This study is dedicated to exploring the ways in which Kierkegaard provides a criterion for thinking about the principles of plurality when, in the context of distinguishing between Socrates and Christ, between different conceptions of difference—between those that support the difference of the other and those that do not—he writes that, just as no one must separate what God has joined, so no one must join what God has separated. When Kierkegaard then makes central to faith the incommensurability of single (...) individuals, he indicates that the inviolable singularity of self and other is the one principle that can be true for all—that can be plural—since it is the one principle that is inclusive of all. In my paper I argue through Kierkegaard that the relationship between the singular and the plural embraces the paradox of absolute difference, the paradox of difference as absolute: the single individual exists only by standing in absolute relation to all others as absolute; the plural exists only insofar as it involves the commitment to the singular standard that, as absolute, preserves the difference of all. (shrink)

Through reading Kierkegaard and Derrida together I argue that deconstruction has its historical origin in, and goes no further than, biblical principles. I begin with an analysis of the complexities in Kierkegaard’s exposition of the biblical command to love the neighbor: in showing the command to express the deconstructionof originary presence, Kierkegaard appears to invoke as central to it the apparent binary opposition between divine and human being. I next turn to the Derridean deconstruction of binary opposites and particularly to (...) Derrida’s insistence that deconstruction is justice and that it never proceeds without love. In then engaging Kierkegaard’s critical distinction between ancient Greek and biblical thought, as well as Derrida’s response to issues posed by such a distinction, I show that the deconstruction of Kierkegaard can proceed only on the basis of principles whose origin is biblical and, therefore, that ideas central to Derridean deconstruction, including différance and the supplement, presuppose and express the biblical concept of neighbor. (shrink)

The extreme asceticism that Kierkegaard attributes to Christianity in The Moment and Late Writings is discussed in this essay as the challenge to the single individual in the present age. His polemic against Christendom is examined in terms of the interrelationship between the concept of neighbor, which he develops in Works of Love, and the concept of history, which he develops in both his pseudonymous and his acknowledged texts and which involves the distinction between the ancient Greek and biblical worlds. (...) I argue that it is the interrelationship of these two concepts—neighborly love and history—that allows us to see the way in which his texts provide the hermeneutical principles through which the extreme asceticism of the Moment and thus his challenge there to the single individual can be appropriated. (shrink)

When, in The Gay Science, Nietzsche poses the question of how the natural sciences are possible, he insists that they depend not on a principle that is natural but on the will to truth, the will not to deceive even oneself, with which, he holds, ?we stand on moral ground.? Yet, that the natural sciences stand on ground that is moral also means, for Nietzsche, that their origin is to be located in ?a faith that is thousands of years old,? (...) a faith that, in the Genealogy of Morals, he develops as presupposing what he calls the ascetic ideals of Judaism and Christianity. Further, in holding that the natural sciences have their origin in principles that are biblical, Nietzsche goes on to indicate that, like the natural sciences, his own critical position, unconditionally honest atheism, is, in forbidding itself ?the lie involved in belief in God,? not opposed to, but is rather an expression of, Judaism and Christianity's ascetic ideals. In addressing the interrelationships among the religious, the secular, and the natural sciences in light of Nietzsche, I argue that the natural sciences have their origin in principles that are not natural but that are no less religious than secular. (shrink)

In my paper I examine the relationship between biblical principles and modern western philosophy. I begin with various biblical passages, including the twice-told tale of the miracle of the loaves and fish from the Gospel of Matthew, the story of creation, and the story of Adam and Eve, contrasting them with what I argue are the non-tales of Plato’s Republic. I then move on to modern philosophical texts—Descartes, Spinoza, Kant, Hegel, and Kierkegaard in order to examine the idea that what (...) constitutes the ‘twice’ in our modern twice-told tales is the biblical, self-reflexive recognition that it is the core of values underpinning our stories—love of neighbor—that is itself the story that we tell. (shrink)

This study is dedicated to examining how the principles and values that mark the difference between the ancient Greco-Roman and biblical traditions help us to think about what (and who) we look at when we look at art. We begin with the Bible's self-reflexive communication to its readers regarding the status of its own images and then consider works by Michelangelo, Tejo Remy, and Charles White—while also calling on Shakespeare, Hegel, and Kierkegaard—to show that art in the biblical tradition presupposes (...) and expresses love as the transformation of life's finite immediacies, the transformation of its finite givens, into the gift. We argue, overall, that the art of the biblical tradition demands that viewers pose the question of whether, where, and how deeply they find in those works—and in themselves—the principle of love as the redemption of finitude, the redemption of existence. (shrink)

Until not too many years ago, all logics except classical logic (and, perhaps, intuitionistic logic too) were considered to be things esoteric. Today this state of a airs seems to have completely been changed. There is a growing interest in many types of nonclassical logics: modal and temporal logics, substructural logics, paraconsistent logics, non-monotonic logics { the list is long. The diversity of systems that have been proposed and studied is so great that a need is felt by many researchers (...) to try to put some order in the present logical jungle. Thus Cl91], Ep90] and Wo88] are three recent books in which an attempt is made to develop a general theoretical framework for the study of logics. On the more pragmatic side, several systems have been developed with the goal of providing a computerized logical framework in which many di erent logical systems can be implemented in a uniform way. An example is the Edinburgh LF( HHP91]). (shrink)

On the basis of Poincaré and Weyl’s view of predicativity as invariance, we develop an extensive framework for predicative, type-free first-order set theory in which $\Gamma _0$ and much bigger ordinals can be defined as von Neumann ordinals. This refutes the accepted view of $\Gamma _0$ as the “limit of predicativity”.

Zemach's arguments have gone to show that terms might be rigid designators in ordinary language even though they are not natural kind terms. It has been argued that his argument is inconclusive. However it has been claimed that Putnam's argument is much too strong for it would preclude interesting scientific hypotheses about identity between what appear to be different substances, solely on the grounds of modal necessity.It has been shown that rigid designators can be disjunctive but that this possibility is (...) not a foregone conclusion, as Zemach would have us believe. We have to allow for differences between how wesingled out the extension of a natural kind termin the past and how we intend to single out such extensionin the future. Language is not inherently conservative. There is much room for reformist manoevres. *** DIRECT SUPPORT *** A0985038 00002. (shrink)

PREFACE This essay developed from ideas in my doctoral thesis submitted to the Hebrew University in 1977. Chapter three has been amended as regards one of ...

We present correct and natural development of fundamental analysis in a predicative set theory we call PZFU\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {PZF}^{\mathsf {U}}}$$\end{document}. This is done by using a delicate and careful choice of those Dedekind cuts that are adopted as real numbers. PZFU\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {PZF}^{\mathsf {U}}}$$\end{document} is based on ancestral logic rather than on first-order logic. Its key feature is that it is definitional in the sense (...) that every object which is shown in it to exist is defined by some closed term of the theory. This allows for a very concrete, computationally-oriented model of it, and makes it very suitable for MKM and ITP. The development of analysis in PZFU\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {PZF}^{\mathsf {U}}}$$\end{document} does not involve coding, and the definitions it provides for the basic notions are the natural ones, almost the same as one can find in any standard analysis book. (shrink)

The notion of bilattice was introduced by Ginsberg, and further examined by Fitting, as a general framework for many applications. In the present paper we develop proof systems, which correspond to bilattices in an essential way. For this goal we introduce the notion of logical bilattices. We also show how they can be used for efficient inferences from possibly inconsistent data. For this we incorporate certain ideas of Kifer and Lozinskii, which happen to suit well the context of our work. (...) The outcome are paraconsistent logics with a lot of desirable properties.1. (shrink)

We provide a general investigation of Logic in which the notion of a simple consequence relation is taken to be fundamental. Our notion is more general than the usual one since we give up monotonicity and use multisets rather than sets. We use our notion for characterizing several known logics (including Linear Logic and non-monotonic logics) and for a general, semantics-independent classi cation of standard connectives via equations on consequence relations (these include Girard's \multiplicatives" and \additives"). We next investigate the (...) standard methods for uniformly representing consequence relations: Hilbert type, Natural Deduction and Gentzen type. The advantages and disadvantages of using each system and what should be taken as.. (shrink)

Linear logic is a new logic which was recently developed by Girard in order to provide a logical basis for the study of parallelism. It is described and investigated in Gi]. Girard's presentation of his logic is not so standard. In this paper we shall provide more standard proof systems and semantics. We shall also extend part of Girard's results by investigating the consequence relations associated with Linear Logic and by proving corresponding str ong completeness theorems. Finally, we shall investigate (...) the relation between Linear Logic and previously known systems, especially Relevance logics. (shrink)

Non-deterministic matrices are multiple-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically out of a certain nonempty set of options. We consider two different types of semantics which are based on Nmatrices: the dynamic one and the static one . We use the Rasiowa-Sikorski decomposition methodology to get sound and complete proof systems employing finite sets of mv-signed formulas for all propositional logics based on such structures with either of the above (...) types of semantics. Later we demonstrate how these systems can be converted into cut-free ordinary Gentzen calculi which are also sound and complete for the corresponding non-deterministic semantics. As a by-product, we get new semantic characterizations for some well-known logics. (shrink)

Paradefinite logics are logics that can be used for handling contradictory or partial information. As such, paradefinite logics should be both paraconsistent and paracomplete. In this paper we consider the simplest semantic framework for introducing paradefinite logics. It consists of the four-valued matrices that expand the minimal matrix which is characteristic for first degree entailments: Dunn–Belnap matrix. We survey and study the expressive power and proof theory of the most important logics that can be developed in this framework.

We define in precise terms the basic properties that an ‘ideal propositional paraconsistent logic’ is expected to have, and investigate the relations between them. This leads to a precise characterization of ideal propositional paraconsistent logics. We show that every three-valued paraconsistent logic which is contained in classical logic, and has a proper implication connective, is ideal. Then we show that for every n > 2 there exists an extensive family of ideal n -valued logics, each one of which is not (...) equivalent to any k -valued logic with k < n. (shrink)

We show by way of example how one can provide in a lot of cases simple modular semantics for rules of inference, so that the semantics of a system is obtained by joining the semantics of its rules in the most straightforward way. Our main tool for this task is the use of finite Nmatrices, which are multi-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically out of a certain nonempty set (...) of options. The method is applied in the area of logics with a formal consistency operator (known as LFIs), allowing us to provide in a modular way effective, finite semantics for thousands of different LFIs. (shrink)

Many efforts have been made in recent years to construct formal systems for mechanizing general mathematical reasoning. Most of these systems are based on logics which are stronger than first-order logic. However, there are good reasons to avoid using full second-order logic for this task. In this work we investigate a logic which is intermediate between FOL and SOL, and seems to be a particularly attractive alternative to both: ancestral logic. This is the logic which is obtained from FOL by (...) augmenting it with the transitive closure operator. While the study of this logic has so far been mostly model-theoretical, this work is devoted to its proof theory. Two natural Gentzen-style proof systems for ancestral logic are presented: one for the reflexive transitive closure, and one for the non-reflexive one. We show that these systems are sound for ancestral logic and provide evidence that they indeed encompass all forms of reasoning for this logic that are used in practice. The two systems are shown to be equivalent by providing translation algorithms between them. We end with an investigation of two main proof-theoretical properties: cut elimination and constructive consistency proof. (shrink)

Canonical Propositional Gentzen-type systems are systems which in addition to the standard axioms and structural rules have only pure logical rules with the sub-formula property, in which exactly one occurrence of a connective is introduced in the conclusion, and no other occurrence of any connective is mentioned anywhere else. In this paper we considerably generalize the notion of a “canonical system” to first-order languages and beyond. We extend the Propositional coherence criterion for the non-triviality of such systems to rules with (...) unary quantifiers and show that it remains constructive. Then we provide semantics for such canonical systems using 2-valued non-deterministic matrices extended to languages with quantifiers, and prove that the following properties are equivalent for a canonical system G: (1) G admits Cut-Elimination, (2) G is coherent, and (3) G has a characteristic 2-valued non-deterministic matrix. (shrink)

We suggest two precise abstract definitions of the notion of ‘relevance logic’ which are both independent of any proof system or semantics. We show that according to the simpler one, R → source is the minimal relevance logic, but R itself is not. In contrast, R and many other logics are relevance logics according to the second definition, while all fragments of linear logic are not.

. The paper presents a method for transforming a given sound and complete n-sequent proof system into an equivalent sound and complete system of ordinary sequents. The method is applicable to a large, central class of (generalized) finite-valued logics with the language satisfying a certain minimal expressiveness condition. The expressiveness condition decrees that the truth-value of any formula φ must be identifiable by determining whether certain formulas uniformly constructed from φ have designated values or not. The transformation preserves the general (...) structure of proofs in the original calculus in a way ensuring preservation of the weak cut elimination theorem under the transformation. The described transformation metod is illustrated on several concrete examples of many-valued logics, including a new application to information sources logics. (shrink)

In order to handle inconsistent knowledge bases in a reasonable way, one needs a logic which allows nontrivial inconsistent theories. Logics of this sort are called paraconsistent. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. Da Costa’s approach has led to the family of logics (...) of formal (in)consistency (LFIs). In this paper we provide in a modular way simple non-deterministic semantics for 64 of the most important logics from this family. Our semantics is 3-valued for some of the systems, and infinite-valued for the others. We prove that these results cannot be improved: neither of the systems with a three-valued non-deterministic semantics has either a finite characteristic ordinary matrix or a two-valued characteristic non-deterministic matrix, and neither of the other systems we investigate has a finite characteristic non-deterministic matrix. Still, our semantics provides decision procedures for all the systems investigated, as well as easy proofs of important proof-theoretical properties of them. (shrink)

Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We show that all reasonable paraconsistent logics based on three-valued deterministic matrices are maximal in our strong sense. This applies to practically all three-valued (...) paraconsistent logics that have been considered in the literature, including a large family of logics which were developed by da Costa's school. Then we show that in contrast, paraconsistent logics based on three-valued properly nondeterministic matrices are not maximal, except for a few special cases (which are fully characterized). However, these non-deterministic matrices are useful for representing in a clear and concise way the vast variety of the (deterministic) three-valued maximally paraconsistent matrices. The corresponding weaker notion of maximality, called premaximal paraconsistency, captures the "core" of maximal paraconsistency of all possible paraconsistent determinizations of a non-deterministic matrix, thus representing what is really essential for their maximal paraconsistency. (shrink)

The logic $G3^{<}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ was introduced in Robles and Mendéz as a paraconsistent logic which is based on Gödel’s 3-valued matrix, except that Kleene–Łukasiewicz’s negation is added to the language and is used as the main negation connective. We show that $G3^{<}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ is exactly the intersection of $G3^{\{1\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ and $G3^{\{1,0.5\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$, the two truth-preserving 3-valued logics which are based on the same truth tables. We then construct a Hilbert-type system which has for $\to $ as its sole rule of inference, and is (...) strongly sound and complete for $G3^{<}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$. Then we show how, by adding one axiom or one new rule of inference, we get strongly sound and complete systems for $G3^{\{1\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ and $G3^{\{1,0.5\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$. Finally, we provide quasi-canonical Gentzen-type systems which are sound and complete for those logics and show that they are all analytic, by proving the cut-elimination theorem for them. (shrink)

A logic \ is called self-extensional if it allows to replace occurrences of a formula by occurrences of an \-equivalent one in the context of claims about logical consequence and logical validity. It is known that no three-valued paraconsistent logic which has an implication can be self-extensional. In this paper we show that in contrast, there is exactly one self-extensional three-valued paraconsistent logic in the language of \ for which \ is a disjunction, and \ is a conjunction. We also (...) investigate the main properties of this logic, determine the expressive power of its language, and provide a cut-free Gentzen-type proof system for it. (shrink)

We investigate two large families of logics, differing from each other by the treatment of negation. The logics in one of them are obtained from the positive fragment of classical logic (with or without a propositional constant ff for “the false”) by adding various standard Gentzen-type rules for negation. The logics in the other family are similarly obtained from LJ+, the positive fragment of intuitionistic logic (again, with or without ff). For all the systems, we provide simple semantics which is (...) based on non-deterministic four-valued or three-valued structures, and prove soundness and completeness for all of them. We show that the role of each rule is to reduce the degree of non-determinism in the corresponding systems. We also show that all the systems considered are decidable, and our semantics can be used for the corresponding decision procedures. Most of the extensions of LJ+ (with or without ff) are shown to be conservative over the underlying logic, and it is determined which of them are not. (shrink)