Non-Classical Negation in the Works of Helena Rasiowa and Their Impact on the Theory of Negation

The paper is devoted to the contributions of Helena Rasiowa to the theory of non-classical negation. The main results of Rasiowa in this area concerns–constructive logic with strong (Nelson) negation,–intuitionistic negation and some of its generalizations: minimal negation of Johansson and semi-negation.We discuss also the impact of Rasiowa works on the theory of non-classical negation.A lecture presented at the International Conference Trends in Logic III : A. Mostowski, H. Rasiowa and C. Rauszer in memoriam, Warsaw, Ruciane-Nida September 23-26, 2005.     

Gentzen-Type Methods for Bilattice Negation

A general Gentzen-style framework for handling both bilattice (or strong) negation and usual negation is introduced based on the characterization of negation by a modal-like operator. This framework is regarded as an extension, generalization or re- finement of not only bilattice logics and logics with strong negation, but also traditional logics including classical logic LK, classical modal logic S4 and classical linear logic CL. Cut-elimination theorems are proved for a variety of proposed sequent calculi including CLS (a conservative extension of CL) and CLS_cw (a conservative extension of some bilattice logics, LK and S4). Completeness theorems are given for these calculi with respect to phase semantics, for SLK (a conservative extension and fragment of LK and CLS_cw, respectively) with respect to a classical-like semantics, and for SS4 (a conservative extension and fragment of S4 and CLScw, respectively) with respect to a Kripke-type semantics. The proposed framework allows for an embedding of the proposed calculi into LK, S4 and CL.     

Brandom,Hegel and Inferentialism

In the course of developing a semantics with epistemological intent, Brandom claims that his inferentialism is Hegelian. This paper argues that, even on a charitable reading, Brandom is an anti-Hegelian.     

The Art of Nature: Hegel and the Critique of Judgment

This essay examines the reasons for Hegel's frequently professed claim that Kant's Critique of Judgment simultaneously reveals the internal limits of critical philosophy and opens the door to his own system of speculative idealism. It evaluates Hegel's contention that the conceptions of aesthetic experience, organic purposiveness, and the intuitive intellect developed in the third Critique together conspire to undermine the epistemological and metaphysical foundations of the theories of nature and freedom advanced in the first and second Critiques . Finally it explains how Hegel understands his logic and real philosophy as a realist and quasi-naturalistic alternative to Kant's subjective idealism, one that purports to generate a system of categories adequate not only to dead matter but also to organic life and free self-conscious spirit.     

Liberal Naturalism: The Curious Case of Hegel

Abstract

My aim in this paper is to defend the claim that the absolute idealism of Hegel is a liberal naturalist position against Sebastian Gardner’s claim that it is not genuinely naturalistic, and also to defend the position of ‘liberal naturalism’ from Ram Neta’s charge that there is no logical space for it to occupy. By ‘liberal naturalism’, I mean a doctrine which is a non-reductive form of philosophical naturalism. Like Fred Beiser, I take the thesis of liberal naturalism to find support in the idealism of Hegel. I begin by first explaining what philosophical naturalism amounts to. I then move on to show, using Finn Spicer’s and Alison Stone’s understandings of philosophical naturalism, how there is a stronger form of philosophical naturalism but also how there is a weaker form as well. Having established the distinction between stronger and weaker variants of philosophical naturalism, I discuss Sebastian Gardner’s recent objections to treating absolute idealism as a genuinely naturalist position. I argue that Gardner is incorrect to claim that absolute idealism is not a genuinely naturalist position on both historical and interpretive grounds, where to do so I bring in features of Hegel’s idealism to show that Hegel was committed to liberal naturalism. In the next section of the paper, I address Ram Neta’s charge that there is no logical space for liberal naturalism. To counter this claim, I offer an Hegelian diagnosis of Neta’s charge and argue that Neta’s concern about the possibility of liberal naturalism is illegitimately motivated.     

Nelson's Negation on the Base of Weaker Versions of Intuitionistic Negation

Constructive logic with Nelson negation is an extension of the intuitionistic logic with a special type of negation expressing some features of constructive falsity and refutation by counterexample. In this paper we generalize this logic weakening maximally the underlying intuitionistic negation. The resulting system, called subminimal logic with Nelson negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices admit representation formalizing the intuitive idea of refutation by means of counterexamples giving in this way a counterexample semantics of the logic in question and some of its natural extensions. Among the extensions which are near to the intuitionistic logic are the minimal logic with Nelson negation which is an extension of the Johansson's minimal logic with Nelson negation and its in a sense dual version — the co-minimal logic with Nelson negation. Among the extensions near to the classical logic are the well known 3-valued logic of Lukasiewicz, two 12-valued logics and one 48-valued logic. Standard questions for all these logics — decidability, Kripke-style semantics, complete axiomatizability, conservativeness are studied. At the end of the paper extensions based on a new connective of self-dual conjunction and an analog of the Lukasiewicz middle value ½ have also been considered.     

Hegel: Death of God and Recognition of the Self

Abstract

This paper covers the theme of the death of God considered from a Hegelian standpoint. For Aristotle, the image of God as ‘thought thinking itself’ was an image of the knowledge aspired to in philosophy. With the notion of God becoming man and his insistence on the icon of the Cross, Hegel challenged the Aristotelian goal of philosophy as immutable knowledge of an ‘ultimate’ reality. Hegel viewed the crisis of normativity (the death of the Cartesian divine guarantor) as strictly linked to the conception of the self. It is Nietzsche who is best known for alluding to the full significance of this image for modern life, but Hegel’s thought on the complex relations of philosophy and religion in the modern world can be regarded as an attempt to think through this same historical phenomenon. In this paper, I focus on the philosophical relevance of Hegel’s notion of the death of God. I argue that unpacking the significance of the ‘truths’ presented symbolically in modern Christianity is crucial in understanding the requirements that an idealistic philosophy must meet.     

A Note on Negation in Categorial Grammar

A version of strong negation is introduced into Categorial Grammar.The resulting syntactic calculi turn out to be systems of connexivelogic.     

Canonical Extensions and Relational Representations of Lattices with Negation

This work is part of a wider investigation into lattice-structured algebras and associated dual representations obtained via the methodology of canonical extensions. To this end, here we study lattices, not necessarily distributive, with negation operations.We consider equational classes of lattices equipped with a negation operation ¬ which is dually self-adjoint (the pair (¬,¬) is a Galois connection) and other axioms are added so as to give classes of lattices in which the negation is De Morgan, orthonegation, antilogism, pseudocomplementation or weak pseudocomplementation. These classes are shown to be canonical and dual relational structures are given in a generalized Kripke-style. The fact that the negation is dually self-adjoint plays an important role here, as it implies that it sends arbitrary joins to meets and that will allow us to define the dual structures in a uniform way.Among these classes, all but one—that of lattices with a negation which is an antilogism—were previously studied by W. Dzik, E. Or?owska and C. van Alten using Urquhart duality.In some cases in which a given axiom does not imply that negation is dually self-adjoint, canonicity is proven with the weaker assumption of antitonicity of the negation.     

Quantized Linear Logic,Involutive Quantales and Strong Negation

A new logic, quantized intuitionistic linear logic (QILL), is introduced, and is closely related to the logic which corresponds to Mulvey and Pelletier's (commutative) involutive quantales. Some cut-free sequent calculi with a new property quantization principle and some complete semantics such as an involutive quantale model and a quantale model are obtained for QILL. The relationship between QILL and Wansing's extended intuitionistic linear logic with strong negation is also observed using such syntactical and semantical frameworks.     

Sacrifice In Hegel's Phenomenology Of Spirit

In this paper I rely on recent literature that emphasises the importance of recognition in Hegel's philosophy in order to apply the recognition-theoretic approach to the notion of sacrifice in the Phenomenology of Spirit. Firstly, I conduct a preliminary analysis by examining the general meaning of sacrifice as a form of determinate negation. Secondly, I focus on two phenomenological moments (the struggle between ‘faith’ and ‘pure insight’, and the cult) in order to answer the question, ‘Is a real (effective and unselfish) sacrifice possible?’ Finally, I argue that sacrifice should be considered as a Darstellung, and I explain the twofold connection between sacrifice and recognition. I conclude that there is no sacrifice without recognition, and the process of recognition is intrinsically sacrificial.     

Normal Modal Substructural Logics with Strong Negation

We introduce modal propositional substructural logics with strong negation, and prove the completeness theorems (with respect to Kripke models) for these logics.     

The Revival of Rejective Negation

Whether assent (acceptance) and dissent (rejection) are thought of as speech acts or as propositional attitudes, the leading idea of rejectivism is that a grasp of the distinction between them is prior to our understanding of negation as a sentence operator, this operator then being explicable as applying to A to yield something assent to which is tantamount to dissent from A. Widely thought to have been refuted by an argument of Frege"s, rejectivism has undergone something of a revival in recent years, especially in writings by Huw Price and Timothy Smiley. While agreeing that Frege"s argument does not refute the position, we shall air some philosophical qualms about it in Section 5, after a thorough examination of the formal issues in Sections 1–4. This discussion draws on – and seeks to draw attention to – some pertinent work of Kent Bendall in the 1970s.     

On Marx,Hegel, and Critical Theory in Postwar Germany: A Conversation with Iring Fetscher

This paper consists of an introduction to the life and work of Iring Fetscher by the interviewer, followed by a conversation with Fetscher, and notes. In the interview, Fetscher discusses his relationship to Marxism, Hegelianism, Lukács, and the Frankfurt School, as well as his critique of Althusser. The contribution of Fetscher, an extremely well-known German specialist on Soviet and Marxist thought, is here discussed in greater detail than anywhere else to date in the English-language scholarly literature.     

Negation in the Context of Gaggle Theory

We study an application of gaggle theory to unary negative modal operators. First we treat negation as impossibility and get a minimal logic system Ki that has a perp semantics. Dunn's kite of different negations can be dealt with in the extensions of this basic logic K_i. Next we treat negation as “unnecessity” and use a characteristic semantics for different negations in a kite which is dual to Dunn's original one. K_u is the minimal logic that has a characteristic semantics. We also show that Shramko's falsification logic FL can be incorporated into some extension of this basic logic K_u. Finally, we unite the two basic logics K_i and K_u together to get a negative modal logic K_-, which is dual to the positive modal logic K₊ in [7]. Shramko has suggested an extension of Dunn's kite and also a dual version in [12]. He also suggested combining them into a “united” kite. We give a united semantics for this united kite of negations.     

Personal Respect, Private Property, And Market Economy: What Critical Theory Can Learn From Hegel

The aim of the present paper is to show that Hegel’s concept of personal respect is of great interest to contemporary Critical Theory. The author first analyzes this notion as it appears in the Philosophy of Right and then offers a new interpretation of the conceptual relation between personal respect and the institutions of (private) property and (capitalist) markets. In doing so, he shows why Hegel’s concept of personal respect allows us to understand markets as possible institutionalizations of this kind of recognition, and why it is compatible with a critique of neoliberal capitalism. He argues that due to these features Hegel’s notion of personal respect is of great interest to theoreticians within the tradition of critical theory.