The aim of this paper is to show that the operations of forming direct products and submatrices suffice to construct exhaustive semantics for all structural strengthenings of the consequence determined by a given class of logical matrices.

In the paper some consequence operations generated by ukasiewicz's matrices are examined.

This is an expository paper on the problem of independent axiomatization of any set of sentences. This subject was investigated in 50's and 60's, and was abandoned later on, though not all fundamental questions were settled then. Besides, some papers written at that time are hardly available today and there are mistakes and misunderstandings there. We would like to get back to that unfinished business to clarify the subject matter, correct mistakes and answer questions left open by others. We shall (...) deal with results of many authors. However, they will be exposed in a different manner, with complete proofs and, often, with refinements and supplements. Some questions will be brought up to date and related to other questions in logic. New results and questions will also be added. Stress will be laid on constructive aspects; that is, we will examine the problem of the possibility of independent axiomatization as well as algebraic means by use of which independent sets of axioms can be given. (shrink)

We consider the notion of structural completeness with respect to arbitrary (finitary and/or infinitary) inferential rules. Our main task is to characterize structurally complete intermediate logics. We prove that the structurally complete extension of any pure implicational in termediate logic C can be given as an extension of C with a certain family of schematically denned infinitary rules; the same rules are used for each C. The cardinality of the family is continuum and, in the case of (the pure implicational (...) fragment of) intuitionistic logic, the family cannot be reduced to a countable one. It means that the structurally complete extension of the intuitionistic logic is not countably axiomatizable by schematic rules. (shrink)

An operation on inferential rules, called H-operation, is used to minimize the axiom basis for classical logic.

This paper, which in its subject matter goes back to works on strongly nite logics , is concerned with the following problems: Let Cn1; Cn2 be two strongly nite logics over the same propositional language. Is the supremum of Cn1 and Cn2 also a strongly nite operation? Is any nite matrix axiomatizable by a nite set of standard rules? The rst question can be found in [9] . The second conjec- ture was formulated by Wolfgang Rautenberg, but investigations into this (...) problem had been carried out earlier in works of many logicians . Moreover, Stephen Bloom [1] posed a conjecture stronger than that: the consequence de- termined by a nite matrix is nitely based, i.e. it is the consequence generated by a nite set of standard rules. This hypothesis was, however, disproved by Andrzej Wronski [10] . In the present paper it is shown that neither nor holds true. The negative answer to can be viewed as a generalization of the result given by Andrzej Wronski [10]. (shrink)

We characterize all finitary consequence relations over S4.3, both syntactically, by exhibiting so-called passive rules that extend the given logic, and semantically, by providing suitable strongly adequate classes of algebras. This is achieved by applying an earlier result stating that a modal logic L extending S4 has projective unification if and only if L contains S4.3. In particular, we show that these consequence relations enjoy the strong finite model property, and are finitely based. In this way, we extend the known (...) results by Bull and Fine, from logics, to consequence relations. We also show that the lattice of consequence relations over S4.3 is countable and distributive and it forms a Heyting algebra. (shrink)

This paper develops a proof theory for logical forms of proofs in the case of monadic languages. Among the consequences are different kinds of generalization of proofs in various schematic proof systems. The results use suitable relations between logical properties of partial proof data and algebraic properties of corresponding sets of linear diophantine equations.

An operation on inferential rules, called H-operation, is used to minimize the axiom basis for classical logic.

We attempt to define the classical propositional logic by use of appropriate derivability conditions called Cn-definitions. The conditions characterize basic properties of propositional connectives.

The paper deals with reconstruction of the unique reductivecounterpart of the deductive logic. The procedure results in the deductivereductive form of logic. This extension is illustrated on the base of intuitionistic logics: Heyting’s, Brouwerian and Heyting-Brouwer’s ones.

ABSTRACT The following text is a large fragment of the lectures on heresy that Leszek Kołakowski (1927–2009) gave between November 1982 and February 1983 on the Polish radio station Radio Free Europe. These lectures have never been published in English. They were only published under the title ‘Herezja’ in Poland after the author’s death in 2010 by the publishing company Znak. Kołakowski raises the universal and timeless issues of tolerance, ideological struggles, protection of doctrine by religious institutions and the changing (...) attitude of the Catholic Church towards heretical challenges (tackling them and attempting to annex and colonize certain ideas). The concept of heresy is an opportunity to describe the history of ideological conflicts, show how they arise and how they can be resolved. The story of struggle between dogmatists and heretics is an endless story of changing social order and giving new shapes to the reality created by people and their ideas. Kołakowski points out that any doctrinal, political or ideological system must allow for criticism and otherness. Nearly 40 years after these lectures were given, they retain a special freshness in the context of the political, ideological and social crisis that Europe is experiencing in the early twenty-first century. (shrink)

The mainstream concept of death—the biological one—identifies death with the cessation of an organism. In this article, I challenge the mainstream position, showing that there is no single well-established concept of an organism and no universal concept of death in biological terms. Moreover, some of the biological views on death, if applied in the context of bedside decisions, might imply unacceptable consequences. I argue the moral concept of death—one similar to that of Robert Veatch—overcomes such difficulties. The moral view identifies (...) death with the irreversible cessation of a patient’s moral status, that is, a state when she can no longer be harmed or wronged. The death of a patient takes place when she is no longer capable of regaining her consciousness. In this regard, the proposal elaborated herein resembles that of Veatch yet differs from Veatch’s original project since it is universal. In essence, it is applicable in the case of other living beings such as animals and plants, provided that they have some moral status. (shrink)

In their insightful article, Brent Kious and Margaret Battin (2019) correctly identify an inconsistency between an involuntary psychiatric commitment for suicide prevention and physician aid in dying (PAD). They declare that it may be possible to resolve the problem by articulating “objective standards for evaluating the severity of others’ suffering,” but ultimately they admit that this task is beyond the scope of their article since the solution depends on “a deep and difficult” question about comparing the worseness of two possible (...) scenarios: letting someone die (who could have been helped) with not letting someone die (whose suffering could only be alleviated by death). In our commentary, we argue that creating such standards is more difficult than the authors assume because of the many types of deep uncertainties we have to deal with: (1) diagnostic, (2) motivational, and (3) existential. (shrink)

Singer claims that there are two ways of challenging the fact that brain-dead patients, from whom organs are usually retrieved, are in fact biologically alive. By means of the first, the so called dead donor rule may be abandoned, opening the way to lethal organ donation. In the second, it might be posited that terms such as “life” and “death” do not have any primary biological meaning and are applicable to persons instead of organisms. This second possibility permits one to (...) acknowledge that brain-dead patients are deceased because they are irreversibly unconscious. In the commentary which follows, I will argue that Singer’s second option is preferable since it provides a higher amount of organs available for transplant, and is better suited to the meaning of “death” which occurs in ordinary language. I will also defend such a concept of death against the objections raised by Michael Nair-Collins in the article Can the brain-dead be harmed or wronged? On the moral status of brain death and its implications for organ transplantation. (shrink)

It seems that nonmonotonicity of our reasoning is an obvious truth. Almost every logician not even believes, but simply knows very well that a human being thinks in a nonmonotonic way. Moreover, a nonmonotonicity of thinking seems to be a phenomenon parallel to the existence of human beings. Examples allegedly illustrating this phenomenon are not even analyzed today. They are simply quoted. Nowadays, this is a standard approach to nonmonotonicity. However, even simple analysis of those “obvious” examples shows that they (...) illustrate various problems of our thinking, among which none concerns nonmonotonicity. (shrink)

The paper is a continuation of A deductive-reductive form oflogic: general theory and intuitionistic case and considers the problem of definability of modal operators on the intuitionistic base. Contrary tothe classical case, it seems that the fact whether the connective is Heyting’sor Brouwerian is essential for the intuitionistic logic. The connective of possibility has the classical interpretation, i.e. w |= ✸α iff ∃t, if it is defined on the base of the logicwith Brouwerian connective of coimplication.

As far as existence of enduring subjects of change is concerned, Roman Ingarden's substance ontology stands in direct opposition to Whitehead's process metaphysics. However, with regard to domain of necessary relations, both systems are deeply platonic. Thus in this paper I pursue towards revealing some parallels and differences between both systems regarding views on ideal entities, necessary relations and pure possibilities. I examine whether having assumed contrary concepts of reality (substantial/procesual) both philosophers are forced to accept diverse conclusions in respect (...) to ideal entities domain. I also point out some crucial problems that are to be found in both systems: what is the ground of necessary relations – is it a nature of so called ideal qualities, or is it content of ideas (Ingarden); is it an individual or a relational essence of eternal objects, so called "primordial nature of God", or evolving actuality (Whitehead). Finally I investigate whether points of convergence between both concepts could serve as a basis for some modifications to overcome relevant problems. (shrink)

On the base of the classical logic the connectives of necessity and possibility have the equivalent positions in this sense that each of them is definable by the other one. The consequence of this fact is the possibility to define of the both modalities using the connective of identity. Thus, the connective of propositional identity defining the congruence of the propositional language has become the base of the reconstruction of necessity operator in some modal systems. Already in 1957 Greniewski [9] (...) obtained system S5 on the base of the propositional calculus with identity. Next, Cresswell [3], [4] reconstructed S4 and S5 in a similar way. Systematic investigations of this problem were undertaken by Suszko within the broader frame of SCI programme [16], [17], [18]. The connections between modal connectives and the connective of identity settle some philosophical sense of both modalities. Something is necessary, if it is identical with a logical truth. If something is different from the negation of some logical truth, it is possible. It is however, the “ontological” characteristics typical for modalities defined on the base of the classical logic. The settling of similar connections between intuitionistic identity and intuitionistic modalities uncover the philosophical sense of the “epistemic” necessity and possibility. Unfortunately, the symmetry between the classical and intuitionistic case is not entire. The acceptance of both equalities: = ¬ ¬ and = ¬ ¬, in the intuitionistic logic yields the classical versions of both modalities . Thus, there is a problem with defining the connective of intuitionistic possibility by the connective of identity. In Sections 2 and 3 two “partial” solutions of the mentioned problem are presented. They are “partial” because the connective of possibility has undesirable properties. The second solution yields the strengthening of the necessity connective, too. The main aim of this paper is obtained in Section 5, where a definition of the intuitionistic possibility is given. This definition uses the new connective of nonidentity, presented in Section 4. The new approach bases on the idea that some connectives like implication, negation, identity, necessity are “Heyting's” ones. The case of possibility is different. Together with coimplication, weak negation, and nonidentity, possibility belongs to the so-called “Brouwerian” connectives. The connectives of the first kind have such property that in Kripke's style semantics their interpretations in each point of this semantics are related to the interpretations in next points. In the case of the “Brouwerian” connectives the interpretations in each point depend on the interpretations in previous points. In this sense, “Heyting's” connectives are interpreted from the point of view of the future, while the interpretations of the “Brouwerian” connectives are from the point of view of the past. The relations between nonidentity and possibility connectives allow to say that in temporal semantics something is possible in the moment a, if before a there was such moment b in which it was different from the contradiction. Thus, “something is possible” does not mean that it held once in the past. A broader philosophical remarks about the nature of intuitionistic possibility are contained in two Comments in the last Section. (shrink)

In this text I take into account Roman Ingarden's theory of general objects. Polish phenomenologist avoided most of paradoxes implied by traditional theories of universals pointing out the difference between two kinds of form inherent in general object, namely form of subject of properties and form of whole and parts. Every idea has twofold construction: as an idea (qua idea) it is a subject of peculiar properties and on the other hand it has a content consisted of so called constants (...) and variables. The content of the idea has a whole-parts form. This however is inconsistent with other analyses of whole-parts form and of the character of constants and variables given by Roman Ingarden. In the consequence I claim that a third form, unique for general object, is needed. I also point out difficulty of indicating positive characteristic of general object (i.e. other than negation of properties of individuals). (shrink)

ABSTRACTThe author traces the impact of Abramowski's ideas on the recent history of Poland. His concepts were not only popular in the Polish Socialist Party and the syndicalist movement in the interwar period, but they also exerted a profound influence on the cooperative movement and democratic left-wing opposition in the 1970s and 1980s. The leaders of the Workers’ Defence Committee were much influenced by Abramowski's ideas and, according to some researchers, the Solidarity movement from 1980 to 1981 in Poland was (...) the culmination of his concepts. Today's anti-systemic movements in Poland are also inspired by Abramowski. The author also draws attention to certain similarities between Abramowski's ideas, Kropotkin's idea, Gramsci's concept of civil society and the thought of the young Marx. The author also outlines Abramowski's social ideas in the context of ideas promoted by the main theoreticians of the Polish left in the late nineteenth and early twentieth centuries. (shrink)

Volume 17, Issue 5, November 2020, Page 566-588.

ABSTRACT The article analyses the language used by the Polish nationalist right in relation to LGBT communities and the right to abortion. The authors show links between the language of Church hierarchs and right-wing columnists as the ideological backbone of the governing right-wing populist right. According to the authors, the attack on gender is the same method of political mobilisation and power management as the campaign against refugees and the anti-immigrant hysteria. On the one hand, the anti-gender discourse may strengthen (...) the narrative against the ‘liberal EU’ and, on the other hand, it is used to show substitute ‘scapegoats’ in Poland. Moral panic makes it possible to release class tensions in the manner that is safe for the system through the artificially induced cultural and ideological conflict. However, the dispersed anti-gender discourse has a real impact on social attitudes – on the one hand, it polarises social sympathies and, on the other hand, it strengthens conservative attitudes. The analysis is based on right-wing press articles, Church hierarchs’ statements, videos on YouTube and a parliamentary debate about the right to abortion. (shrink)

Interpretations of logics with only truth-functional connectives create a number of problems regarding the understanding of interpreted sentences. A particular problem is caused by the understanding of a sentence that is the negation of another. What is the meaning of sentence ¬p, for a particular sentence p? Even when we know what the semantic correlate of the sentence p is, we still do not know how to understand the semantic correlate of the sentence ¬p. The standard algebraic approach does not (...) explain much. The problem is still open, because it is still unclear how to understand the element of algebra −v(p). The propositional logic with the connective of the content implication, as it is a non-Fregean logic, proposes a simple solution to this problem. The key to understanding the content of the sentence ¬p is to understand the content of the sentence p. Because, the content of the natural language sentence usually has a partial understanding, so it can be understood differently. Therefore, the negation ¬p is here understood adequately to the current and partial understanding of the sentence p. It seems that the proposed approach is consistent with our daily thinking. (shrink)

Epistemicism seems to be the most dominating approach to vagueness in the recent twenty years. In the logical and philosophical tradition, e.g. Peirce, vagueness does not depend on human knowledge. Epistemicists deny this fact and contend that vagueness is merely the result of our imperfect mind, our dearth of knowledge, sort of phantom, finally, that it simply does not exist. In my opinion, such a stance not only excludes vagueness comprehended in terms of human knowledge, but which is worse, stems (...) from spurious logical arguments. The part of arguments called Sorensen’s Arguments or even Proofs were the subject of my analysis in the book Paradoksy and in the paper “Epistemicism and Roy Sorensen Arguments” published in the Bulletin of the Section of Logic. Here I shall only briefly refer to these works and focus mainly on the arguments launched by Tymothy Williamson. One of them is to uncover why we are not able to recognize the alleged sharp boundary between positive and negative extensions of any vague predicates. Williamson’s reasoning is based on his margin for error principle. Another argumentation of Williamson aims at the refutation of the principle I know that I know. It should be emphasized that all the aforementioned arguments are fundamental for epistemicism and all of them are fallacious because of either formal or false-premise fallacy. There is the circumstance that we cannot deem epistemicism logical. Finally, we show that within the epistemic frame the following thesis is valid: if what epistemicism states is the case, then what epistemicism states is not the case. This immediately implies → ¬p’) that it is not the case what epistemicism states. So, either epistemicism or logic. (shrink)

The idea of belief revision is strictly connected with the notion of contraction given by the set of postulates formulated by Alchourrón, Gärdenfors and Makinson. In the present paper expansion and contraction are defined by Tarski's consequence relation and Tarski-like elimination relation. The logic of falsehood (i.e. a logic dual in Wójcicki's sense to the given logic of truth) plays a key role for defining the elimination relation. A decision of adding or refusing of some sentences is arbitrary and depends (...) on our wish only. This decision cannot be logical and logic cannot justify it. In our approach logic is a tool for faultless and precise realization of extension or reducing of the set of our beliefs. Step forward extends the set of our beliefs and it is used when some new belief appears. Step backward reduces the set of our beliefs and it is used when we reject from some previously accepted belief. But why some "initial" sentences should be added or refused depends on extralogical reasons. The logic for the back-reasoning uses the class of models adequate for the logic extending the set of our beliefs. However, the class is used in a specific i.e. dual form. That is why the step forward (expansion) and the step backward (contraction) constitute the one whole. Procedure of contraction satisfies the well known AGM postulates. We limit our considerations to first six conditions for contraction. Satisfaction of almost every postulate is a good sign that our approach is reasonable. The only exception we make for the controversial fifth postulate. (shrink)

A few years ago, believing that human thinking is nonmonotonic, I tried to reconstruct a nonmonotonic reasoning by application of two monotonic procedures. I called them “step forward” and “step backward” . The first procedure is just a consequence operation responsible for an extension of the set of beliefs. The second one, defined on the base of the logic of falsehood reconstructed for the given logic of truthfulness, is responsible for a reduction of the set of beliefs. Both procedures taken (...) together were successfully verified by using so-called AGM , postulates for expansion, contraction and revision formulated by Alchourrón, Gärdenfors and Makinson . Reasoning composed of the mutual application of both procedures seemed to be quite natural for modeling our thinking. At that time, I supposed that it should be nonmonotonic but I was wrong. It turned out impossible to satisfy a definition of the nonmonotonic inference by reasoning composed both steps. To understand why this is impossible, I began to analyze how nonmonotonicity is obtainable in some well-known cases in the literature. I analyzed the problem from two points of view: non-formal examples for nonmonotonicity and formal constructions of nonmonotonic operations/relations. The result of those investigations was astonishing: none of the considered by me cases of nonmonotonicity belonging to point and almost none belonging to satisfies the definition of nonmonotonic inference. Arguments against the nonmonotonic character of well-known examples for nonmonotonicity of human thinking are more precisely presented in [6]. I present them below an abbreviated version of them. (shrink)

The ancient paradox of Protagoras had the opinion of an unsolved problem. The two solutions proposed in the 20th century by W. Lenzen and L. Aqvist are considered to be the best. In fact none of them may be treated as proper. In the paper we show that both of authors avoid contradiction solely by means of mere neglect. However quite a simple solution seems to be feasible when the paradox is approached as an amphibolic construction, thus an ambiguity.

This book, provides a critical approach to all major logical paradoxes: from ancient to contemporary ones. There are four key aims of the book: 1. Providing systematic and historical survey of different approaches – solutions of the most prominent paradoxes discussed in the logical and philosophical literature. 2. Introducing original solutions of major paradoxes like: Liar paradox, Protagoras paradox, an unexpected examination paradox, stone paradox, crocodile, Newcomb paradox. 3. Explaining the far-reaching significance of paradoxes of vagueness and change for philosophy (...) and ontology. 4. Proposing a novel, well justified and, as it seems, natural classification of paradoxes. (shrink)

As shown in (Łukowski, Gensler, 2013), the paradox of the stone is a failed attempt to show that “omnipotence” is a contradictory concept. An element of the argument presented there is that God, while unable to lift the stone, can nevertheless annihilate it. This work considers the amplification of the paradox of the stone to the form generated by the question: can God create a stone which He will not be able to lift, nor, once created, will He be able (...) to destroy. (shrink)