N4-lattices provide algebraic semantics for the logic N4, the paraconsistent variant of Nelson's logic with strong negation. We obtain the representation of N4-lattices showing that the structure of an arbitrary N4-lattice is completely determined by a suitable implicative lattice with distinguished filter and ideal. We introduce also special filters on N4-lattices and prove that special filters are exactly kernels of homomorphisms. Criteria of embeddability and to be a homomorphic image are obtained for N4-lattices in terms of the above mentioned representation. (...) Finally, subdirectly irreducible N4-lattices are described. (shrink)

The relationships between various modal logics based on Belnap and Dunn’s paraconsistent four-valued logic FDE are investigated. It is shown that the paraconsistent modal logic \, which lacks a primitive possibility operator \, is definitionally equivalent with the logic \, which has both \ and \ as primitive modalities. Next, a tableau calculus for the paraconsistent modal logic KN4 introduced by L. Goble is defined and used to show that KN4 is definitionally equivalent with \ without the absurdity constant. Moreover, (...) a tableau calculus is defined for the modal bilattice logic MBL introduced and investigated by A. Jung, U. Rivieccio, and R. Jansana. MBL is a generalization of BK that in its Kripke semantics makes use of a four-valued accessibility relation. It is shown that MBL can be faithfully embedded into the bimodal logic \ over the non-modal vocabulary of MBL. On the way from \ to MBL, the Fischer Servi-style modal logic \ is defined as the set of all modal formulas valid under a modified standard translation into first-order FDE, and \ is shown to be characterized by the class of all models for \. Moreover, \ is axiomatized and this axiom system is proved to be strongly sound and complete with respect to the class of models for \. Moreover, the notion of definitional equivalence is suitably weakened, so as to show that \ and \ are weakly definitionally equivalent. (shrink)

Earlier algebraic semantics for Belnapian modal logics were defined in terms of twist-structures over modal algebras. In this paper we introduce the class of BK -lattices, show that this class coincides with the abstract closure of the class of twist-structures, and it forms a variety. We prove that the lattice of subvarieties of the variety of BK -lattices is dually isomorphic to the lattice of extensions of Belnapian modal logic BK . Finally, we describe invariants determining a twist-structure over a (...) modal algebra. (shrink)

Let K be the least normal modal logic and BK its Belnapian version, which enriches K with ‘strong negation’. We carry out a systematic study of the lattice of logics containing BK based on: • introducing the classes of so-called explosive, complete and classical Belnapian modal logics; • assigning to every normal modal logic three special conservative extensions in these classes; • associating with every Belnapian modal logic its explosive, complete and classical counterparts. We investigate the relationships between special extensions (...) and counterparts, provide certain handy characterisations and suggest a useful decomposition of the lattice of logics containing BK. (shrink)

The present paper is devoted to computational aspects of propositional inconsistency-adaptive logics. In particular, we prove (relativized versions of) some principal results on computational complexity of derivability in such logics, namely in cases of CLuN r and CLuN m , i.e., CLuN supplied with the reliability strategy and the minimal abnormality strategy, respectively.

Inspired by Hintikka’s ideas on constructivism, we are going to ‘effectivize’ the game-theoretic semantics for independence-friendly first-order logic, but in a somewhat different way than he did in the monograph ‘The Principles of Mathematics Revisited’. First we show that Nelson’s realizability interpretation—which extends the famous Kleene’s realizability interpretation by adding ‘strong negation’—restricted to the implication-free first-order formulas can be viewed as an effective version of GTS for FOL. Then we propose a realizability interpretation for IF-FOL, inspired by the so-called ‘trump (...) semantics’ which was discovered by Hodges, and show that this trump realizability interpretation can be viewed as an effective version of GTS for IF-FOL. Finally we prove that the trump realizability interpretation for IF-FOL appropriately generalises Nelson’s restricted realizability interpretation for the implication-free first-order formulas. (shrink)

This paper sheds light on the relationship between the logic of generalized truth values and the logic of bilattices. It suggests a definite solution to the problem of axiomatizing the truth and falsity consequence relations, \ and \ , considered in a language without implication and determined via the truth and falsity orderings on the trilattice SIXTEEN 3 . The solution is based on the fact that a certain algebra isomorphic to SIXTEEN 3 generates the variety of commutative and distributive (...) bilattices with conflation. (shrink)

The article is devoted to the systematic study of the lattice εN4⊥ consisting of logics extending N4⊥. The logic N4⊥ is obtained from paraconsistent Nelson logic N4 by adding the new constant ⊥ and axioms ⊥ → p, p → ∼ ⊥. We study interrelations between εN4⊥ and the lattice of superintuitionistic logics. Distinguish in εN4⊥ basic subclasses of explosive logics, normal logics, logics of general form and study how they are relate.

This is a review of the book by Leo S. Klein “The Tortures of Science: A Scientist and Power, a Scientist and Money, a Scientist and Morals” (Moscow: Novoe literaturnoe obozreniye, 2017. 576 pp). The review presents the book’s structure and the key topics; marks some controversial points and conclusions made by the author. The review is dedicated to the recent 90th anniversary of Professor Leo S. Klein.

We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for ‘neither’ or/and ‘both’ leads to quite unexpected results. To be more (...) precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics. (shrink)

It was shown in the previous work of the author that one can avoid the paradox of minimal logic { ϕ , ¬ ϕ } ¬ ψ defining the negation operator via reduction not a constant of absurdity, but to a unary operator of absurdity. In the present article we study in details what does it mean that negation in a logical system can be represented via an absurdity or contradiction operator. We distinguish different sorts of such presentations. Finally, we (...) consider the possibility to represent the negation via absurdity and contradiction operators in such well known systems of paraconsistent logic as D.Batens's logic CLuN and Sette's logic P 1. (shrink)

This work treats the problem of axiomatizing the truth and falsity consequence relations, ⊨ t and ⊨ f, determined via truth and falsity orderings on the trilattice SIXTEEN 3 (Shramko and Wansing, 2005). The approach is based on a representation of SIXTEEN 3 as a twist-structure over the two-element Boolean algebra.

This work presents the two surviving letters preserved from the extensive correspondence of V.E. Sesemann, a professor of philosophy at Vilnius University at the time (1961), and B.D. Dandaron, a Buddhist teacher who was a researcher at the Buryat Integrated Research Institute at that time. The letters discuss the authors’ current work and creative plans, as well as everyday life and resettling after release from prison in 1956. In his letter, B.D. Dandaron devotes significant attention to a list of literature (...) that he was using in writing a study on aesthetics and/or that was mentioned in the records of the lectures of V.E. Sesemann in the prison camp. The letters give some idea of the nature of the relations between the correspondents and the areas of their research interests. The publication is preceded by a preface, which offers reflections on the context of the appearance of lectures on aesthetics that formed the basis of B.D. Dandaron’s research interest in this issue. The preface also provides context on the prison camp “seminar” on philosophy, in which the correspondents were active participants. V.E. Sesemann lectured on a wide range of areas in Western philosophy, including aesthetics, the history of philosophy, and philosophical anthropology. B.D. Dandaron lectured on the philosophy and practice of Buddhism. B.D. Dandaron’s particular interest in aesthetic issues was due to the fact that in the system of Buddhist teachings he developed, he paid great attention to the development of emotional intuition (associated with aesthetic perception) as an intermediate link between the rational intuition inherent in every person and the emotional-mystical intuition developed during yogic practice. (shrink)

The study serves as an introduction to the publication of B. Baradin’s (1878-1937) theses for the lecture by A. Dorzhiev (1853-1938), which was to be read at the international Buddhist exhibition planned in Leningrad in 1927. The author dwells in detail on the biographies of the Buryat academic scientist B. Baradin, as well as his Buddhist mentor Geshe A. Dorzhiev, at whose request he compiled theses. Turning to the history of the first Buddhist exhibition, which took place during the Civil (...) War in 1919, the author notes the contribution of academic scientists S.F. Oldenburg, B.Ya. Vladimirtsov, F.I. Shcherbatsky and O.O. Rosenberg. The role of Shcherbatsky, who sought to present Buddhism as a completely unique religion, consonant with modern science and not based on blind faith, is particularly emphasized. This was supposed to put Buddhism in a favorable light before the Bolsheviks and the Soviet government and prevent persecution against it. Shcherbatsky’s contribution to the ideology of the Buddhist renewal movement, whose leaders sought to cooperate with the atheistic authorities, was noted. The historical circumstances in which the second, already international, exhibition was being prepared and the reasons why it could not take place are discussed. Focusing on the content of Baradin’s theses, the author emphasizes that the main thing in them was the proximity of Buddhism to the ideology of the West and modern science. In conclusion, the author notes that the theme of the convergence of Buddhism and science has anticipated modern Western scientific research on Buddhist meditation practices, as well as discussions between Western scientists and the Dalai Lama about the benefits of Buddhist psychopractics for the modern world. (shrink)

Two logics L1 and L2 are negatively equivalent if for any set of formulas X and any negated formula ¬, ¬ can be deduced from the set of hypotheses X in L1 if and only if it can be done in L2. This article is devoted to the investigation of negative equivalence relation in the class of extensions of minimal logic.

The variety of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf N4}^\perp}$$\end{document}-lattices provides an algebraic semantics for the logic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf N4}^\perp}$$\end{document}, a version of Nelson’s logic combining paraconsistent strong negation and explosive intuitionistic negation. In this paper we construct the Priestley duality for the category of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf N4}^\perp}$$\end{document}-lattices and their homomorphisms. The obtained duality naturally extends the Priestley duality for Nelson (...) algebras constructed by R. Cignoli and A. Sendlewski. (shrink)

In a rather general setting, we prove a number of basic theorems concerning computational complexity of derivability in adaptive logics. For that setting, the so-called standard format of adaptive logics is suitably adopted, and the corresponding completeness results are established in a very uniform way.

On the 28th of October, 2006, Alexander Vladimirovich Kuznetsov, so is his full name, would have turned 80. Although belated, the editorial board of Logic and Logical Philosophy, we, the editors and contributors of the present issue, and other members of the logic community mark this event with the present issue. Most of those who contributed to it knew Kuznetsov in person and/or were influenced by him or by his ideas, which very often resided in somebody else’s papers or became (...) part of a scientific folklore. But, to be sure, there are even more mathematicians and logicians who have been witnesses to his scientific activity and/or have taken advantage of its results. (shrink)

In this paper we show that axiomatic extensions of H. Wansing’s connexive logic $$\textsf{C}$$ ( $$\textsf{C}^{\perp }$$ ) are algebraizable (in the sense of J.W. Blok and D. Pigozzi) with respect to sub-varieties of $$\textsf{C}$$ ( $$\textsf{C}^{\perp }$$ )-algebras. We develop the structure theory of $$\textsf{C}$$ ( $$\textsf{C}^{\perp }$$ )-algebras, and we prove their representability in terms of twist-like constructions over implicative lattices (Heyting algebras). As a consequence, we further clarify the relationship between the aforementioned classes. Finally, taking advantage of (...) the above machinery, we provide some preliminary remarks on the lattice of axiomatic extensions of $$\textsf{C}$$ ( $$\textsf{C}^{\perp }$$ ) as well as on some properties of their equivalent algebraic semantics. (shrink)

Today a change is imperative in approaching global problems: what is needed is not arm-twisting and power politics, but searching for ways of co-evolution in the complex social and geopolitical systems of the world. The modern theory of self-organization of complex systems provides us with an understanding of the possible forms of coexistence of heterogeneous social and geopolitical structures at different stages of development regarding the different paths of their sustainable co-evolutionary development. The theory argues that the evolutionary channel to (...) the observed increasing complexity is extremely narrow and only certain discrete spectra of relatively stable self-maintained structures are feasible in complex systems. There exists a restricted set of ways of assembling a complex evolutionary whole from diverse parts. The law of nonlinear synthesis of complex structures reads: the integration of structures in more complex ones occurs due to the establishment of a common tempo of their evolution. On the basis of the theory, we can see not only desirable but also attainable futures. (shrink)

The publication presents for the first time the B. Baradin’s theses to A. Dorzhiev’s lecture that was supposed to be delivered at the international Buddhist exhibition in Leningrad in 1927. A. Dorzhiev was a famous Buryat lama who received the academic title of Geshe (the highest philosophical academic degree in the Gelug tradition of Tibetan Buddhism) upon completion of his philosophical education in the monasteries of Mongolia and Tibet. After 1918, he was involved in organizational issues of the Buddhist Sangha (...) in Russia. B. Baradin was an academic scientist and public figure, a professor at St. Petersburg University, and the author of studies on the history and teachings of Tibetan Buddhism. After the revolution of 1917, he began working in Buryatia in various administrative positions, focusing on the organization of the scientific and cultural life of the Republic. Their main idea is that the modern world consists of the two equal parts - East and West, unification of which is possible and even necessary for the further development of humanity. The system-forming teaching of the East is Buddhism; and the lecture hold on those aspects of Buddhist philosophy that could be compared with the teachings of the West. Four points are highlighted: 1) Buddhism is comparable to Darwin’s theory in that both consider humans in the context of all living beings in an evolutionary manner; 2) Buddhism, like science, accepts the determinism principle; 3) the Buddhist concept of shunya is similar to the relativistic principles of physics; and 4) the system of Buddhist psychotechniques is similar to the theory of reflexology. B. Baradin concludes that Indo-Buddhist culture and European culture are complementary and their rapprochement will bring us closer to a truly universal culture. The theses mentioned here are, apparently, the first systematic exposition of the views of neo-Buddhists-renewalists on the points of convergence between Buddhist and Western cultures. (shrink)

Saul Aaron Kripke, the most influential philosopher and logician of his generation, died on September 15, 2022, at the age of 81.

In this paper individual proofs are integrated into provability logic. Systems of axioms for a logic with operators “A is provable” and “p is a proof of A” are introduced, provided with Kripke semantics and decision procedure. Completeness theorems with respect to the arithmetical interpretation are proved.

Paradigms of historiography of philosophy in India have being changed since late 19th c. till present, depending on the social and cultural context of the history of Indian philosophy as a part of contemporary Indian culture. This change manifests itself in the conceptions of Indian historians concerning the teaching of Buddhist Mahāyāna school of Yogācāra (4th c. and later). Historians of colonial times, basing themselves on the philosophy of Neovedаntism (S. Radhakrishnan, S. Dasgupta), regarded Buddhism as a derivate of late (...) Vedic culture and Yogācāra as a teaching that reflected - though in an essentially transmuted form - the ideology of Upaniṣads. The latter, according to Neovedantists, was based on the postulate of the existence of the single cosmic soul - Brahman, the true human Self (Ātman) being identical to it. Historians of the late colonial and early postcolonial times (P.T. Raju, D.P. Chattopadhyaya, A.K. Chatterjee, partly also S. Dasgupta) brought Yogācāra closer to the teachings of European idealism, mainly to conceptions of G. Berkeley, G.W.F. Hegel, F. Bradley, J.E. McTaggart, trying to demonstrate a principal identity of fundamental problems in Indian and Western philosophy. At the same time, they brought Yogācāra together with the teaching of Brāhmaṇic school, Advaita Vedānta, regarded as another form of Indian idealism. In later times, following the evolution of contemporary Indian culture and changings in its social and political context, historians like D.J. Kalupahana became to analyze Yogācāra as a kind of philosophy of mind. All these facts show the dependence of strategies of historico-philosophical studies in India on its social, political and cultural context: in the Yogācāra teaching mainly those aspects call attention that a historian sees as the closest to the problematic field of contemporary philosophy. (shrink)

The article examines the use of texts by Church Fathers in esoteric constructions, specifically Tertium Organum, an early work of P. D. Ouspensky created in 1911 before his acquaintance with George I. Gurdjieff. The author analyzes fragments from The Philokalia, the well-known collection of texts by Orthodox ascetic writers of the Middle Ages. Despite the difference between the esoteric system developed by Ouspensky and the Orthodox tradition, the esotericist considers it possible to use the texts of this tradition to illustrate (...) and confirm his own constructions. To do this, Ouspensky uses a special strategy of interpretation, which consists of the following: (1) selecting fragments of the interpreted text closest to his own constructions; and (2) rethinking the selected fragments within the framework of these constructions. As a result, the identity of various religious and philosophical traditions becomes vivid; these traditions turn out to be, to one degree or another, imperfect expressions of the doctrine being represented perfectly only in the esotericist’s own work. This strategy leads to the formation of doubles of interpreted texts and, more broadly, doubles of religious traditions to which the texts being interpreted belong. (shrink)

We study completely decomposable torsion-free abelian groups of the form $\mathcal{G}_S := \oplus_{n \in S} \mathbb{Q}_{p_n}$ for sets $S \subseteq \omega$. We show that $\mathcal{G}_S$has a decidable copy if and only if S is $\Sigma^0_2$and has a computable copy if and only if S is $\Sigma^0_3$.

We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst nonforking types.

This article uses the personal diaries and memoirs of Archpriest Sergius (Sergei) Bulgakov to examine the circumstances of his expulsion from Bolshevik-occupied Crimea in late 1922. At the time, he was rector of the Alexander Nevsky Cathedral in Yalta. The expulsion of Fr. Sergius was part of a large-scale operation to expel the humanist intelligentsia, who did not fit within the ideological contours of the new government. We will examine the political aspects of the condemnations of Fr. Sergius’s doctrine of (...) Sophia, the theological development of which he began in exile. We will also examine aspects of Fr. Sergius’s attitude to monarchy and his belief in the “white tsar,” noting that Bulgakov’s position as an émigré could hardly be characterized as monarchical. Sophiology may be the political equivalent of ecclesiastical democracy, of faith in the ecclesiastical people and the Christian community, which was alien to the conservative monarchism of Orthodox believers abroad, centered in Serbia’s Sremski Karlovci. We will examine the circumstances of the church’s condemnation of Fr. Sergius, which took place against the context of canonical uncertainty, and we will touch upon some aspects of the contemporary reception of Sophiology in Russia in a political context (V. Bibikhin, O. Kirichenko). (shrink)

This article examines the personal and creative relationship between the theologian, philosopher, and priest Sergei Nikolaevich Bulgakov and the artist and icon painter Yulia Reitlinger (Sister Ioa...

This article discusses development of the content of the concept “Philosophy Steamer,” which refers to the 1922 expulsion from Russia of a group of intelligentsia who sharply criticized the authorities. The author shows that the group of exiled philosophers was united both by their previous philosophical and social activity and by their joint activity as émigrés. She analyzes the concepts of “historical collective individuality,” “collective person,” and “communal person” introduced by Lev P. Karsavin in order to determine the holistic nature (...) of this exiled social group and its characteristic self-consciousness, behavior, aspirations, and feelings (moods). Examining the event known as the “Philosophy Steamer” through the concepts listed above allows the author to cognize the historical, social, and cultural reality more broadly and productively. The article demonstrates that the concept of a collective person can also be applied to other social groups, thereby extending the methodology of human-studies cognition beyond the study of facts and statistical data. (shrink)

Most commonly, the scientific realism debate is seen as dividing those who do and do not think that the striking empirical and practical successes of at least our best scientific theories indicate with high probability that those theories are ‘approximately true’. But I want to suggest that this characterization of the debate has far outlived its usefulness. Not only does it obscure the central differences between two profoundly different types of contemporary scientific realist, but even more importantly it serves to (...) disguise the most substantial points of actual disagreement between these two kinds of realists and those who instead think the historical record of scientific inquiry itself reveals that such realism is untenable in either form. (shrink)

Reply in Book Symposium on James Ladyman, Don Ross: 'Everything must go: metaphysics naturalized', Oxford: Oxford University Press, 2007.

The article examines the views of the prominent Russian politician and publicist Vasily Vitalyevich Shulgin, whom the author considers to be the largest ideologist of the “chieftain” political subculture in Russian political culture. Following Shulgin, the author distinguishes two fundamentally different models of power: “monarchical” type of power and “chieftain” type of power. V.V. Shulgin was one of the first Russian thinkers who, after Alexander Pushkin and Sergei Solovyov, considered the “golden age” of the Russian society to be under the (...) rule of “leaders-heroes”. Shulgin explained many of the problems of Russian statehood revealed in the early 20th century by the degradation of the Russian ruling class and specifically the Romanov dynasty. Under these conditions, the national leader P.A. Stolypin, able to bring the country out of crisis by evolution, had appeared “next to the monarch,” but he has not been appreciated by Russian society and it has caused a national catastrophe. The First World War has accelerated the degradation of the Russian government. The “democratic forces” that came to power in Russia for a short time could not nominate a new “leader” from their ranks. Shulgin foresaw that “intermediate figures” like the White generals or the Red diarchy of Lenin and Trotsky would eventually give way to the autocratic rule of an all-Russian “Chief,” who would combine the ideology of the Whites and the will of the Reds. (shrink)

We analyse the extent of possible computations following Hogarth ([2004]) conducted in Malament–Hogarth (MH) spacetimes, and Etesi and Németi ([2002]) in the special subclass containing rotating Kerr black holes. Hogarth ([1994]) had shown that any arithmetic statement could be resolved in a suitable MH spacetime. Etesi and Németi ([2002]) had shown that some relations on natural numbers that are neither universal nor co-universal, can be decided in Kerr spacetimes, and had asked specifically as to the extent of computational limits there. (...) The purpose of this note is to address this question, and further show that MH spacetimes can compute far beyond the arithmetic: effectively Borel statements (so hyperarithmetic in second-order number theory, or the structure of analysis) can likewise be resolved: Theorem A. If H is any hyperarithmetic predicate on integers, then there is an MH spacetime in which any query ? n H ? can be computed. In one sense this is best possible, as there is an upper bound to computational ability in any spacetime, which is thus a universal constant of that spacetime. Theorem C. Assuming the (modest and standard) requirement that spacetime manifolds be paracompact and Hausdorff, for any spacetime there will be a countable ordinal upper bound, , on the complexity of questions in the Borel hierarchy computable in it. Introduction 1.1 History and preliminaries Hyperarithmetic Computations in MH Spacetimes 2.1 Generalising SADn regions 2.2 The complexity of questions decidable in Kerr spacetimes An Upper Bound on Computational Complexity for Each Spacetime CiteULike Connotea Del.icio.us What's this? (shrink)