In this article, we defend that incorporating a rejection operator into a paraconsistent language involves fully specifying its inferential characteristics within the logic. To do this, we examine a recent proposal by Berto for a paraconsistent rejection, which — according to him — avoids paradox, even when introduced into a language that contains self-reference and a transparent truth predicate. We will show that this proposal is inadequate because it is too incomplete. We argue that the reason it avoids trouble is (...) that the inferential characteristics of the new operator are left unspecified, exporting the task of specifying them to metaphysicians. Additionally, we show that when completing this proposal with some plausible rules for the rejection operator, paradoxes do arise. Finally, we draw some more general implications from the study of this example. (shrink)

Paraconsistent and dialetheist approaches to a theory of truth are faced with a problem: the expressive resources of the logic do not suffice to express that a sentence is just true—i.e., true and not also false—or to express that a sentence is consistent. In his recent book, Spandrels of Truth, Jc Beall proposes a ‘just true’-operator to identify sentences that are true and not also false. Beall suggests seven principles that a ‘just true’-operator must fulfill, and proves that his operator (...) indeed fulfills all of them. He concludes that just true has been expressed in the language. I argue that, while the seven conditions may be necessary for an operator to express just true, they are not jointly sufficient. Specifically, first, I prove that a further plausible desideratum for necessary conditions on ‘just true’ is not fulfilled by Beall's proposal, namely that ‘just true’ ascriptions should themselves be just true, and not also false (or equivalently, that the ‘just true’-operator iterates). Second, I show that Beall's operator does not adequately express just true, but that it merely captures an arbitrary proper subset of the just true sentences. Further, there is no prospect of extending the proposal in order to encompass a more reasonable subset of the just true sentences without presupposing that we have antecedent means to characterize the class of just true sentences. (shrink)

The goal of the paper is to discuss whether substructural non-contractive accounts of the truth-theoretic paradoxes can be philosophically motivated. First, I consider a number of explanations that have been offered to justify the failure of contraction and I argue that they are not entirely compelling. I then present a non-contractive theory of truth that I’ve proposed elsewhere. After looking at some of its formal properties, I suggest an explanation of the failure of structural contraction that is compatible with it.

According to Ole Hjortland, Timothy Williamson, Graham Priest, and others, anti-exceptionalism about logic is the view that logic “isn’t special”, but is continuous with the sciences. Logic is revisable, and its truths are neither analytic nor a priori. And logical theories are revised on the same grounds as scientific theories are. What isn’t special, we argue, is anti-exceptionalism about logic. Anti-exceptionalists disagree with one another regarding what logic and, indeed, anti-exceptionalism are, and they are at odds with naturalist philosophers of (...) logic, who may have seemed like natural allies. Moreover, those internal battles concern well-trodden philosophical issues, and there is no hint as to how they are to be resolved on broadly scientific grounds. We close by looking at three of the founders of logic who may have seemed like obvious enemies of anti-exceptionalism—Aristotle, Frege, and Carnap—and conclude that none of their positions is clearly at odds with at least some of the main themes of anti-exceptionalism. We submit that, at least at present, anti-exceptionalism is too vague or underspecified to characterize a coherent conception of logic, one that stands opposed to more traditional approaches. (shrink)

The idea of classical recapture has played a prominent role for non-classical logicians. In the specific case of non-classical theories of truth, although we know that it is not possible to retain classical logic for every statement involving the truth predicate, it is clear that for many such statements this is in principle feasible, and even desirable. What is not entirely obvious or well-known is how far this idea can be pushed. Can the non-classical theorist retain classical logic for every (...) non-paradoxical statement? If not, is she forced to settle for a very weak form of Classical Recapture, or are there robust versions of classical recapture available to her? These are the main questions that I will address in this paper. As a test case I will consider a paracomplete account of the truth-theoretic paradoxes and I will argue for two claims. First, that it is not possible to retain the law of excluded middle for every non-paradoxical statement. Secondly, that there are no robust versions of classical recapture available to the paracomplete logician. (shrink)

In Heck, Richard Heck presents variants on the familiar liar paradox, intended to reveal limitations of theories of transparent truth. But all existing theories of transparent truth can respond to Heck's variants in just the same way they respond to the liar. These new variants thus put no new pressure on theories of transparent truth.

Graham Priest defends the use of a nonmonotonic logic, LPm, in his analysis of reasoning in the face of true contradictions, such as those arising from the paradoxes of self-reference. In the course of defending this choice of logic in the face of the criticism that this logic is not truth preserving, Priest argued that requirement is too much to ask: since LPm is a nonmonotonic logic, it necessarily fails to preserve truth. In this article, I show that this assumption (...) is incorrect, and I explain why nonmonotonic logics can nonetheless be truth preserving. Finally, I diagnose Priest's error, to explain when nonmonotonic logics do indeed fail to preserve truth. (shrink)

The paradoxes of self reference have to be dealt with by anyone seeking to give a satisfactory account of the logic of truth, of properties, and even of sets of numbers. Unfortunately, there is no widespread agreement as to how to deal with these paradoxes. Some approaches block the paradoxical inferences by rejecting as invalid a move that classical logic counts as valid. In the recent literature, this deviant logic analysis of the paradoxes has been called into question.This disagreement motivates (...) a re-examination of the philosophy of formal logic and the status of logical truths and rules. In this paper I do some of this work, and I show that this gives us the means to defend the deviant approaches against such criticisms. As a result I hope to show that these analyses of the paradoxes are worthy of more serious consideration than they have so far received. (shrink)

Tato stať je zamýšlena jako úvod do spojité problematiky explikace pravdivosti a sémantic-kých paradoxů, které onu explikaci pronásledují. Tyto paradoxy jsou viděny jako prostředek ukázání, že nedostatečně promyšlená explikace našeho pojmu pravdivosti (významu pravdi-vostního predikátu) je explikací nevyhovující. Poté jsou předvedeny rysy obou klasických přístupů, Russellovy teorie typů a Tarského hierarchie metajazyků. Nakonec jsou podány zá-kladní ideje hlavních názorových škol, které přišly poté.

It is known that a semantically closed theory with description may well be trivial if the principles concerning denotation and descriptions are formulated in certain ways, even if the underlying logic is paraconsistent. This paper establishes the nontriviality of a semantically closed theory with a natural, but non-extensional, description operator.

This paper articulates Sylvan's theory of mathematical objects as non-existent, by improving (arguably) his treatment of the Characterisation Postulate. It then defends the theory against a number of natural objections, including one according to which the account is just platonism in disguise.

A motivation behind one kind of logical pluralism is the thought that there are different kinds of objects, and that reasoning about situations involving these different kinds requires different kinds of logics. Given this picture, a natural question arises: what kind of logical apparatus is appropriate for situations which concern more than one kind of objects, such as may arise, for example, when considering the interactions between the different kinds? The paper articulates an answer to this question, deploying the methodology (...) of Chunk and Permeate, developed in a different context by Brown and Priest (J Philos Log 33:379–388, 2004). (shrink)

The paper concerns interpretations of the paraconsistent logic LP which model theories properly containing all the sentences of first order arithmetic. The paper demonstrates the existence of such models and provides a complete taxonomy of the finite ones.

In ?Definability and the Structure of Logical Paradoxes? (Australasian Journal of Philosophy, this issue) Haixia Zhong takes issue with an account of the paradoxes of self-reference to be found in Beyond the Limits of Thought [Priest 1995. The point of this note is to explain why the critique does not succeed. The criterion for distinguishing between the set-theoretic and the semantic paradoxes offered does not get the division right; the semantic paradoxes are not given a uniform solution; no reason is (...) provided as to why the naïve denotation relation is ?indefinite? (other than that its definiteness leads to contradiction); and the account of the denotation relation given clearly misses the mark, even by consistent standards. (shrink)

This is a brief comment on Restall concerning my use of nonmontonic logics.

In this paper, I investigate the prospects for using the distinction between rejection and denial to resolve Saul Kripke’s puzzle about belief. One puzzle Kripke presents in A Puzzle About Belief poses what would have seemed a fairly straightforward question about the beliefs of the bilingual Pierre, who is disposed to sincerely and reflectively assent to the French sentence Londres est jolie, but not to the English sentence London is pretty, both of which he understands perfectly well. The question to (...) be answered is whether Pierre believes that London is pretty, and Kripke argues, of each answer, that it is unacceptable. On my proposal, either answer to the question is to be rejected, but neither answer is to be denied, using the resource of partially-defined predicates. After demonstrating how this serves as a solution to the puzzle, I illustrate some philosophical motivations—independent of Kripke’s puzzle —for adopting a view on which belief is a partially defined predicate. I conclude that there are decent prospects for the proposed response to Kripke’s puzzle. (shrink)

A number of theorists hold that literal, linguistic meaning is determined by the cognitive mechanism that underpins semantic competence. Borg and Larson and Segal defend a version of the view on which semantic competence is underpinned by the cognition of a truth-conditional semantic theory—a semantic theory which is true. Let us call this view the “cognitivist account of meaning”. In this paper, I discuss a surprisingly serious difficulty that the cognitivist account of meaning faces in light of the liar paradox. (...) I raise an argument to the effect that, in light of linguistic evidence concerning the liar paradox, the cognised semantic theory is inconsistent. This contradicts the cognitivist account. I consider a range of possible responses to the difficulty, raising problems for each. The liar paradox poses a serious difficulty to the cognitivist account of meaning, and it is unclear whether the difficulty can be resolved. (shrink)

The Analytic Tradition in Philosophy is an excellent successor to an excellent book : It is a fine an example of the necromantic style in the history of philosophy where the object of the exercise is to resurrect the mighty dead in order to get into an argument with them, either because we think them importantly right or instructively wrong. However what was a pardonable a simplification and a reasonable omission in the earlier book has now metamorphosed into a sin (...) of omission and an oversimplification in its successor—the book lacks is an adequate discussion of the Ramified Theory of Types and Russell’s reasons for adopting it. I conclude with a brief meditation on Open Question Arguments and the threats that they pose to some of the analyses proposed by Russell and Moore themselves. (shrink)

Davidson conjectured that suitably formulated Tarski-style theories of truth can “do duty” as theories of meaning for the spoken languages that humans naturally acquire. But this conjecture faces a pair of old objections that are, in my view, fatal when combined. Foster noted that given any theory of the sort Davidson envisioned, for a language L, there will be many equally true theories whose theorems pair endlessly many sentences of L with very different specifications of whether or not those sentences (...) are true. And if L includes words ‘true’, then for reasons stressed by Tarski, it’s hard to see how any truth theory for L could be correct. Moreover, each of these concerns amplifies the other. Appealing to possible worlds will not help with Foster’s Problem, for reasons that Chomsky discussed in the 1950s, and appealing to trivalent models of truth will not avoid concerns illustrated with Liar Sentences. (shrink)

In a recent article M. Colyvan has argued that Quinean forms of scientific realism are faced with an unexpected upshot. Realism concerning a given class of entities, along with this route to realism, can be vindicated by running an indispensability argument to the effect that the entities postulated by our best scientific theories exist. Colyvan observes that among our best scientific theories some are inconsistent, and so concludes that, by resorting to the very same argument, we may incur a commitment (...) to inconsistent entities. Colyvan's argument could be interpreted, and in part is presented, as a reductio of Quinean scientific realism; yet, Colyvan in the end manifests some willingness to bite the bullet, and provides some reasons why we shouldn't feel too uncomfortable with those entities. In this paper we wish to indicate a way out to the scientific realist, by arguing that no indispensability argument of the kind suggested by Colyvan is actually available. To begin with, in order to run such an indispensability argument we should be justified in believing that an inconsistent theory is true; yet, in so far as the logic we accept is a consistent one it is arguable that our epistemic predicament could not be possibly one in which we are justified in so believing. Moreover, also if our logic admitted true contradictions, as Dialetheism does, it is arguable that Colyvan's indispensability argument could not rest on a true premise. As we will try to show, dialetheists do not admit true contradictions for cheap: they do so just as a way out of paradox, namely whenever we are second-level ignorant as to the metaphysical possibility of evidence breaking the parity among two or more inconsistent claims; Colyvan's examples, however, are not of this nature. So, even under the generous assumption that Dialetheism is true, we will conclude that Colyvan's argument doesn't achieve its surprising conclusion. (shrink)

This paper portrays the later Wittgenstein’s conception of contradictions and his therapeutic approach to them. I will focus on and give relevance to the Lectures on the Foundations of Mathematics, plus the Remarks on the Foundations of Mathematics. First, I will explain why Wittgenstein’s attitude towards contradictions is rooted in: a rejection of the debate about realism and anti-realism in mathematics; and Wittgenstein’s endorsement of logical pluralism. Then, I will explain Wittgenstein’s therapeutic approach towards contradictions, and why it means that (...) a contradiction is not a problem for logic and mathematics. Rather, contradictions are problematic when we do not know what to infer from them. Once a meaning is established through a new rule of inference, the contradiction becomes a usable expression like many others in our inferential apparatus. Thus, the apparent problem is dissolved. Finally, I will take three examples of dissolved contradictions from Wittgenstein to clarify further his notion. I will conclude considering why his position on contradictions led him to clash with Alan Turing, and whether the latter was convinced by the Wittgensteinian proposal. (shrink)

This paper offers a discussion of metaphysical pluralism, alethic pluralism, and logical pluralism. According to the metaphysical pluralist, there are several ways of being. According to the alethic pluralist, there are several ways of being true, and according to the logical pluralist, there are several ways of being valid. Each of these three forms of pluralism will be considered on its own, but the ambition of the paper is to explore possible connections between them. My primary objective is to present (...) and develop a positive account according to which the different forms of pluralism are intimately related. I will proceed in two steps. First, I will investigate the connection between alethic pluralism and logical pluralism. I will argue that a certain version of alethic pluralism supports logical pluralism. Second, I will connect alethic pluralism and logical pluralism to metaphysical pluralism. I will suggest that the former two are at least partly founded on the latter. (shrink)

This is a discussion of different ways of working out the idea that the semantic paradoxes show that natural languages are somehow “inconsistent”. I take the workable form of the idea to be that there are expressions such that a necessary condition of understanding them is that one be inclined to accept inconsistent claims (an conception also suggested by Matti Eklund). I then distinguish “simple” from “complex” forms of such views. On a simple theory, such expressions are meaningless, while on (...) a complex theory they are not. I argue that complex theories are incompatible with truth conditional semantics and that simple theories are only coherent when the inconsistent claims are metalingusitic attributions of meaning. I close with a discussion of the version of the simple metalinguistic theory I have defended in “Understanding the Liar” and other papers. (shrink)

We take seriously the idea that paradoxes come in quantifiable degree by offering an exact measure of paradox. We consider three factors relevant to the degree of paradox, which are a function of the degree of belief in each of the individual propositions in the paradox set and the degree of belief in the set as a whole. We illustrate the proposal with a particular measure, and conclude the discussion with some critical remarks.

We give a complete characterization of Priest's Finite Inconsistent Arithmetics observing that his original putative characterization included arithmetics which cannot in fact be realized.

In a famous and controversial paper, B. H. Slater has argued against the possibility of paraconsistent logics. Our reply is centred on the distinction between two aspects of the meaning of a logical constant *c* in a given logic: its operational meaning, given by the operational rules for *c* in a cut-free sequent calculus for the logic at issue, and its global meaning, specified by the sequents containing *c* which can be proved in the same calculus. Subsequently, we use the (...) same strategy to counter Quine's meaning variance argument against deviant logics. In a nutshell, we claim that genuine rivalry between (similar) logics *L* and *L'* is possible whenever each constant in *L* has the same operational meaning as its counterpart in *L'* although differences in global meaning arise in at least one case. (shrink)

I discuss paradoxes of implication in the setting of a proof-conditional theory of meaning for logical constants. I argue that a proper logic of implication should be not only relevant, but also constructive and nonmonotonic. This leads me to select as a plausible candidate LL, a fragment of linear logic that differs from R in that it rejects both contraction and distribution.

It has been argued recently that dialetheist theories are unable to express the concept of naive validity. In this paper, we will show that LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}$$\end{document} can be non-trivially expanded with a naive validity predicate. The resulting theory, LPVal\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}^{\mathbf {Val}}$$\end{document} reaches this goal by adopting a weak self-referential procedure. We show that LPVal\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf (...) {LP}^{\mathbf {Val}}$$\end{document} is sound and complete with respect to the three-sided sequent calculus SLPVal\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {SLP}^{\mathbf {Val}}$$\end{document}. Moreover, LPVal\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}^{\mathbf {Val}}$$\end{document} can be safely expanded with a transparent truth predicate. We will also present an alternative theory LPVal∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}^{\mathbf {Val}^{*}}$$\end{document}, which includes a non-deterministic validity predicate. (shrink)

_T__S_ is a logic that has no valid inferences. But, could there be a logic without valid metainferences? We will introduce _T__S_ _ω_, a logic without metainferential validities. Notwithstanding, _T__S_ _ω_ is not as empty—i.e., uninformative—as it gets, because it has many antivalidities. We will later introduce the two-standard logic [_T__S_ _ω_, _S__T_ _ω_ ], a logic without validities and antivalidities. Nevertheless, [_T__S_ _ω_, _S__T_ _ω_ ] is still informative, because it has many contingencies. The three-standard logic [ \(\mathbf {TS}_{\omega (...) }, \mathbf {ST}_{\omega }, [{\overline {\emptyset }}{\emptyset }, {\emptyset } {\overline {\emptyset }}]\) ] that we will further introduce, has no validities, no antivalidities and also no contingencies whatsoever. We will also present two more validity-empty logics. The first one has no supersatisfiabilities, unsatisfabilities and antivalidities ∗. The second one has no invalidities nor non-valid-nor-invalid (meta)inferences. All these considerations justify thinking of logics as, at least, three-standard entities, corresponding, respectively, to what someone who takes that logic as correct, accepts, rejects and suspends judgement about, just because those things are, respectively, validities, antivalidities and contingencies of that logic. Finally, we will present some consequences of this setting for the monism/pluralism/nihilism debate, and show how nihilism and monism, on one hand, and nihilism and pluralism, on the other hand, may reconcile—at least according to how Gillian Russell understands nihilism, and provide some general reasons for adopting a multi-standard approach to logics. (shrink)

This paper argues that the traditional characterization of the notion of paradox — an apparently valid argument with apparently true premises and an apparently false conclusion — is too narrow; there are paradoxes that do not satisfy it. After discussing, and discarding, some alternatives, an outline of a new characterization of the notion of paradox is presented. A paradox is found to be an apparently valid argument such that, apparently, it does not present the kind of commitment to the conclusion (...) that should be implied by an acceptance of the truth of the premises and the validity of the argument. (shrink)

Can a present or future event bring about a past event? An answer to this question is demanded by many other interesting questions. Can anybody, even a god, do anything about what has already occurred? Should we plan for the past, as well as for the future? Can anybody precognise the future in a way quite different from normal prediction? Do the causal laws and the past jointly preclude free action? Does current physical theory entail a consistent version of backwards (...) causation? Recent articles on the problem of backwards causation have drawn attention to the importance of the principle of the fixity of the past: that the past is now fixed. It can be shown that the standard argument against backwards causation (the bilking experiment) simply builds in the assumption of past fixity. A fixed past deprives future events of past efficacy. This has naturally led to the speculation that by abandoning past fixity real power over the past may be possible.In this paper I show that in order to have an interesting thesis of backwards causation it is not enough simply to drop past fixity. More must go. In particular, to ensure what could be called future-to-past efficacy we must abandon two entrenched principles of permanence: the principle of permanent fixity, and the principle of permanent truth. The only alternative for backwards causal theorists is to embrace real contradictions in nature. (shrink)

En este artículo discutimos la tesis de Jc Beall según la cual no hay negación lógica. Evaluamos la solidez del argumento con el que defiende su tesis y presentamos dos razones para rechazar una de sus premisas: que la negación tiene que ser excluyente o exhaustiva. La primera razón involucra una presentación alternativa de las reglas de la negación en sistemas de secuentes diferentes al que Beall presupone. La segunda razón establece que la negación no tiene que ser excluyente o (...) exhaustiva. (shrink)

Tarski's Undefinability of Truth Theorem comes in two versions: that no consistent theory which interprets Robinson's Arithmetic (Q) can prove all instances of the T-Scheme and hence define truth; and that no such theory, if sound, can even express truth. In this note, I prove corresponding limitative results for validity. While Peano Arithmetic already has the resources to define a predicate expressing logical validity, as Jeff Ketland has recently pointed out (2012, Validity as a primitive. Analysis 72: 421-30), no theory (...) which interprets Q closed under the standard structural rules can define nor express validity, on pain of triviality. The results put pressure on the widespread view that there is an asymmetry between truth and validity, viz. that while the former cannot be defined within the language, the latter can. I argue that Vann McGee's and Hartry Field's arguments for the asymmetry view are problematic. (shrink)

Beall and Murzi :143–165, 2013) introduce an object-linguistic predicate for naïve validity, governed by intuitive principles that are inconsistent with the classical structural rules. As a consequence, they suggest that revisionary approaches to semantic paradox must be substructural. In response to Beall and Murzi, Field :1–19, 2017) has argued that naïve validity principles do not admit of a coherent reading and that, for this reason, a non-classical solution to the semantic paradoxes need not be substructural. The aim of this paper (...) is to respond to Field’s objections and to point to a coherent notion of validity which underwrites a coherent reading of Beall and Murzi’s principles: grounded validity. The notion, first introduced by Nicolai and Rossi, is a generalisation of Kripke’s notion of grounded truth, and yields an irreflexive logic. While we do not advocate the adoption of a substructural logic, we take the notion of naïve validity to be a legitimate semantic notion that points to genuine expressive limitations of fully structural revisionary approaches. (shrink)

Since Saul Kripke’s influential work in the 1970s, the revisionary approach to semantic paradox—the idea that semantic paradoxes must be solved by weakening classical logic—has been increasingly popular. In this paper, we present a new revenge argument to the effect that the main revisionary approaches breed new paradoxes that they are unable to block.

We cast doubts on the suggestion, recently made by Graham Priest, that glut theorists may express disagreement with the assertion of A by denying A. We show that, if denial is to serve as a means to express disagreement, it must be exclusive, in the sense of being correct only if what is denied is false only. Hence, it can’t be expressed in the glut theorist’s language, essentially for the same reasons why Boolean negation can’t be expressed in such a (...) language either. We then turn to an alternative proposal, recently defended by Beall (in Analysis 73(3):438–445, 2013; Rev Symb Log, 2014), for expressing truth and falsity only, and hence disagreement. According to this, the exclusive semantic status of A, that A is either true or false only, can be conveyed by adding to one’s theory a shrieking rule of the form A & ~A |- \bot, where \bot entails triviality. We argue, however, that the proposal doesn’t work either. The upshot is that glut theorists face a dilemma: they can either express denial, or disagreement, but not both. Along the way, we offer a bilateral logic of exclusive denial for glut theorists—an extension of the logic commonly called LP. (shrink)

A principle of continuity due to Leibniz has recently been revived by Graham Priest in arguing for an inconsistent account of motion. This paper argues that the Leibniz Continuity Condition has a reasonable interpretation in a different, though still inconsistent, class of dynamical systems. The account is then applied to the quantum mechanical description of the hydrogen atom.

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a (...) “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “ n -opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m ” ( m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid). (shrink)

Priest and others have presented their “most telling” argument for paraconsistent logic: that only paraconsistent logics allow non-trivial inconsistent theories. This is a very prevalent argument; occurring as it does in the work of many relevant and more generally paraconsistent logicians. However this argument can be shown to be unsuccessful. There is a crucial ambiguity in the notion of non-triviality. Disambiguated the most telling reason for paraconsistent logics is either question-begging or mistaken. This highlights an important confusion about the role (...) of logic in our development of our theories of the world. Does logic chart good reasoning or our commitments? We also consider another abductive argument for paraconsistent logics which also is shown to fail. (shrink)