The famous theory of undecidable sentences created by Kurt Godel in 1931 is presented as clearly and as rigorously as possible. Introductory explanations beginning with the necessary facts of arithmetic of integers and progressing to the theory of representability of arithmetical functions and relations in the system (S) prepare the reader for the systematic exposition of the theory of Godel which is taken up in the final chapter and the appendix.

This paper suggests that some climate denialism is bullshit. Those who spread it do not display a proper concern for the truth. This paper also shows that this bullshit is harmful in some significant ways. It undermines the epistemic demands imposed on us by what we care about, by the social roles we occupy, and by morality. It is also harmful because it corrodes epistemic trust.

A new layer of complexity, constituted of networks of information token recurrence, has been identified in socio-technical systems such as the Wikipedia online community and the Zooniverse citizen science platform. The identification of this complexity reveals that our current understanding of the actual structure of those systems, and consequently the structure of the entire World Wide Web, is incomplete, which raises novel questions for data science research but also from the perspective of social epistemology. Here we establish the principled foundations (...) and practical advantages of analyzing information diffusion within and across Web systems with Transcendental Information Cascades, and outline resulting directions for future study in the area of socio-technical systems. We also suggest that Transcendental Information Cascades may be applicable to any kind of time-evolving system that can be observed using digital technologies, and that the structures found in such systems comprise properties common to all naturally occurring complex systems. (shrink)

This review pertains to the book _Evolutionary Theology (Teologia ewolucyjna)_ written by Wojciech P. Grygiel and Damian Wąsek. The book presents a distinct and modern viewpoint on theology by offering a comprehensive analysis of the characteristics of theological language and utilizing it to reevaluate certain theological beliefs, such as the concept of original sin, within the framework of the ever-changing understanding of the Universe. This approach contributes significantly to the restoration of theology’s credibility in modern culture by bridging the gap (...) between science and theology. (shrink)

In this paper, the concept of structural information is presented. The mathematical foundation of the concept is put forward, and the nature of information encoded in a structure is studied. A method for calculating the amount of structural information is introduced. An application to the analysis of cognitive maps is also presented and discussed.

This article highlights the limitations of typicality accounts of thermodynamic behavior so as to promote an alternative line of research: understanding and accounting for the success of the techniques and equations physicists use to model the behavior of systems that begin away from equilibrium. This article also takes steps in this promising direction. It examines a technique commonly used to model the behavior of an important kind of system: a Brownian particle that has been introduced to an isolated fluid at (...) equilibrium. It also accounts for the success of the model, by identifying and grounding the technique’s key assumptions. (shrink)

Scientific models are frequently discussed in philosophy of science. A great deal of the discussion is centred on approximation, idealisation, and on how these models achieve their representational function. Despite the importance, distinct nature, and high presence of toy models, they have received little attention from philosophers. This paper hopes to remedy this situation. It aims to elevate the status of toy models: by distinguishing them from approximations and idealisations, by highlighting and elaborating on several ways the Kac ring, a (...) simple statistical mechanical model, is used as a toy model, and by explaining why toy models can be used to successfully carry out important work without performing a representational function. (shrink)

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)

I highlight that the aim of using statistical mechanics to underpin irreversible processes is, strictly speaking, ambiguous. Traditionally, however, the task of underpinning irreversible processes has been thought to be synonymous with underpinning the Second Law of thermodynamics. I claim that contributors to the foundational discussion are best interpreted as aiming to provide a microphysical justification of the Minus First Law, despite the ways their aims are often stated. I suggest that contributors should aim at accounting for both the Minus (...) First Law and Second Law. (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)

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)

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.

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 →.

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.

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)

Motivated by the significant amount of successful collaborative problem solving activity on the Web, we ask: Can the accumulated information propagation behavior on the Web be conceived as a giant machine, and reasoned about accordingly? In this paper we elaborate a thesis about the computational capability embodied in information sharing activities that happen on the Web, which we term socio-technical computation, reflecting not only explicitly conditional activities but also the organic potential residing in information on the Web.

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 book is a study in the logic of questions (sometimes called erotetic logic). The central topics in erotetic logic have been the structure of questions and the question-answer relationship. This book doesn't neglect these problems, but much of it is focussed on other issues. The main subject is the logical analysis of certain relations between questions and the contexts of their appearance. And our aim is to elaborate the conceptual apparatus of the inferential approach to the logic of questions. (...) Questions are asked for many reasons and for different purposes. Yet, before a question is asked or posed, a questioner must arrive at it. In many cases arriving at a question resembles coming to a conclusion: there are some premises involved and some inferential thought processes take place. If we agree that a conclusion need not be "conclusive", we may say that sometimes questions can play the role of conclusions. But questions can also perform the role of premises: we often pass from some "initial" question to another question. In other words, there are inferential thought processes - we shall call them erotetic inferences - in which questions play the roles of conclusions or conclusions and premises. The inferential approach to the logic of questions focusses its attention on the analysis of erotetic inferences. This book consists of eight chapters. (shrink)

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)

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