### Book

## The Road to Universal Logic

This is the first volume of a collection of papers in honor of the fiftieth birthday of Jean-Yves Béziau. These 25 papers have been written by internationally distinguished logicians, mathematicians, computer scientists, linguists and philosophers, including Arnon Avron, John Corcoran, Wilfrid Hodges, Laurence Horn, Lloyd Humbertsone, Dale Jacquette, David Makinson, Stephen Read, and Jan Woleński. It is a state-of-the-art source of cutting-edge studies in the new interdisciplinary field of universal logic. The papers touch upon a wide range of topics including combination of logic, non-classical logic, square and other geometrical figures of opposition, categorical logic, set theory, foundation of logic, philosophy and history of logic (Aristotle, Avicenna, Buridan, Schröder, MacColl). This book offers new perspectives and challenges in the study of logic and will be of interest to all students and researchers interested the nature and future of logic.

Recent developments in international football governance seem to be progressively leading toward an increasing use of technological devices for refereeing purposes. Opponents to change are often portrayed as old-fashioned or conservative. Philosophy might be of some help to overcome the dispute. In this paper, we first explore several concepts that are central to the current debate on football refereeing. Then, we determine the business of referees in relation to rules. We assess different arguments displayed regarding the role of chance and skills in competitions. Finally, we argue for the idea of referees as full players in football games.

The conception of logical pluralism claims that there is not one true logic but there are many. The conception of metalogical relativism is based on the assumption that there is not one correct answer as to whether a given argument is deductively valid, but there are many—many non-classical answers depending of which non-classical metalogic we exploit: intuitionistic, relevant, quantum, many-valued, etc. Since this leads to the interplay between logics and metalogics the question arises: What is the nature of this interplay? The Universal Logics approach gives us hints at some answers to this question but at the expense of the exploitation of different combination of non-classical logical systems leading to the transition from metalogical pluralism to metalogical monism. The only problem in this case is the impossibility of the exploitation of an infinite combination of non-classical systems. There are also some semantic keys to the issue under consideration which are connected with the problem of the interplay of classical and non-classical universes: non-classical logics would be interpreted in the classical universe, and vice versa, the classical logic would be interpreted in non-classical universes.

This paper proposes a definition of categorical model of the deep inference system BV, defined by Guglielmi. Deep inference introduces the idea of performing a deduction in the interior of a formula, at any depth. Traditional sequent calculus rules only see the roots of formulae. However in these new systems, one can rewrite at any position in the formula tree. Deep inference in particular allows the syntactic description of logics for which there is no sequent calculus. One such system is BV, which extends linear logic to include a noncommutative self-dual connective. This is the logic our paper proposes to model. Our definition is based on the notion of a *linear functor*, due to Cockett and Seely. A BV-*category* is a linearly distributive category, possibly with negation, with an additional tensor product which, when viewed as a bivariant functor, is linear with a degeneracy condition. We show that this simple definition implies all of the key isomorphisms of the theory. We consider Girard’s category of *probabilistic coherence spaces* and show that it contains a self-dual monoidal structure in addition to the *-autonomous structure exhibited by Girard. This structure makes the category a BV-category. We believe this structure is also of independent interest, as well-behaved noncommutative operators generally are.

The text introduces all main subjects of "naive" (nonaxiomatic) set theory: functions, cardinalities, ordered and well-ordered sets, transfinite induction and its applications, ordinals, and operations on ordinals. Included are discussions and proofs of the Cantor-Bernstein Theorem, Cantor's diagonal method, Zorn's Lemma, Zermelo's Theorem, and Hamel bases. With over 150 problems, the book is a complete and accessible introduction to the subject.

We construct a model category (in the sense of Quillen) for set theory, starting from two arbitrary, but natural, conventions. It is the simplest category satisfying our conventions and modelling the notions of niteness, countability and innite equi-cardinality. We argue that from the homotopy theoretic point of view our construction is essentially automatic following basic existing methods, and so is (almost all) the verication that the construction works.

We use the posetal model category to introduce homotopy-theoretic intu- itions to set theory. Our main observation is that the homotopy invariant version of cardinality is the covering number of Shelah's PCF theory, and that other combinatorial objects, such as Shelah's revised power function - the cardinal function featuring in Shelah's revised GCH theorem | can be obtained using similar tools. We include a small \dictionary" for set theory in QtNaamen, hoping it will help in nding more meaningful homotopy-theoretic intuitions in set theory.

This paper proposes a definition of categorical model of the deep inference system BV, defined by Guglielmi. Deep inference introduces the idea of performing a deduction in the interior of a formula, at any depth. Traditional sequent calculus rules only see the roots of formulae. However in these new systems, one can rewrite at any position in the formula tree. Deep inference in particular allows the syntactic description of logics for which there is no sequent calculus. One such system is BV, which extends linear logic to include a noncommutative self-dual connective. This is the logic our paper proposes to model. Our definition is based on the notion of a *linear functor*, due to Cockett and Seely. A BV-*category* is a linearly distributive category, possibly with negation, with an additional tensor product which, when viewed as a bivariant functor, is linear with a degeneracy condition. We show that this simple definition implies all of the key isomorphisms of the theory. We consider Girard’s category of *probabilistic coherence spaces* and show that it contains a self-dual monoidal structure in addition to the *-autonomous structure exhibited by Girard. This structure makes the category a BV-category. We believe this structure is also of independent interest, as well-behaved noncommutative operators generally are.

In the monography we consider theoretic and methodic origins of fundamental notions in the theory of functions of real variable. The text is designed for future and active school math teachers.

In the same way that universal algebra is a general theory of algebraic structures, universal logic is a general theory of logical structures. During the 20th century, numerous logics have been created: intuitionistic logic, deontic logic, many-valued logic, relevant logic, linear logic, non monotonic logic, etc. Universal logic is not a new logic, it is a way of unifying this multiplicity of logics by developing general tools and concepts that can be applied to all logics. One aim of universal logic is to determine the domain of validity of such and such metatheorem (e.g. the completeness theorem) and to give general formulations of metatheorems. This is very useful for applications and helps to make the distinction between what is really essential to a particular logic and what is not, and thus gives a better understanding of this particular logic. Universal logic can also be seen as a toolkit for producing a specific logic required for a given situation, e.g. a paraconsistent deontic temporal logic.

The article considers the Views of L. N. Tolstoy not only as a representative, but also as a accomplisher of the Enlightenment. A comparison of his philosophy with the ideas of Spinoza and Diderot made it possible to clarify some aspects of the transition to the unique Tolstoy’s religious and philosophical doctrine. The comparison of General and specific features of the three philosophers was subjected to a special analysis. Special attention is paid to the way of thinking, the relation to science and the specifics of the worldview by Tolstoy and Diderot. An important aspect is researched the contradiction between the way of thinking and the way of life of the three philosophers.

Tolstoy's transition from rational perception of life to its religious and existential bases is shown. Tolstoy gradually moves away from the idea of a natural man to the idea of a man, who living the commandments of Christ. Starting from the educational worldview, Tolstoy ended by creation of religious and philosophical doctrine, which were relevant for the 20th century.

This important new book offers the first full-length interpretation of the thought of Martin Heidegger with respect to irony. In a radical reading of Heidegger's major works (from *Being and **Time* through the ‘Rector's Address' and the ‘Letter on Humanism' to ‘The Origin of the Work of Art' and the *Spiegel* interview), Andrew Haas does not claim that Heidegger is simply being ironic. Rather he argues that Heidegger's writings make such an interpretation possible - perhaps even necessary.

Heidegger begins *Being and Time* with a quote from Plato, a thinker famous for his insistence upon Socratic irony. *The Irony of Heidegger *takes seriously the apparently curious decision to introduce the threat of irony even as philosophy begins in earnest to raise the question of the meaning of being. Through a detailed and thorough reading of Heidegger's major texts and the fundamental questions they raise, Haas reveals that one of the most important philosophers of the 20th century can be read with as much irony as earnestness.* The Irony of Heidegger* attempts to show that the essence of this irony lies in uncertainty, and that the entire project of onto-heno-chrono-phenomenology, therefore needs to be called into question.

The article is concerned with the notions of technology in essays of Ernst and Friedrich Georg Jünger. The special problem of the connection between technology and freedom is discussed in the broader context of the criticism of culture and technocracy discussion in the German intellectual history of the first half of the 20th century.