding the emergence of the modern cosmology are discussed. The mathematical field equations enabled for the first time in the history to set a cosmological problem as a physical one. The “power” of mathematics was not easy to be read by the pioneers of cosmology. Also the philosophical perspective on the Universe changed in front of this “power” expressed by Einstein and de Sitter in their models. In the paper we propose also some new (...) ideas related to the mathematics of cosmology as a physical science. (shrink)

We propose a new schema for the deduction theorem and prove that the deductive system S of a prepositional logic L fulfills the proposed schema if and only if there exists a finite set A(p, q) of propositional formulae involving only prepositional letters p and q such that A(p, p) L and p, A(p, q) s q.

W tym artykule zaproponuję konstrukcję trójwartościowej logiki przekonań, którą nazywam: LSB3_1. Podam również i udowodnię kompletność LSB3_1 w odniesieniu do danej semantyki. LSB3_1 opiera się na preformalnych założeniach i intuicjach, które przedstawiono w sekcji 1. Sekcja 2 zawiera składnię i podział instrukcji LSB3_1 na wewnętrzne i zewnętrzne. Rozdział 3 przedstawia semantykę LSB3_1, a także szereg tautologii i formuł nietautologicznych w LSB3_1 wraz z ich intuicyjną interpretacją. System aksjomatyczny dla LSB3_1 i jego porównanie z silną logiką Kleene'a są przedstawione w sekcji (...) 4. Twierdzenie o kompletności dla LSB3_1 jest przedstawione w rozdziale 5. Zdefiniuję termin koniunkcyjna forma normalna i przedstawię lematy, które prowadzą do udowodnienia redukcji języka LSB3_1 wzory przed udowodnieniem twierdzenia o zupełności. (shrink)

Peter Millican and Branden Thornhill-Miller have recently argued that contradictions between different religious belief systems, in conjunction with the host of defeaters based on empirical research concerning alleged sources of evidence for ‘perceived supernatural agency’, render all ‘first-order’, that is actual, religious traditions positively irrational, and a source of discord on a global scale. However, since the authors recognise that the ‘secularisation thesis’ appears to be incorrect, and that empirical research provides evidence that religious belief also has beneficial individual and (...) social effects, they put forward a hypothesis of a ‘second-order religious belief ’, with Universalist overtones and thus free of intergroup conflict, and free of irrationality, since supported (solely) by the Fine-Tuning Argument. While granting most of their arguments based on empirical research and embracing the new paradigm of the atheism/religion debate implicit in their paper, I contend that Millican’s and Thornhill-Miller’s proposal is unlikely to appeal to religious believers, because it misconstrues the nature and grounds of religious belief. I suggest that their hypothesis may be refined by taking into account a view of axiologically grounded religious belief that I refer to as ‘agatheism’, since it identifies God or the Ultimate Reality with the ultimate good (to agathon). (shrink)

The paper presents reflections on human existence and the nature of human being. The author of the paper presents his own standpoint and compares it with other philosophical and religious conceptions of the human being, inter alia those formed in Christianity, Marxism and Cartesianism. The primary concern of the essay is the existential-anthropological significance of human being’s creativity.

Discursive (or discussive) logic, D₂, introduced by Jaśkowski, is widely recognized as a first formal approach to paraconsistency. Jaśkowski applied a quite extraordinary technique at that time to describe his logic. He neither gave a set of the axiom schemata nor presented a direct semantics for D₂ but used a translation function to express his philosophical and logical intuitions. Discursive logic was defined by an interpretation in the language of S₅ of Lewis. The aim of this paper is to present (...) a modified system of the discursive logic that allows some of the weaker versions of Duns Scotus's thesis to be valid. The initial idea is to consider a different characteristic of the connective of negation. We introduce both a direct semantics and an axiomatization of the new system, prove the key metatheorems, and describe labeled tableaux for the system. (shrink)

This book celebrates the work of Don Pigozzi on the occasion of his 80th birthday. In addition to articles written by leading specialists and his disciples, it presents Pigozzi’s scientific output and discusses his impact on the development of science. The book both catalogues his works and offers an extensive profile of Pigozzi as a person, sketching the most important events, not only related to his scientific activity, but also from his personal life. It reflects Pigozzi's contribution to the rise (...) and development of areas such as abstract algebraic logic, universal algebra and computer science, and introduces new scientific results. Some of the papers also present chronologically ordered facts relating to the development of the disciplines he contributed to, especially abstract algebraic logic. The book offers valuable source material for historians of science, especially those interested in history of mathematics and logic. (shrink)

Dear Readers,After a long break caused, on the one hand, by staff changes in the Editorial Board and, on the other, by the lingering coronavirus pandemic, another issue of the Ignatianum Philosophical Yearbook is coming out. The new editorial team has decided to expand the journal’s formula to include texts from the general humanities; thus, there will be articles not only on philosophy, but also on history, cultural and religious studies, and theology, on top of Polish and foreign literature. The (...) journal is also open to a wide range of issues that take into account the mission of the Jesuit University Ignatianum as a Catholic research center. An important element of this approach is the issue of the Mediterranean civilizational tradition and cultural heritage, which in our understanding are the most important sources of knowledge and inspiration for the study of contemporary problems of individual people, societies and states as well. (shrink)

We examine the notion of primitive satisfaction in logical matrices. Theorem II. 1, being the matrix counterpart of Baker's well-known result for congruently distributive varieties of algebras (cf [1], Thm. 1.5), links the notions of primitive and standard satisfaction. As a corollary we give the matrix version of Jónsson's Lemma, proved earlier in [4]. Then we investigate propositional logics with disjunction. The main result, Theorem III. 2, states a necessary and sufficient condition for such logics to be finitely based.

ABSTRACTAlthough the British Idealists did not provide a systematic account of language as a distinct philosophical phenomenon, language is nonetheless a fundamental element of Idealist social and political philosophy. This is seen mostly in the Idealist treatment of the concept of general will, which resulted in a Hegelian theory of community, constituted by shared understandings and a shared account of the common good and common interest. This article contains analysis of the relations between language and socio-political institutions in British Idealist (...) writings. The narrative focuses on Thomas Hill Green and Bernard Bosanquet – two major political thinkers representing the said philosophical tradition. The argument proceeds as follows: firstly, possible linguistic interpretations of the Idealist philosophical tradition are introduced. Then the author turns to Green’s conception of general will as the foundation of socio-political order and political sovereignty. In the third section, he... (shrink)

The paper is a critical review of Leszek Kołakowski’s book Metaphysical Horror. According to Kołakowski, the starting-point of metaphysical horror is the awareness of changeability, transience, contingency and fragility of the world and human existence in face of the overwhelming and abysmal face of Nothingness. According to Kołakowski, the inevitable urge to overcome metaphysical horror leads to the idea of the Absolute, which can appear in two forms: God and cogito.What underlies the present paper is disagreement with Kołakowski’s perspective of (...) metaphysical horror that leads to question about reasons for praise for human mortality and human relationship towards Nothingness, reasons for an acceptance of the awaiting existential horror. (shrink)

The Center of Interdisciplinary Studies in Cracow has a very rich tradition that has been studied by many and recently by Kamil Trombik. The very difficult period for the Church and for philosophy during the materialistic Marxist ideology was an opportunity for card. K. Wojtyła to outline a new mode of dialog between science and religion. The future Center, organized by Michał Heller and Józef Życiński not only captured this idea but transformed it into an academic institution centered on the (...) idea of philosophy in science, which was also developed at the Vatican Observatory by W.G. Stoeger et al. Trombik’s book is an excellent study of the historical context and initial years of activity of the Center and its development. (shrink)

The classesMatr( ) of all matrices (models) for structural finitistic entailments are investigated. The purpose of the paper is to prove three theorems: Theorem I.7, being the counterpart of the main theorem from Czelakowski [3], and Theorems II.2 and III.2 being the entailment counterparts of Bloom's results [1]. Theorem I.7 states that if a classK of matrices is adequate for , thenMatr( ) is the least class of matrices containingK and closed under the formation of ultraproducts, submatrices, strict homomorphisms and (...) strict homomorphic pre-images. Theorem II.2 in Section II gives sufficient and necessary conditions for a structural entailment to be finitistic. Section III contains theorems which characterize finitely based entailments. (shrink)

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)

Sociology has been often defined as a science of "social relations". The aim of this article is to contribute to the clarification of this concept. I take into account only two classic analytical sociologies — those developed by Max Weber and by Florian Znaniecki. These sociologies seem to me only partly useful for the analysis of macroscale (ethnic, racial, industrial, and international) problems. They refer to human individual interactions within social collectivities, and not between them. If we follow expressis verbis (...) the individualistic aspects of Weber's model and the analytical aspects of Znaniecki's theory, it would be difficult to base empirical investigations of social relations. The situation changes when we depart from the classics' ways of construction of the concepts of social relation and concentrate on the implication of the concept presented by the same authors. Fortunately, the research practices of both scholars encourage us to abandon a literal interpretation of their models. They present a variety of interesting typologies of social relations and social objects between which these relations can exist. (shrink)

The idea of obtaining a pilot-wave quantum theory on a lattice with discrete time is presented. The motion of quantum particles is described by a \-distributed Markov chain. Stochastic matrices of the process are found by the discrete version of the least-action principle. Probability currents are the consequence of Hamilton’s principle and the stochasticity of the Markov process is minimized. As an example, stochastic motion of single particles in a double-slit experiment is examined.

The article begins from a consideration of the relationships between ethics and method, dignity and trustworthiness of the scholar in his normal activity. Then the views of Stanisław Ossowski are presented concerning the conditions of the permanence and development of the normal academic life, and his commandments for the academic professions are quoted safeguarding the normal academic life. A discussion of the question of freedom and responsibility in academic research is combined with Ossowski’s conception of philosophy of science. Ossowski had (...) a conception of the historical mission and function of science. With it was connected a notion of indispensable freedom of the scientists which they must possess to be able to perform their professional tasks normally. Ossowski also had a conception of the duties and professional responsibility of the scientists. The article ends with a review of threats to the normal functioning of science and of the countermeasures that should be used, according to Ossowski, in order to ensure growth of science and unencumbered performance of scholarly tasks. (shrink)

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.