Yes,Virginia, there Really are Paraconsistent Logics

B. H. Slater has argued that there cannot be any truly paraconsistent logics, because it's always more plausible to suppose whatever negation symbol is used in the language is not a real negation, than to accept the paraconsistent reading. In this paper I neither endorse nor dispute Slater's argument concerning negation; instead, my aim is to show that as an argument against paraconsistency, it misses (some of) the target. A important class of paraconsistent logics — the preservationist logics — are not subject to this objection. In addition I show that if we identify logics by means of consequence relations, at least one dialetheic logic can be reinterpreted in preservationist (non-dialetheic) terms. Thus the interest of paraconsistent consequence relations — even those that emerge from dialetheic approaches — does not depend on the tenability of dialetheism. Of course, if dialetheism is defensible, then paraconsistent logic will be required to cope with it. But the existence (and interest) of paraconsistent logics does not depend on a defense of dialetheism.     

Dialectic and Dialetheism

In this article, I consider the possibility of interpreting Hegel's dialectic as dialetheism. After a first basic recapitulation about the meaning of the words ‘dialetheism’ and ‘dialectic’ and a consideration of Priest's own account of the relation between dialectical and dialetheic logic in 1989, I discuss some controversial issues, not directly considered by Priest. As a matter of fact, the reflection on paraconsistent logics and dialetheism has enormously grown in recent years. In addition, the reception of Hegel's logic and metaphysics has also impressively improved. So I suggest that the discussion about the binomial dialectic/dialetheism should be reopened, on these new bases.     

Dialetheism and the Impossibility of the World

This paper first offers a standard modal extension of dialetheic logics that respect the normal semantics for negation and conjunction, in an attempt to adequately model absolutism, the thesis that there are true contradictions at metaphysically possible worlds. It is shown, however, that the modal extension has unsavoury consequences for both absolutism and dialetheism. While the logic commits the absolutist to dialetheism, it commits the dialetheist to the impossibility of the actual world. A new modal logic AV is then proposed which avoids these unsavoury consequences by invalidating the interdefinability rules for the modal operators with the use of two valuation relations. However, while using AV carries no significant cost for the absolutist, the same isn't true for the dialetheist. Although using AV allows her to avoid the consequence that the actual world is an impossible world, it does so only on the condition that the dialetheist admits that she cannot give a dialetheic solution to all self-referential semantic paradoxes. Thus, unless there are any further available modal logics that don't commit her to the impossibility of the actual world, the dialetheist is faced with a dilemma. Either admit that the actual world is an impossible world, or admit that her research programme cannot give a comprehensive solution to the self-referential paradoxes.     

Measuring coherence using LP-models

This paper introduces a technique for measuring the degree of (in)coherence of inconsistent sets of propositional formulas. The coherence of these sets of formulas is calculated using the minimal models of those sets in G. Priest's Logic of Paradox. The compatibility of the information expressed by a set of formulas with the background or domain knowledge can also be measured with this technique. In this way, Hunter's objections to many-valued paraconsistent logics as instruments for measuring (in)coherence are addressed.     

Further Remarks on Truth and Contradiction

We address an issue recently discussed by Graham Priest: whether the very nature of truth (understood as in correspondence theories) rules out true contradictions, and hence whether a correspondence-theoretic notion of truth rules against dialetheism. We argue that, notwithstanding appearances to the contrary, objections from within the correspondence theory do not stand in the way of dialetheism. We close by highlighting, but not attempting to resolve, two further challenges for dialetheism which arise out of familiar philosophical theorizing about truth.     

Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics

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.     

Against classical dialetheism

Dialetheism is the view that there are true contradictions. Classical dialetheism holds further the view that the law of excluded middle is indeed a logical law. Most famous dialetheists, such as G. Priest and J. Beall, are classical dialetheists; they take classical dialetheism to be the only plausible solution to the semantic paradoxes. The main contention of the paper is, however, that their views should be rejected. Based on inspecting Priest’s and Beall’s dialetheist theories from a special perspective, this paper contends that classical dialetheism has no natural and plausible way to assign truth values to various truth-ineliminable sentences, i.e., sentences whose truth-conditions essentially involve the property of being true. Several examples of such truth-ineliminable sentences are given in the paper, and two classical dialetheist strategies for assigning them truth values are inspected. This paper argues that none of these strategies is successful.     

Paraconsistent Logic

In some logics, anything whatsoever follows from a contradiction; call these logics explosive. Paraconsistent logics are logics that are not explosive. Paraconsistent logics have a long and fruitful history, and no doubt a long and fruitful future. To give some sense of the situation, I’ll spend Section 1 exploring exactly what it takes for a logic to be paraconsistent. It will emerge that there is considerable open texture to the idea. In Section 2, I’ll give some examples of techniques for developing paraconsistent logics. In Section 3, I’ll discuss what seem to me to be some promising applications of certain paraconsistent logics. In fact, however, I don’t think there’s all that much to the concept ‘paraconsistent’ itself; the collection of paraconsistent logics is far too heterogenous to be very productively dealt with under a single label. Perhaps that will emerge as we go.     

A Hierarchy of Weak Double Negations

In this paper, a way of constructing many-valued paraconsistent logics with weak double negation axioms is proposed. A hierarchy of weak double negation axioms is addressed in this way. The many-valued paraconsistent logics constructed are defined as Gentzen-type sequent calculi. The completeness and cut-elimination theorems for these logics are proved in a uniform way. The logics constructed are also shown to be decidable.     

Commands as Divine Attributes

Theories of ethics that attempt to incorporate divine speech or commands as necessary elements in the construction of moral obligations are often viewed as vulnerable to a challenge based on the so‐called Euthyphro dilemma. According to this challenge, opponents of theistic ethics suppose that divine speech either informs one of a preexisting set of values and obligations, which makes it inconsequential, or is entirely arbitrary, which makes it irrational. This essay analyzes some of the debates on the nature of divine commands in eleventh‐century works of Islamic jurisprudence (u?ūl al‐fiqh). I show that Mu?tazilī jurisprudents advanced the view that divine commands were actions performed in time that had concrete manifestations, while Ash?arīs argued that divine speech in general, and commands in particular, were eternal divine attributes. After exposing certain weaknesses in the Euthyphro‐inspired objections to theistic ethics, I argue that the Ash?arī idea of commands as divine attributes is a promising move for scholars interested in defending a divine command view of moral obligation.     

Particular Divine Action: A Challenge To Intellectual Integrity In A Post-Christian Age

Abstract

The fact that certain configurations of problems and the philosophical antinomies, paradoxes and confusions they contain regularly return in the history of the rational exposition of these problems points to more than the limitations of human reason and the inexhaustibility of the subject matter; it is indicative of a structural problem (Ibid.). If we agree that integrity is defmed as the quality of being unimpaired based on unity or wholeness, then holding beliefs based on theories compromised by structural problems jeopardises one’s intellectual integrity. The Christian notion of particular divine action founders on more than one structural problem. On the one hand, divine action is held to be constitutive of the Christian faith. On the other hand, it is held to be inconceivable (unthinkable), improbable (unlikely to happen), impossible (incapable of occurring or happening), unnecessary (not required or desired), and indefensible (not justifiable or excusable) by the way many interpret current scientific, philosophical and theological theories. Together, these objections assert that divine action is implausible. This paper outlines five objections to particular divine action that poses a challenge to intellectual integrity. In doing so, the minimum requirements for a rationally justifiable theory of divine action is delineated.     

Socratic Proofs and Paraconsistency: A Case Study

This paper develops a new proof method for two propositional paraconsistent logics: the propositional part of Batens' weak paraconsistent logic CLuN and Schütte's maximally paraconsistent logic Φ_v. Proofs are de.ned as certain sequences of questions. The method is grounded in Inferential Erotetic Logic.     

Ideal Paraconsistent Logics

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.     

Paraconsistent logic from a modal viewpoint

In this paper we study paraconsistent negation as a modal operator, considering the fact that the classical negation of necessity has a paraconsistent behavior. We examine this operator on the one hand in the modal logic S5 and on the other hand in some new four-valued modal logics.     

The revenger's paradox

Conclusion Unlike the more commonly discussed paradoxes of deontic logic, the revenger's paradox is present not only in strong logics such as SDL, but also in much weaker logics that have been constructed to avoid the paradoxes of SDL. The paradox shows that any deontic logic with intersubstitutivity makes counter-intuitive truth assignments to some deontic expressions.However, the revenger's paradox does not pose as serious a problem as does Ross's paradox to the logics in which it occurs. A deontic logic in which Ross's paradox does not arise is preferable to one in which it does arise, even if the latter is subject to the revenger's paradox.     

Islam,the Divine Command Theory,and Religious Fundamentalism

ABSTRACT

In 1980, George F. Hourani and, more recently in 2010, Mariam al-Attar argued that Islam is not compatible with divine command theory. They rehearsed some standard objections (namely the objectivity of morality, divine goodness, moral knowledge) targeted against meta-ethical divine command theory taking the Qur’an into consideration, and argued that it should be discarded from an Islamic point of view. In addition, they criticized divine command theory since it involves obedience to God and, therefore, does not allow moral reasoning but rather is the source of religious fundamentalism and violence. Although these objections are powerful when applied to the early Ash?arite version of divine command theory, they are not powerful against recent formulations of it presented by Christian philosophers. For example, a divine command theory like that of Robert Merrihew Adams is well suited to respond to these objections. Thus, divine command theory, in its recent formulations, is a genuine option for Muslims and should be given due consideration.     

Anti-intuitionism and paraconsistency

This paper aims to help to elucidate some questions on the duality between the intuitionistic and the paraconsistent paradigms of thought, proposing some new classes of anti-intuitionistic propositional logics and investigating their relationships with the original intuitionistic logics. It is shown here that anti-intuitionistic logics are paraconsistent, and in particular we develop a first anti-intuitionistic hierarchy starting with Johansson's dual calculus and ending up with Gödel's three-valued dual calculus, showing that no calculus of this hierarchy allows the introduction of an internal implication symbol. Comparing these anti-intuitionistic logics with well-known paraconsistent calculi, we prove that they do not coincide with any of these. On the other hand, by dualizing the hierarchy of the paracomplete (or maximal weakly intuitionistic) many-valued logics (Iⁿ)_nω we show that the anti-intuitionistic hierarchy (I^n*)_nω obtained from (Iⁿ)_nω does coincide with the hierarchy of the many-valued paraconsistent logics (Pⁿ)_nω. Fundamental properties of our method are investigated, and we also discuss some questions on the duality between the intuitionistic and the paraconsistent paradigms, including the problem of self-duality. We argue that questions of duality quite naturally require refutative systems (which we call elenctic systems) as well as the usual demonstrative systems (which we call deictic systems), and multiple-conclusion logics are used as an appropriate environment to deal with them.     

Combining classical logic, paraconsistency and relevance

We present a logic with has both a simple semantics and a cut-free Gentzen-type system on one hand, and which combines relevance logics, da Costa's paraconsistent logics, and classical logic on the other. We further show that the logic has many other nice properties, and that its language is ideal from the semantic point of view.     

Luther and Shankara: Two Ways of Salvation in the Indian Context

Abstract : This article compares and contrasts the soteriology of Reformer Martin Luther with Advaita philosopher Shankara. Luther's emphasis on the communication of attributes (communicatio idiomatum) in the two natures of Christ gets doubled in faith, where the indwelling Christ takes on our human nature while giving the believer the fruits of his divine nature, such as eternal life. Conversely, our human finite history replete with suffering is taken up into the divine life, dignifying what is mundanely human. In the Indian tradition of the Upanishads and nonduality in philosophy, Shankara seeks the union of the Self (atman) with the highest reality, the Absolute (Brahman). The realization of the oneness of Self with Brahman requires the shedding of all historical or personal attributes. The result is that the suffering of oppressed untouchables and other lower castes is dubbed unreal. A healthy soteriology in the context of Indian spirituality—a Dalit soteriology—could benefit from Luther's exchange of attributes, because the mundane sufferings of humble people are dignitifed by receiving a place in God's reality.