In the paper, pre-Hilbert algebras are defined as a generalization of Hilbert algebras (namely, a Hilbert algebra is just a pre-Hilbert algebra satisfying the property of antisymmetry). Pre-Hilbert algebras have been inspired by Henkin's Positive Implicative Logic. Their properties and characterizations are investigated. Some important results and examples are given. Moreover, positive implicative pre-Hilbert algebras are introduced and studied, their connections with some algebras of logic are presented. The hierarchies existing between the classes of algebras considered here are shown.

Recent years have seen the appearance of many English-language hand books of logic and numerous monographs on topical discoveries in the foundations of mathematics. These publications on the foundations of mathematics as a whole are rather difficult for the beginners or refer the reader to other handbooks and various piecemeal contribu tions and also sometimes to largely conceived "mathematical fol klore" of unpublished results. As distinct from these, the present book is as easy as possible systematic exposition of the now (...) classical results in the foundations of mathematics. Hence the book may be useful especially for those readers who want to have all the proofs carried out in full and all the concepts explained in detail. In this sense the book is self-contained. The reader's ability to guess is not assumed, and the author's ambition was to reduce the use of such words as evident and obvious in proofs to a minimum. This is why the book, it is believed, may be helpful in teaching or learning the foundation of mathematics in those situations in which the student cannot refer to a parallel lecture on the subject. This is also the reason that I do not insert in the book the last results and the most modem and fashionable approaches to the subject, which does not enrich the essential knowledge in founda tions but can discourage the beginner by their abstract form. A. G. (shrink)

During the global financial crisis of 2007–2009 and the currentCovid-19 debacle, central banks acted quickly, boldly, and effectively. The paper argues that they did so thanks to the lessons learned from the past financial crises, which provided them with opportunities to reconsider their previous beliefs. A case in point is the banking crisis in the United States during the Great Depression of the 1930s that taught central banks to act rapidly and decisively in order to prevent an initial liquidity crisis (...) from escalating into a solvency crisis leading to bankruptcies of banks with potentially disastrous consequences for the entire economy. The paper revisits the consequences of the currency crises of the 1990s, which transformed exchange rate regimes across the world: the EMS one of 1992–1993 that accelerated the launch of the euro, and the Asian one of 1997–1998 resulted in the proliferation of floating exchange rates in emerging economies. (shrink)

The main aim of the present paper is to explain a nature of relationships exist between Nelson and Heyting algebras. In the realization, a topological duality theory of Heyting and Nelson algebras based on the topological duality theory of Priestley for bounded distributive lattices are applied. The general method of construction of spaces dual to Nelson algebras from a given dual space to Heyting algebra is described. The algebraic counterpart of this construction being a generalization of the Fidel-Vakarelov construction is (...) also given. These results are applied to compare the equational category N of Nelson algebras and some its subcategories with the equational category H of Heyting algebras. It is proved that the category N is topological over the category H. (shrink)

Natural Deduction Natural Deduction is a common name for the class of proof systems composed of simple and self-evident inference rules based upon methods of proof and traditional ways of reasoning that have been applied since antiquity in deductive practice. The first formal ND systems were independently constructed in the 1930s by G. Gentzen and S. Jaśkowski and … Continue reading Natural Deduction →.

In this article, the author notes that Thomas Aquinas, in his brief work entitled De Ente et Essentia, proved that at the base of understanding the world, the human being, and God in particular, there is our understanding of being and its essence. When we make a small mistake at the beginning in our understanding of being and its essence, it will turn to be a big one in the end. And what is “at the end” of our knowledge is (...) the discovery of the First and Ultimate Cause of all things, known as: Ipsum Esse, God, the Absolute, The Most Perfect Substance, on whom everything depends, and who depends not on anything else. These present inquiries about the proper understanding of being and its essence are aimed at formulating proof of the necessity of existence of a Being that is the First Cause, and which, existing as Ipsum Esse, is the source and reason of existence of all beings. Without these inquiries, the proof itself would be incomprehensible, and more importantly it would be a purely a priori one. Furthermore, without the existential conception of being, which Thomas first formulated, one could not discover the First Cause which, as Ipsum Esse, is the source of the existence of every being. This issue seems to have escaped the attention of the author of the book Aquinas’s Way to God. The Proof in “De Ente et Essentia.”. (shrink)

This book provides a detailed exposition of one of the most practical and popular methods of proving theorems in logic, called Natural Deduction. It is presented both historically and systematically. Also some combinations with other known proof methods are explored. The initial part of the book deals with Classical Logic, whereas the rest is concerned with systems for several forms of Modal Logics, one of the most important branches of modern logic, which has wide applicability.

We examine some extensions of the constructive propositional logic with strong negation in the setting of varieties of $\mathcal{N}$ -lattices. The main aim of the paper is to give a description of all pretabular, primitive and preprimitive varieties of $\mathcal{N}$ -lattices.

This summary was delivered at the closing session of the “Korczak: A New Anthropology of Education” conference on September 21, 2002 in the Polish Culture Institute of Warsaw University’s Polish Studies Faculty. Author points that the primary goal of the conference was to set Korczak’s work against modern-day thought trends and cultural dilemmas. Summing up the result of the conference author notices that the deepest roots of the Korczak’s work lies in the Judaic and Christian traditions but it also draws (...) on European enlightenment, Polish romanticism, and positivism. (shrink)

A new type of formalization of classical first-order logic with equality is introduced on the basis of the sequent calculus. It serves to justify the claim that equality is a logical constant characterised by well-behaved rules satisfying properties usually regarded as essential. The main feature of this approach is the application of sequents built not only from formulae but also from terms. Two variants of sequent calculus are examined, a structural and a logical one. The former is defined in accordance (...) with Dos̆en’s criteria for logical constants. The latter is a standard Gentzen’s sequent calculus and satisfies Hacking’s criteria for logicality, including cut elimination. It is also shown that provided rules are harmonious in the sense advocated by Gratzl and Orlandelli. (shrink)

The paper presents a uniform proof-theoretic treatment of several kinds of free logic, including the logics of existence and definedness applied in constructive mathematics and computer science, and called here quasi-free logics. All free and quasi-free logics considered are formalised in the framework of sequent calculus, the latter for the first time. It is shown that in all cases remarkable simplifications of the starting systems are possible due to the special rule dealing with identity and existence predicate. Cut elimination is (...) proved in a constructive way for sequent calculi adequate for all logics under consideration. (shrink)

Values are relational properties that can be defined only in relation to a goal-directed system. Biological values originated with living systems and subjective values originated with the origin of vertebrate (and possibly others’) mind through a conversion (subjectivization) of biological values. While this conversion is understandable in adaptive (functional) terms, the evolutionary mechanism whereby positive and negative meanings in the mind were assigned to molecular and/or neuronal configurations in the brain, so far defies our comprehension. Whatever their origin, the primary (...) subjective values are experienced by most if not all vertebrates, and the hominid conceptual values are all ultimately derived from experiential values. (shrink)

This paper is concerned with n-localization property introduced by Newelski and Rosłanowski in [10] and getting it for CS iterations of forcing notions.

Most Enlightenment thinkers believed that the World’s order (as ultimately based on divine laws) is good and thus every gain of knowledge will have good consequences. Scientific process was assumed to entail moral progress. In fact some moral progress did occur in the Western civilization and science contributed to it, but it is widely incommensurate with the progress of science. The Enlightenment’s concept of a concerted scientific and moral progress proved largely wrong for several reasons. (1) Public morality and science (...) evolve largely independently and may either enhance or inhibit each other. (2) There are no objective values to be read in the World’s order and simply followed. Instead, our real, subjective values and the moral systems they fuel have all been generated and shaped by evolution rather than designed to be universally good, and thus ought to be managed rather than simply followed. (3) Our evolved morality is flawed, deficient, prone to doctrinal manipulation and refractory to progress. (4) The majority of people show metaethical incompetence in failing to take a reasoned critical stand toward the principles and assumptions of received morals. This makes moral progress largely dependent on those who reach metaethical competence by transcending the conventional stages of moral development. (shrink)

DARWIN’s (1859, 1871) discoveries have profound ethical implications that continue to be misrepresented and/or ignored. In contrast to socialdarwinistic misuses of his theory, Darwin was a great humanitarian who paved the way for an integrated scientific and ethical world view. As an ethical doctrine, socialdarwinism is long dead ever since its defeat by E. G. Moore although the socialdarwinistic thought is a hard-die in the biological community. The accusations of sociobiology for being socialdarwinistic are unfounded and stem from the moralistic (...) fallacy that is, a false assumption that morality is good by definition. Both social and developmental psychology demonstrate that the moral agency is a motivational device for executing reciprocity that remains at the core of any morality across all studied societies and throughout the ontogeny of moral judgment. The level of true universalizing ethical reflection (KOHLBERG’s postconvential stages or GIBBS’s existential phase) is achieved by a small minority of humans, thus showing that Homo sapiens is a moral but not an ethical animal. While the origin of reciprocity has been perfectly explained by sociobiology, the evolutionary assembly of affective and cognitive elements that make up the moral agency is being successfully studied by the social/personality/developmental psychology as extended to non-human primates. As DARWIN (1871) expected, the key innovation for the evolution of moral agency was the emergence of empathy that evolved independently at least three times: in elephants, dolphins and primates. Empathy has a motivational power of its own; it is also necessary for moral agency that requires two cognitive abilities: reflective self-consciousness and understanding of causality; the two make possible the attribution of responsibility. All these requirements are met by the chimpanzees whose moral agency operates in dyads. In contrast, the human moral agency allows for a third party intervention that opens up vast opportunities for ideologies, especially religions, to use and misuse the moral agency to enforce a reciprocation that may be harmful to both individuals and the entire group. Also, the moral agency is known to enforce enhanced intragroup cohesion and loyalty in response to conflict and war, which suggests that the two prima facie opposed human universals, morality and warfare, may have coevolved. The most important ethical consequence that follows from the increasing understanding of the primate moral agency is that every received morality is ethically flawed, none can be taken as a paragon of goodness, and each needs corrections by science-informed ethics. In fact, Darwin pioneered the integration of science and ethics, an approach that has come to be appreciated only recently under the heading of consilience. (shrink)

Aside from the local (mostly Western) efforts to subject animal experimentation to public scrutiny, the extent of animal experimentation, the acceptance of alternative methods and the fate of animals in laboratories depend on experimenters’ morality (as defined by social psychology), whose shaping is of crucial importance for the future of animal use in science. Meanwhile, sociological and ethnographic research in laboratories demonstrates that in the matter of animal use the experimenters are unreflective, ethically incompetent, and incapable of taking a critical (...) view of the received morality. This appears to result largely from their higher education which is known to cause a regression in moral development, inculcate a routine exploitation and/or objectification of animals, and ignore both the scientific premises and moral consequences of animal subjectivity. From all we know, it is researchers’ morality rather than pressing needs and expectations of humanity that interferes with a major restriction of invasive animal research in general and a wide, proactive acceptance of alternative methods in particular. (shrink)

True Darwinian Ethics -/- Darwin’s model for the evolution of morality as presented in Descent of Man (1871) is shown to comprise three major stages that are here referred to as empathic premorality, tribal morality, and universalizing morality. Empathy, the key component of Darwin’s “social instincts” that started moral evolution, is here recognized as the principal cognitive device that conveys epistemic credibility to moral agency. The two constitutive elements of the tribal morality are conscience that Darwin conceived of as a (...) conflict between a lasting social motive and an immediate selfish impulse, and true altruism that evolved as a result of group (community) selection. Darwin’s genius recognized the uniqueness of group selection in human evolution as being dependent on mental faculties and speech that facilitated habit formation through praise and blame form other group members, with empathy as a gauge of others’ emotional reactions. Contrary to repeated claims, Darwin did not derive morality from reciprocity that he (and his contemporaries) considered to be “a low motive”, and adduced it only as a mechanism of habit formation. The universalizing morality was brought about by the increasing powers of intellect that led to respecting individual lives (rather than group interests only) and extending humane concerns beyond one’s group, ultimately to all sentient beings, which places Darwin among the spiritual fathers of modern humanitarianism. In terms of substantive ethics, Darwin’s views support moral individualism that requires to treat each subject according to his/her individual characteristics rather than any group membership. Darwin’s moral individualism and universalism have been elaborated by Peter Singer as the principle of equal consideration of interests and the expanding circle of ethics, respectively. Darwin’s model of moral evolution, which starts with intuitive but epistemically reliable moral agency and then allows for its rational improvement, provides a way out of the moral subjectivism of evolutionary ethics that reduces morality to an adaptation to social life. The primary or core morality that relies on empathy and implements reciprocity supports welfare (wellbeing) of a group and its members. Welfare (which has been defined by Darwin) is a basic “terminal” (ultimate) value that constitutes an objective, measurable fact. Since the primary morality supports welfare, it is objectively good, which justifies an ethical reasoning that such morality ought to be extended (with appropriate adjustments) to all sentient beings. Confusion over the status of morality is largely due to the lack of appreciation of its complexity: not only had its motivational apparatus (known as the moral agency) evolved by superposition of many emotional and pure cognitive mechanisms, but it has subsequently been co-opted to implement prudential, religious and possibly other norms that are accidental and sometimes contrary to its original social function. Those secondary morality norms are now enforced by the moral agency but lack moral objectivity and thus may support the perception of moral subjectivism and relativism. In short, true Darwinian ethics is based on the scientific axiology that recognizes evolutionary origins of all values, requires respecting all values that are experienced by each and every subject according to its individual characteristics, and calls for a critical assessment of each and every received morality. (shrink)

In the evolution of the vertebrates and probably a few other animals (Metazoa), biological values have been translated (subjectivized) into affective experience that necessarily involves the consciousness of external objects/events (as different from one’s body), which is tantamount to the origins of subjectivity. Mammals, birds and other vertebrates are experiencing subjects even though their negative and positive experience greatly vary in scope. Some mammals are capable of vicarious experience and may act as empathic agents, and some of them, at least (...) the great apes, use true reflective self-consciousness and thus perceive themselves as agents, which generates attributive feelings. The balance of positive and negative experience determines the intrinsic value of (an individual) life. Every subjective life is autotelic in striving to maximize this balance. A subjective life may also have a social value that is measured in terms of the impact on intrinsic values of other lives. The social value of life is contingent upon an agency, that is, conscious intentional actions that impact intrinsic value of other lives. The difference between the social value of a subjective life and an instrumental value of an individual comes down to the difference between agent causation and event causation. The moral agency refers to a subcategory of agent causation that involves the awareness of impact on another subject’s wellbeing (interests). The moral agency is at present known only in the hominids (including the chimpanzees) which understand causality and use reflective self-consciousness, which in turn enables them to perceive themselves as agents and attribute to themselves responsibility. The moral agency originated as a motivational mechanism of reciprocity execution , and thus as an adaptation for group life, which is why it favors ingroup members and often promotes norms that harm other groups. The human moral agency is frequently used by third parties to implement group norms that are based on ideologies (religious or secular) and may actually harm both group members and the entire groups. Any received, bioculturally evolved morality is unlikely to be good in either absolute or universal sense. Therefore, the moral capacity alone does not make a human life categorically more valuable, even if human motivation were dominated by moral agency (which is true only of some people). The life of a vertebrate has, therefore, two values, intrinsic and social. Each can be negative or positive, and may vary to a large extent independently of the other, whereby a joint value of a subjective life cannot be sensibly assessed and compared without prior assessments of the intrinsic and social component separately. In the absence of objective reasons to rate the intrinsic value of a human life as categorically higher than that of other mentally advanced mammals, and in consideration of the observed range of social values of human lives, some of which approach zero and some are highly negative (lower than any thinkable negative values of non-human lives), the doctrine of extraordinary and inalienable (innate) human „dignity” is groundless and unethical, as it leads to a depreciation of the lives of non-human subjects, and often preempts the need to impart a real social value to one’s own life. (shrink)

Having positive experience is, by defintion, in the interest of every subject. Whether being alive per se, in addition to having positive experiences, is in a subject's interest depends of her/his cognitive development. Only a reflectively self-conscious subject can take and thus have an interest in one's own individual existence and may not want to die regardless of the expected experience. Since most non-human subjects (except for a few mammalian and avian species) are not aware of their subjective existence they (...) cannot take and thus have interest in their lives per se. However, they do have interest in maximizing rewarding experience and minimizing suffering. (shrink)