The fundamental problem of Christology is the apparent contradiction of Christ as recorded at Chalcedon. Christ is human and Christ is divine. Being divine entails being immutable. Being human entails being mutable. Were Christ two different persons there’d be no apparent contradiction. But Chalcedon rules as much out. Were Christ only partly human or only partly divine there’d be no apparent contradiction. But Chalcedon rules as much out. Were the very meaning of ‘mutable’ and/or ‘immutable’ other than what they are, (...) there’d be no apparent contradiction. But the meaning is what it is, and changing the meaning of our terms to avoid the apparent contradiction of Christ is an apparent flight from reality. What, in the end, is the explanation of the apparent contradiction of Christ? Theologians and philosophers have long advanced many consistency-seeking answers, all of which increase the metaphysical or semantical complexity of the otherwise strikingly simple but radical core of Christianity’s GodMan. In this paper, I put the simplest explanation on the theological table: namely, Christ appears to be contradictory because Christ is contradictory. This explanation may sound complicated to the many who are steeped in the mainstream account of logic according to which logic precludes the possibility of true contradictions. But the mainstream account of logic can and should be rejected. Ridding theology of the dogma of mainstream logic illuminates the simple though striking explanation of the apparent contradiction of Christ — namely, that Christ is a contradictory being. Just as the simplest explanation to the apparent roundness of the earth has earned due acceptance, so too should the simplest explanation of the apparent contradiction of Christ. (shrink)

Consequence is at the heart of logic; an account of consequence, of what follows from what, offers a vital tool in the evaluation of arguments. Since philosophy itself proceeds by way of argument and inference, a clear view of what logical consequence amounts to is of central importance to the whole discipline. In this book JC Beall and Greg Restall present and defend what thay call logical pluralism, the view that there is more than one genuine deductive consequence relation, a (...) position which has profound implications for many linguists as well as for philosophers. We should not search for one true logic, since there are many. (shrink)

Consequence is at the heart of logic; an account of consequence, of what follows from what, offers a vital tool in the evaluation of arguments. Since philosophy itself proceeds by way of argument and inference, a clear view of what logical consequence amounts to is of central importance to the whole discipline. In this book JC Beall and Greg Restall present and defend what thay call logical pluralism, the view that there is more than one genuine deductive consequence relation, a (...) position which has profound implications for many linguists as well as for philosophers. We should not search for one true logic, since there are many. (shrink)

1. In both Benedetto Croce's and Hegel's own terminology, dialectics can be understood as dottrina degli opposti (the doctrine of the opposites – Lehre der Gegensätze).1 In the dialectical process,...

This paper briefly defends theses in Beall 2014 against objections advanced in Weber et al. 2014. The second part of the paper both defends and fortifies an objection to the ‘inclosure’ argument for glut theory, spelling an end to the inclosure strategy (or at least its application to the sorites).

Deflationist accounts of truth are widely held in contemporary philosophy: they seek to show that truth is a dispensable concept with no metaphysical depth. However, logical paradoxes present problems for deflationists that their work has struggled to overcome. In this volume of fourteen original essays, a distinguished team of contributors explore the extent to which, if at all, deflationism can accommodate paradox. The volume will be of interest to philosophers of logic, philosophers of language, and anyone working on truth. Contributors (...) include Bradley Armour-Garb, Jody Azzouni, JC Beall, Hartry Field, Christopher Gauker, Michael Glanzberg, Dorothy Grover, Anil Gupta, Volker Halbach, Leon Horsten, Paul Horwich, Graham Priest, Greg Restall, and Alan Weir. (shrink)

This paper presents a simple but, by my lights, effective argument for a subclassical account of logic—an account according to which logical consequence is (properly) weaker than the standard, so‐called classical account. Alas, the vast bulk of the paper is setup. Because of the many conflicting uses of ‘logic’ the paper begins, following a disclaimer on logic and inference, by fixing the sense of ‘logic’ in question, and then proceeds to rehearse both the target subclassical account of logic and its (...) well‐known relative (viz., classical logic). With background in place the simple argument—which takes up less than five pages—is advanced. My hope is that the minimal presentation will help to get ‘the simple argument’ plainly on the table, and that subsequent debate can move us forward. (shrink)

In this paper, we distinguish two versions of Curry's paradox: c-Curry, the standard conditional-Curry paradox, and v-Curry, a validity-involving version of Curry's paradox that isn’t automatically solved by solving c-curry. A unified treatment of curry paradox thus calls for a unified treatment of both c-Curry and v-Curry. If, as is often thought, c-Curry paradox is to be solved via non-classical logic, then v-Curry may require a lesson about the structure—indeed, the substructure—of the validity relation itself.

One of the most dominant approaches to semantics for relevant (and many paraconsistent) logics is the Routley-Meyer semantics involving a ternary relation on points. To some (many?), this ternary relation has seemed like a technical trick devoid of an intuitively appealing philosophical story that connects it up with conditionality in general. In this paper, we respond to this worry by providing three different philosophical accounts of the ternary relation that correspond to three conceptions of conditionality. We close by briefly discussing (...) a general conception of conditionality that may unify the three given conceptions. (shrink)

Three leading philosopher-logicians present a clear and concise overview of formal theories of truth, explaining key logical techniques. Truth is as central topic in philosophy: formal theories study the connections between truth and logic, including the intriguing challenges presented by paradoxes like the Liar.

Weber, Colyvan, and Priest have advanced glutty approaches to the sorites, on which the truth about the penumbral region of a soritical series is inconsistent. The major benefit of a glut-based approach is maintaining the truth of all sorites premisses while none the less avoiding, in a principled fashion, the absurdity of the sorites conclusion. I agree that this is a major virtue of the target glutty approach; however, I think that it can be had without gluts. If correct, this (...) result weighs heavily against the proposed glutty approach, at least given the default-consistency principle that all target glutty philosophers accept: posit gluts only if there is no consistent theory that enjoys the same virtues as the would-be glutty solution. (shrink)

This paper offers a new and very simple alternative to Bochvar's well known nonsense -- or meaninglessness -- interpretation of Weak Kleene logic. To help orient discussion I begin by reviewing the familiar Strong Kleene logic and its standard interpretation; I then review Weak Kleene logic and the standard interpretation. While I note a common worry about the Bochvar interpretation my aim is only to give an alternative -- and I think very elegant -- interpretation, not necessarily a replacement.

In this paper I advance and defend a very simple position according to which logic is subclassical but is weaker than the leading subclassical-logic views have it.

This chapter attempts to give a brief overview of nonclassical (-logic) theories of truth. Due to space limitations, we follow a victory-through-sacrifice policy: sacrifice details in exchange for clarity of big-picture ideas. This policy results in our giving all-too-brief treatment to certain topics that have dominated discussion in the non-classical-logic area of truth studies. (This is particularly so of the ‘suitable conditoinal’ issue: §4.3.) Still, we present enough representative ideas that one may fruitfully turn from this essay to the more-detailed (...) cited works for further study. Throughout – again, due to space – we focus only on the most central motivation for standard non-classical-logic-based truth theories: namely, truth-theoretic paradox (specifically, due to space, the liar paradox). (shrink)

This paper applies what I call the shrieking method (a refined version of an idea with roots in Priest's work) to one of – if not the – issues confronting glut-theoretic approaches to paradox (viz., the problem of ‘just true’ or, what comes to the same, ‘just false’). The paper serves as a challenge to formulate a problem of ‘just true’ that isn't solved by shrieking (as advanced in this paper), if such a problem be thought to exist.

We shed light on an old problem by showing that the logic LP cannot define a binary connective $\odot$ obeying detachment in the sense that every valuation satisfying $\varphi$ and $(\varphi\odot\psi)$ also satisfies $\psi$ , except trivially. We derive this as a corollary of a more general result concerning variable sharing.

Philosophical applications of familiar paracomplete and paraconsistent logics often rely on an idea of . With respect to the paraconsistent logic LP (the dual of Strong Kleene or K3), such is standardly cashed out via an LP-based nonmonotonic logic due to Priest (1991, 2006a). In this paper, I offer an alternative approach via a monotonic multiple-conclusion version of LP.

This paper is a sequel to Beall (2011), in which I both give and discuss the philosophical import of a result for the propositional (multiple-conclusion) logic LP+. Feedback on such ideas prompted a spelling out of the first-order case. My aim in this paper is to do just that: namely, explicitly record the first-order result(s), including the collapse results for K3+ and FDE+.

Traditional monotheism has long faced logical puzzles. We argue that such puzzles rest on the assumed logical truth of the Law of Excluded Middle, which we suggest there is little theological reason to accept. By way of illustration we focus on God's alleged stone problem, and present a simple but plausible ‘gappy’ framework for addressing this puzzle. We assume familiarity with the proposed logic but an appendix is offered as a brief review.

The first sentence in this essay is a lie. There is something odd about saying so, as has been known since ancient times. To see why, remember that all lies are untrue. Is the first sentence true? If it is, then it is a lie, and so it is not true. Conversely, suppose that it is not true. As we (viz., the authors) have said it, presumably with the intention of you believing it when it is not true, it is (...) a lie. But then it is true! (shrink)

I believe that, for reasons elaborated elsewhere (Beall, 2009; Priest, 2006a, 2006b), the logic LP (Asenjo, 1966; Asenjo & Tamburino, 1975; Priest, 1979) is roughly right as far as logic goes.1 But logic cannot go everywhere; we need to provide nonlogical axioms to specify our (axiomatic) theories. This is uncontroversial, but it has also been the source of discomfort for LP-based theorists, particularly with respect to true mathematical theories which we take to be consistent. My example, throughout, is arithmetic; but (...) the more general case is also considered. (shrink)

The Law of Non-Contradiction - that no contradiction can be true - has been a seemingly unassailable dogma since the work of Aristotle, in Book G of the Metaphysics. It is an assumption challenged from a variety of angles in this collection of original papers. Twenty-three of the world's leading experts investigate the 'law', considering arguments for and against it and discussing methodological issues that arise whenever we question the legitimacy of logical principles. The result is a balanced inquiry into (...) a venerable principle of logic, one that raises questions at the very centre of logic itself. The aim of this volume is to present a comprehensive debate about the Law of Non-Contradiction, from discussions as to how the law is to be understood, to reasons for accepting or re-thinking the law, and to issues that raise challenges to the law, such as the Liar Paradox, and a 'dialetheic' resolution of that paradox. The editors contribute an introduction which surveys the issues and serves to frame the debate, and a useful bibliography offering a guide to further reading. This volume will be of interest to anyone working on philosophical logic, and to anyone who has ever wondered about the status of logical laws and about how one might proceed to mount arguments for or against them. (shrink)

English summary: Criticism is often levelled at Hegel's philosophy of the absolute spirit for its being conceived at the expense of the world. In view of this, Hegel's regular reference to a world in the Phenomenology of Spirit is somewhat puzzling. How then are we to understand the presence of the world, or even numerous worlds? Tereza Matejckova contends that at every stage of its journey, consciousness designs a new world-model, each built on central concepts of the respective form of (...) consciousness. Consciousness thus finds itself in a world of contradiction, of life, of struggle, or one of unhappy diremption. The point of absolute knowledge is not that the world is now derived from consciousness itself, but rather that it conceives a world-model that corresponds with reality, as whose reflection it itself now grasps. German description: Oft wird gegen Hegel der Vorwurf erhoben, seine Philosophie des absoluten Geistes sei auf Kosten der Welt konzipiert. Umso auffallender ist, dass Hegel in der Phanomenologie des Geistes nahezu auf jeder Ebene auf eine Welt zu sprechen kommt. Wie ist diese Gegenwart der Welt, sogar einer Mehrzahl von Welten zu deuten? Tereza Matejckova behauptet, dass das Bewusstsein auf jeder Ebene seiner Genese ein Weltmodell entwirft, in dessen Zentrum Schlusselbegriffe der einschlagigen Bewusstseinsgestalt stehen. Das Bewusstsein findet sich somit in einer Welt des Widerspruchs, des Lebens, des Kampfes oder der unglucklichen Entzweiung wieder. Pointe des absoluten Wissens ist nicht, dass das Bewusstsein nun die Welt aus sich selbst ableitet, sondern dass es ein der Wirklichkeit entsprechendes Weltmodell konzipiert, als dessen Widerschein es sich selbst erfasst. (shrink)