This paper explores allowing truth value assignments to be undetermined or "partial" and overdetermined or "inconsistent", thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including ukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, and I investigate (...) twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson 's "constructible falsity". (shrink)

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.

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)

ABSTRACTDo truth tables—the ordinary sort that we use in teaching and explaining basic propositional logic—require an assumption of consistency for their construction? In this essay we show that truth tables can be built in a consistency-independent paraconsistent setting, without any appeal to classical logic. This is evidence for a more general claim—that when we write down the orthodox semantic clauses for a logic, whatever logic we presuppose in the background will be the logic that appears in the foreground. Rather than (...) any one logic being privileged, then, on this count partisans across the logical spectrum are in relatively similar dialectical positions. (shrink)

This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem.

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,...

Ultralogic as Universal? is a seminal text in non-classcial logic. Richard Routley presents a hugely ambitious program: to use an 'ultramodal' logic as a universal key, which opens, if rightly operated, all locks. It provides a canon for reasoning in every situation, including illogical, inconsistent and paradoxical ones, realized or not, possible or not. A universal logic, Routley argues, enables us to go where no other logic—especially not classical logic—can. Routley provides an expansive and singular vision of how a universal (...) logic might one day solve major problems in set theory, arithmetic, linguistics, physics, and more. It circulated in typescript in the late 1970s before appearing as the Appendix to Exploring Meinong's Jungle and Beyond. With engaging, forceful prose, unsparing criticism of entrenched institutions, and many tantalizing proof sketches, Ultralogic? has had a major influence on the development of paraconsistent and relevant logic. This new edition makes this work available for a modern audience, newly typeset and corrected, along with extensive notes, and new commentary essays. (shrink)

The aim of this paper is to lay bare the roots of dialetheism in discussions about dialectics and dialectical logic at the time of the first development of paraconsistent logics. In other words, th...

In this first volume of The Sylvan Jungle, the editors present a scholarly edition of the first chapter, "Exploring Meinong's Jungle," of Richard Routley's 1000-plus page book, Exploring Meinong's Jungle and Beyond. Going against the Quinean orthodoxy, Routley’s aim was to support Meinong’s idea that we can truthfully refer to non-existent and even impossible objects, like Superman, unicorns and the round-square cupola on Berkeley College. The tools of non-classical logic at Routley’s disposal enabled him to update Meinong’s project for a (...) new generation. This volume begins with an Introduction from Dominic Hyde, “The ‘Jungle Book’ in Context,” an essay that situates Exploring Meinong’s Jungle and Beyond historically. We provide the original Preface by Routley, followed by Chapter 1: “Exploring Meinong’s Jungle and Beyond.” In Chapter 2, Nicholas Griffin argues that Sylvan’s project was insufficiently radical with his essay, “Why the Original Theory of Items Didn’t Go Far Enough.” Sylvan revisits his position from this time in Chapter 3, with his article, “Re-Exploring Item-Theory.” Filippo Casati, who has worked in the Routley Archives then takes up the question of the future of Sylvan’s research program in his essay, “The Future Perfect of Exploring Meinong’s Jungle.” Iconic and iconoclastic Australian philosopher Richard Routley published Exploring Meinong’s Jungle and Beyond in 1980. This work has fallen out of print, yet without great fanfare it has influenced two generations of philosophers and logicians. (shrink)

The naive set theory problem is to begin with a full comprehension axiom, and to find a logic strong enough to prove theorems, but weak enough not to prove everything. This paper considers the sub-problem of expressing extensional identity and the subset relation in paraconsistent, relevant solutions, in light of a recent proposal from Beall, Brady, Hazen, Priest and Restall [4]. The main result is that the proposal, in the context of an independently motivated formalization of naive set theory, leads (...) to triviality. (shrink)

In this article, our aim is to take a step towards a full understanding of the notion of paraconsistency in the context of metainferential logics. Following the work initiated by Barrio et al. [2018], we will consider a metainferential logic to be paraconsistent whenever the metainferential version of Explosion (or meta-Explosion) is invalid. However, our contribution consists in modifying the definition of meta-Explosion by extending the standard framework and introducing a negation for inferences and metainferences. From this new perspective, Tarskian (...) paraconsistent logics such as LP will not turn out to be metainferentially paraconsistent, in contrast to, for instance, non-transitive logics like ST. Finally, we will end up by defining a logic which is metainferentially paraconsistent at every level, and discussing whether this logic is uniform through translations. (shrink)

In this paper a logic for reasoning disquotationally about truth is presented and shown to have a standard model. This work improves on Hartry Field's recent results establishing consistency and omega-consistency of truth-theories with strong conditional logics. A novel method utilising the Banach fixed point theorem for contracting functions on complete metric spaces is invoked, and the resulting logic is shown to validate a number of principles which existing revision theoretic methods have heretofore failed to provide.

In "Semantic paradoxes and abductive methodology", Williamson presents a new Quinean argument based on central ingredients of common pragmatism about theory choice (including logical theory, as is common). What makes it new is that, in addition to avoiding Quine's unfortunate charge of mere terminological squabble, Williamson's argument explicitly rejects at least for purposes of the argument Quine's key conservatism premise. In this paper I do two things. First, I argue that Williamson's new Quinean argument implicitly relies on Quine's conservatism principle. (...) Second, by way of answering his charges against nonclassical logic I directly defend a particular subclassical account of logical consequence. (shrink)

The idea of relevant logic—that irrelevant inferences are invalid—is appealing. But the standard semantics for relevant logics involve baroque metaphysics: a three-place accessibility relation, a star operator, and ‘bad’ worlds. In this article we propose that these oddities express a mismatch between non-classical object theory and classical metatheory. A uniformly relevant semantics for relevant logic is a better fit.

In his famous work on vagueness, Russell named “fallacy of verbalism” the fallacy that consists in mistaking the properties of words for the properties of things. In this paper, I examine two (clusters of) mainstream paraconsistent logical theories – the non-adjunctive and relevant approaches –, and show that, if they are given a strongly paraconsistent or dialetheic reading, the charge of committing the Russellian Fallacy can be raised against them in a sophisticated way, by appealing to the intuitive reading of (...) their underlying semantics. The meaning of “intuitive reading” is clarified by exploiting a well-established distinction between pure and applied semantics. If the proposed arguments go through, the dialetheist or strong paraconsistentist faces the following Dilemma: either she must withdraw her claim to have exhibited true contradictions in a metaphysically robust sense – therefore, inconsistent objects and/or states of affairs that make those contradictions true; or she has to give up realism on truth, and embrace some form of anti-realistic (idealistic, or broadly constructivist) metaphysics. Sticking to the second horn of the Dilemma, though, appears to be promising: it could lead to a collapse of the very distinction, commonly held in the literature, between a weak and a strong form of paraconsistency – and this could be a welcome result for a dialetheist. (shrink)

The paper concerns time, change and contradiction, and is in three parts. The first is an analysis of the problem of the instant of change. It is argued that some changes are such that at the instant of change the system is in both the prior and the posterior state. In particular there are some changes from p being true to p being true where a contradiction is realized. The second part of the paper specifies a formal logic which accommodates (...) this possibility. It is a tense logic based on an underlying paraconsistent prepositional logic, the logic of paradox. (See the author's article of the same name Journal of Philosophical Logic 8 (1979).) Soundness and completeness are established, the latter by the canonical model construction, and extensions of the basic system briefly considered. The final part of the paper discusses Leibniz's principle of continuity: Whatever holds up to the limit holds at the limit. It argues that in the context of physical changes this is a very plausible principle. When it is built into the logic of the previous part, it allows a rigorous proof that change entails contradictions. Finally the relation of this to remarks on dialectics by Hegel and Engels is briefly discussed. (shrink)

Da Costa's C systems are surveyed and motivated, and significant failings of the systems are indicated. Variations are then made on these systems in an attempt to surmount their defects and limitations. The main system to emerge from this effort, system CC , is investigated in some detail, and dual-intuitionistic semantical analyses are developed for it and surrounding systems. These semantics are then adapted for the original C systems, first in a rather unilluminating relational fashion, subsequently in a more illuminating (...) way through the introduction of impossible situations where and and or change roles. Finally other attempts to break out of impasses for the original and expanded C systems, by going inside them, are looked at, and further research directions suggested. (shrink)

The idea at the basis of this special issue is that reopening the old debate about the logical status of Hegel’s dialectics is extremely interesting, for various reasons.1 The first reason is that...

Val Plumwood and Richard Sylvan argued from their joint paper The Semantics of First Degree Entailment and onward that the variable sharing property is but a mere consequence of a good entailment relation, indeed they viewed it as a mere negative test of adequacy of such a relation, the property itself being a rather philosophically barren concept. Such a relation is rather to be analyzed as a sufficiency relation free of any form of premise suppression. Suppression of premises, therefore, gained (...) center stage. Despite this, however, no serious attempt was ever made at analyzing the concept. This paper shows that their suggestions for how to understand it, either as the Anti-Suppression Principle or as the Joint Force Principle, turn out to yield properties strictly weaker than that of variable sharing. A suggestion for how to understand some of their use of the notion of suppression which clearly is not in line with these two mentioned principles is given, and their arguments to the effect that the Anderson and Belnap logics T, E and R are suppressive are shown to be both technically and philosophically wanting. Suppression-freedom, it is argued, cannot do the job Plumwood and Sylvan intended it to do. (shrink)

This paper challenges the widespread view that contradictions have semantic content. I argue that contradictory sentences in natural language, taken literally and occurring within assertoric contexts, lack content. I present an extended twofold argument, which rests on a set of considerations about the fundamental connection between meaning-constitutivity in natural language and the semantic status of contradictions. First, I argue that the contradictory negations of analytic statements cannot have semantic content as a necessary condition for the possibility of meaning-constitutive facts in (...) natural language. Second, I argue that the special role of contradictions in the constitution of sameness and difference of content in natural language is incompatible with the view that non-analytic contradictions could have semantic content. These two arguments jointly imply that all contradictions in natural language lack content. (shrink)

Richard Sylvan (né Routley) was one of Australasia's most prolific and systematic philosophers. Though known for his innovative work in logic and metaphysics, the astonishing breadth of his philosophical endeavours included almost all reaches of philosophy. Taking the view that very basic assumptions of mainstream philosophy were fundamentally mistaken, he sought radical change across a wide range of theories. However, his view of the centrality of logic and recognition of the possibilities opened up by logical innovation in the fundamental areas (...) of metaphysics resulted in his working primarily in these two, closely connected fields. It is this work in logic and metaphysics that is the main focus of what follows. (shrink)

Dialetheists believe some sentences are both true and false. Objectors have argued that this makes it unclear how people can disagree with each other because, given the dialetheist’s commitments, if I make a claim and you tell me my claim is false, we might both be correct. Graham Priest (2006a) thinks that people disagree by rejecting or denying what is said rather than ascribing falsehood to it. We build on the work of Julien Murzi and Massimiliano Carrara (2015) and show (...) that Priest’s approach cannot succeed: given the same dialetheist’s commitments you may be correct to reject a claim that I correctly believe. We argue further that any attempt to solve the problem by identifying a new attitude of disagreement will also fail. The culprit, we claim, is the attempt to find a pair of attitudes that satisfy ‘exclusivity’—that is, attitudes such that both cannot be simultaneously correct. Instead of identifying disagreement by the kinds of attitudes involved, we propose dialetheists focus on the normative landscape and identify it in part by whether parties have reasons to change their attitudes. We offer our own normative theory of disagreement to help dialetheists with this challenge. (shrink)

The view that contradictions cannot be true has been part of accepted philosophical theory since at least the time of Aristotle. In this regard, it is almost unique in the history of philosophy. Only in the last forty years has the view been systematically challenged with the advent of dialetheism. Since Graham Priest introduced dialetheism as a solution to certain self-referential paradoxes, the possibility of true contradictions has been a live issue in the philosophy of logic. Yet, despite the arguments (...) advanced by dialetheists, many logicians and philosophers still hold the opinion that contradictions cannot be true. -/- Rather than advocating the truth of certain contradictions, this thesis offers a different challenge to the classical logician. By showing that it can be philosophically coherent to propose that true contradictions are metaphysically possible, the thesis suggests that the classical logician must do more than she currently has to justify her confidence in the impossibility of true contradictions. Simply fighting off the dialetheist’s putative examples of true contradictions at the actual world isn’t enough to justify the classical logician’s conclusion that true contradictions are impossible. -/- To aid the thesis dialectically, we introduce a new position, absolutism, which hypothesises that it’s metaphysically possible for at least one contradiction to be true, contrasting with the dialetheic hypothesis that some contradictions are true in the actual world. We demonstrate that absolutism can be given a philosophically coherent interpretation, an appropriate logic, and that certain criticisms are completely toothless against absolutism. The challenge put to the classical logician is then: On what logical or philosophical grounds can we rule out the metaphysical possibility of true contradictions? (shrink)

No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient moments. Once the problems are resolved, consistency (in some sense or other) is restored. Even when one accepts this view, what remains is the question what (...) mathematicians do during such a transient moment? This requires some method or other to reason with inconsistencies. But there is more: what if one accepts the view that mathematics is always in a phase of transience? In short, that mathematics is basically inconsistent? Do we then not need a mathematics of inconsistency? This paper wants to explore these issues, using classic examples such as infinitesimals, complex numbers, and infinity. (shrink)

Introduction to the Routley/Sylvan Issue.

L’approche la plus influente de la logique des non-existences est celle provenant de la tradition Frege-Russell. L’un des plus importants dissidents à cette tradition, à ses débuts, était Hugh MacColl. C’est en relation avec la notion d’existence et avec les arguments impliquant des fictions, que le travail de MacColl montre une grande différence avec celui de ses contemporains. En effet, MacColl fut le premier à implémenter dans un système formel l’idée qu’introduire des fictions dans le domaine de la logique revient (...) à fournir un langage muni de sous-domaines avec différents types d’objets. Dans cet article, nous avançons quelques remarques sur la portée de la logique de MacColl sur les non-existences. Plus précisément, nous suggérons qu’il y a un lien conceptuel fort entre la notion de subsistance chez Russell et la notion d’existencesymbolique chez MacColl. (shrink)

The claim that nature is self-consistent has recently been contested by a number of paraconsistent logicians. In this paper I will survey the arguments which paraconsistent logicians have presented for the thesis that nature is actually inconsistent. My conclusion is that these arguments all fail.The paraconsistency programme has to date been concerned primarily with outlining the philosophical inadequacy of classical logic, and detailed discussions of issues bearing upon the philosophical adequacy of the paraconsistency position itself are not to be found (...) as yet in the literature. This inadequacy will be illustrated here with respect to the question of the self-consistency of nature. (shrink)

A review of John Woods, Truth in Fiction: Rethinking its Logic. Cham: Springer, 2018.