We give a straightforward computable-model-theoretic definition of a property of \Delta^0_2 sets called order-computability. We then prove various results about these sets which suggest that, simple though the definition is, the property defies any easy characterization in pure computability theory. The most striking example is the construction of two computably isomorphic c.e. sets, one of which is order-computable and the other not.

This article addresses the relationship between ontology and politics in Jean-Luc Nancy’s theory of democracy by probing the implications of his latest ontological innovation, the concept of struction. We argue that Nancy’s democracy is a mode of politics that makes the radical pluralism of struction legitimate, opening and guarding a political space for the coexistence of the incommensurable. From this perspective, and despite Nancy’s own skepticism about the concept of biopolitics, the notion of struction opens a pathway for theorizing democracy (...) in a biopolitical key as the regime of coexistence of radically incommensurable forms of life in the absence of any coordinating principle. We nonetheless take issue with Nancy’s prescription for democracy to remain devoid of any political affirmation of its own. Instead, we suggest that the prescriptive content of democracy consists in the affirmation of the contingency of all the forms of life that coexist in it, which implies their freedom, equality, and comm... (shrink)

In 1933 Godel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that Godel's provability calculus is nothing but the forgetful projection of LP. This also achieves Godel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which (...) resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and λ-calculus. (shrink)

We outline an intuitionistic view of knowledge which maintains the original Brouwer–Heyting–Kolmogorov semantics for intuitionism and is consistent with the well-known approach that intuitionistic knowledge be regarded as the result of verification. We argue that on this view coreflectionA→KAis valid and the factivity of knowledge holds in the formKA→ ¬¬A‘known propositions cannot be false’.We show that the traditional form of factivityKA→Ais a distinctly classical principle which, liketertium non datur A∨ ¬A, does not hold intuitionistically, but, along with the whole of (...) classical epistemic logic, is intuitionistically valid in its double negation form ¬¬(KA¬A).Within the intuitionistic epistemic framework the knowability paradox is resolved in a constructive manner. We argue that this paradox is the result of an unwarranted classical reading of constructive principles and as such does not have the consequences for constructive foundations traditionally attributed it. (shrink)

We compare the logic HYPE recently suggested by H. Leitgeb as a basic propositional logic to deal with hyperintensional contexts and Heyting-Ockham logic introduced in the course of studying logical aspects of the well-founded semantics for logic programs with negation. The semantics of Heyting-Ockham logic makes use of the so-called Routley star negation. It is shown how the Routley star negation can be obtained from Dimiter Vakarelov’s theory of negation and that propositional HYPE coincides with the logic characterized by the (...) class of all involutive Routley star information frames. This result provides a much simplified semantics for HYPE and also a simplified axiomatization, which shows that HYPE is identical with the modal symmetric propositional calculus introduced by G. Moisil in 1942. Moreover, it is shown that HYPE can be faithfully embedded into a normal bi-modal logic based on classical logic. Against this background, we discuss the notion of hyperintensionality. (shrink)

Let ???? be a computable structure and let R be a new relation on its domain. We establish a necessary and sufficient condition for the existence of a copy ℬ of ???? in which the image of R (¬R, resp.) is simple (immune, resp.) relative to ℬ. We also establish, under certain effectiveness conditions on ???? and R, a necessary and sufficient condition for the existence of a computable copy ℬ of ???? in which the image of R (¬R, resp.) (...) is simple (immune, resp.). (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)

We define a novel interpretation R of second order arithmetic into Explicit Mathematics. As a difference from standard D-interpretation, which was used before and was shown to interpret only subsystems proof-theoretically weaker than T 0 , our interpretation can reach the full strength of T 0 . The R-interpretation is an adaptation of Kleene's recursive realizability, and is applicable only to intuitionistic theories.

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)

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.

The paper proves a predicate version of Solovay's well-known theorem on provability interpretations of modal logic: If a closed modal predicate-logical formula R is not valid in some finite Kripke model, then there exists an arithmetical interpretation f such that $PA \nvdash fR$ . This result implies the arithmetical completeness of arithmetically correct modal predicate logics with the finite model property (including the one-variable fragments of QGL and QS). The proof was obtained by adding "the predicate part" as a specific (...) addition to the standard Solovay construction. (shrink)

We study unification problem and problem of admissibility for inference rules in minimal Johanssonsʼ logic J and positive intuitionistic logic IPC+. This paper proves that the problem of admissibility for inference rules with coefficients is decidable for the paraconsistent minimal Johanssonsʼ logic J and the positive intuitionistic logic IPC+. Using obtained technique we show also that the unification problem for these logics is also decidable: we offer algorithms which compute complete sets of unifiers for any unifiable formula. Checking just unifiability (...) of formulas with coefficients also works via verification of admissibility. (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)

Here is an account of recent investigations into the two main concepts of negation developed in the constructive logic: the negation as reduction to absurdity, and the strong negation. These concepts are studied in the setting of paraconsistent logic.

We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t: F that read t is a justification for F. Justification Logic absorbs basic principles originating from both mainstream epistemology and the mathematical theory of proofs. It contributes to the studies of the well-known Justified True Belief vs. Knowledge problem. We state a general Correspondence Theorem showing that behind each epistemic modal logic, there is a (...) robust system of justifications. This renders a new, evidence-based foundation for epistemic logic. As a case study, we offer a resolution of the GoldmanRed Barns in Justification Logic. Furthermore, we formalize the well-known Gettier example and reveal hidden assumptions and redundancies in Gettier’s reasoning. (shrink)

The problematic of biopolitics has become increasingly important in the social sciences. Inaugurated by Michel Foucault's genealogical research on the governance of sexuality, crime and mental illness in modern Europe, the research on biopolitics has developed into a broader interdisciplinary orientation, addressing the rationalities of power over living beings in diverse spatial and temporal contexts. The development of the research on biopolitics in recent years has been characterized by two tendencies: the increasingly sophisticated theoretical engagement with the idea of power (...) over and the government of life that both elaborated and challenged the Foucauldian canon and the detailed and empirically rich investigation of the concrete aspects of the government of life in contemporary societies. Unfortunately, the two tendencies have often developed in isolation from each other, resulting in the presence of at least two debates on biopolitics: the historico-philosophical and the empirical one. This Handbook brings these two debates together, combining theoretical sophistication and empirical rigour. The volume is divided into five sections. While the first two deal with the history of the concept and contemporary theoretical debates on it, the remaining three comprise the prime sites of contemporary interdisciplinary research on biopolitics: economy, security and technology. Featuring previously unpublished articles by the leading scholars in the field, this wide-ranging and accessible companion will both serve as an introduction to the diverse research on biopolitics for undergraduate students and appeal to more advanced audiences interested in the current state of the art in biopolitics studies. (shrink)

This article focuses on Foucault’s and Agamben’s readings of Augustine’s account of human nature and original sin. Foucault’s analysis of Augustine’s account of sexual acts in paradise, subordinated to will and devoid of lust, highlights the way it constitutes the model for the married couple, whose sexual acts are only acceptable if diverted by the will away from desire and towards the tasks of procreation. While Agamben rejects Augustine’s doctrine of original sin and reclaims paradise as the original homeland of (...) humanity, his reappropriation of paradise remains conditioned by our turn towards our true nature, from which we have been estranged by sin. Agamben’s politics of reclaiming paradise necessarily involves the demand for obedience to this originary model of human nature. It therefore follows to the letter Augustine’s description of paradisiacal sex, in which the will prevails over desire by applying itself to and curtailing itself. (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.

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)

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)

This article examines a still contentious question: how conservative and liberal elements are combined in Boris N. Chicherin’s worldview and political doctrine. It considers several points of view...

Why does the theory of law have such a significant role in Russian liberalism, and how is this related to the state of the legal system in Russia and to the public’s legal consciousness? This intro...

Veritism is the thesis that the truth is the fundamental epistemic good. According to Duncan Pritchard, the most pressing objections to veritism are the trivial truths objection and the trivial inquiry problem. The former states that veritism entails that trivial truths are as important as deep and important truths. The latter is a problem that a veritist must prefer trivial inquiry that generates many trivial truths to the serious inquiry with the hope but no guarantee to discover some deep and (...) important truth. Both objections arise from the inability of veritism prima facie to properly rate the different types of truths. Pritchard's solution is to approach the truth from the perspective of the intellectually virtuous inquirer who would prefer weighty truth over trivial truth. In my commentary, I criticise the proposed solution as circular reasoning. The necessary virtue for an intellectually virtuous inquirer is that they would prefer the weighty truth over the trivial one and at the same time, the weighty truth is superior because it is the goal for intellectually virtuous inquirer. I suggest another path to substantiate veritism in the face of the two sibling objections. I argue that truth is the fundamental epistemic good as it makes the epistemic realm practically valuable more than any other epistemic good. The weighty truths are preferable to the trivial ones because the practical value of the deep and important truths is usually higher. The suggested path goes away from the attempts to prove the epistemic value of truth only within the epistemic realm, yet I argue it does not compel the intellectually virtuous inquirer to seek the truth only for the sake of practical reasons. (shrink)

While Foucault’s work on biopolitics continues to inspire diverse studies in a variety of disciplines, it has largely been missing from the debates on the possibility of “affirmative biopolitics” which have been primarily influenced by the work of Agamben and Esposito. This article restores Foucault’s work to these debates, proposing that his final lecture course at the Collège de France in 1983–1984 developed a paradigm of affirmative biopolitics in the reading of the Cynic practice of truth-telling. The Cynic problematization of (...) the relation between truth and life and their transvaluation of conventional truths by relocating them to the domain of bare life not only seeks to transform one’s life in accordance with the truth but also, through the confrontation with the existing conventions and norms, to transform the world as such. Cynic parrhesia is thus biopolitical, insofar as it reclaims the power of one’s life from the social order and its rationalities of government and applies it to oneself, investing one’s existence with truth. Since Foucault developed this reading of Cynicism in the context of his political engagement on behalf on East European dissidents, the article proceeds to analyse the resonances between parrhesia and Václav Havel’s idea of “living within the truth,” elaborating the biopolitical significance of both practices. We conclude by addressing the implications of our interpretation for Foucault scholarship and the wider debates on biopolitics. (shrink)

Various connexive FDE-based modal logics are studied. Some of these logics contain a conditional that is both connexive and strict, thereby highlighting that strictness and connexivity of a conditional do not exclude each other. In particular, the connexive modal logics cBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document}, cKN4, scBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document}, scKN4, cMBL, and scMBL are introduced semantically by means of classes of Kripke models. The logics cBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} (...) \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document} and cKN4 are connexive variants of the FDE-based modal logics BK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document} and KN4 with a weak and a strong implication, respectively. The system cMBL is a connexive variant of the modal bilattice logic MBL. The latter is a modal extension of Arieli and Avron’s logic of logical bilattices and is characterized by a class of Kripke models with a four-valued accessibility relation. In the systems scBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document}, scKN4, and scMBL, the conditional is both connexive and strict. Sound and complete tableau calculi for all these logics are presented and used to show that the entailment relations of the systems under consideration are decidable for finite premise set. Moreover, the logics cBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cBK}^-$$\end{document} and cMBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cMBL}$$\end{document} are shown to be algebraizable. The algebraizability of cMBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cMBL}$$\end{document} is derived from proving cMBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cMBL}$$\end{document} to be definitionally equivalent to MBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {MBL}$$\end{document}. All connexive modal logics studied in this paper are decidable, paraconsistent, and inconsistent but non-trivial logics. (shrink)

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)

Tracing how the logic of inoperativity works in the domains of language, law, history and humanity, 'Agamben and Politics' systematically introduces the fundamental concepts of Agamben's political thought and a critically interprets his insights in the wider context of contemporary philosophy.

This paper introduces a theory of ownership that is rooted in Israel Kirzner’s theory of entrepreneurship – The Entrepreneurial Theory of Ownership. Its central idea is that natural resources are not available to us automatically as other approaches to justice implicitly assume. Before we can use a resource, we need to do preparatory work in the form of making an entrepreneurial judgement on it. This fact, as I argue, makes it possible to put private ownership as a natural right on (...) a firm normative ground and answer many traditional challenges to private property. (shrink)

The book represents a comprehensive account of the principles of the philosophy of the pseudoabsolute and is intended for readers interested in the issues of theoretical philosophy. It is also meant for a wide range of readers interested in the relative and absolute nature of human mentality.

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)

_ Source: _Volume 28, Issue 1, pp 50 - 63 The article addresses the attempts of contemporary continental philosophy to develop a politics that would move beyond the Hobbesian logic of the constitution of political community. In their readings of Hobbes, Roberto Esposito and Giorgio Agamben emphasize the nihilistic character of Hobbes’s approach to community. For Esposito, Hobbes’s commonwealth is legitimized by a prior negation of the originary human community in the construction of the state of nature as the state (...) of war. Yet, as Agamben shows, this negative state of nature is never fully transcended by the commonwealth, which persistently reproduces it in the state of exception. These critiques emphasize the complex relation between nature and artifice in Hobbes’s thought, which have profound implications for the attempts to arrive at a ‘post-Hobbesian’ mode of political community. Neither a facile search for a truer, more fundamental state of nature nor an affirmation of artifice and denaturation as constitutive of human community are sufficient to evade the Hobbesian constellation. A genuine move beyond Hobbes would rather consist in thoroughly deactivating the very relation between nature and artifice whereby they become indistinct and no longer negate each other. (shrink)

Justification Logic provides an axiomatic description of justifications and delegates the question of their nature to semantics. In this note, we address the conceptual issue of the logical type of justifications: we argue that justifications in the logical setting are naturally interpreted as sets of formulas which leads to a class of epistemic models that we call modular models . We show that Fitting models for Justification Logic naturally encode modular models and can be regarded as convenient pre-models of the (...) former. (shrink)

The publication of The Use of Bodies, the final volume in Giorgio Agamben’s Homo Sacer series, makes it possible to take stock of Agamben’s project as a whole. Having started with a powerful critique of the biopolitical sovereignty as the essence of modern politics, Agamben concludes his project with an affirmative vision of inoperative politics of form-of-life, in which life is not negated or sacrificed to the privileged form it must attain, but rather remains inseparable from the form that does (...) nothing but express it. The article begins by reconstituting the non-relational logic that Agamben develops in order to render inoperative the existing apparatuses of ontology, ethics and politics. We then address the dimension of lifestyle as a new key domain of Agamben’s work, in which biopolitics may be recast in an affirmative key of form-of-life. While Agamben is better known for sceptical and scornful statements about contemporary liberal democracies, we shall argue that his affirmative biopolitics, characterized by destituent power, resonates with Claude Lefort’s understanding of democracy as structured around the ontological void and epistemic indeterminacy. In the conclusion we question the viability of this biopolitical democracy, focusing on Agamben’s example of the Nocturnal Council in Plato’s Laws. (shrink)