In this paper we discuss the extent to which conjunction and disjunction can be rightfully regarded as such, in the context of infectious logics. Infectious logics are peculiar many-valued logics whose underlying algebra has an absorbing or infectious element, which is assigned to a compound formula whenever it is assigned to one of its components. To discuss these matters, we review the philosophical motivations for infectious logics due to Bochvar, Halldén, Fitting, Ferguson and Beall, noticing that none of them discusses (...) our main question. This is why we finally turn to the analysis of the truth-conditions for conjunction and disjunction in infectious logics, employing the framework of plurivalent logics, as discussed by Priest. In doing so, we arrive at the interesting conclusion that —in the context of infectious logics— conjunction is conjunction, whereas disjunction is not disjunction. (shrink)

In this paper, we introduce a new logic, which we call AM3. It is a connexive logic that has several interesting properties, among them being strongly connexive and validating the Converse Boethius Thesis. These two properties are rather characteristic of the difference between, on the one hand, Angell and McCall’s CC1 and, on the other, Wansing’s C. We will show that in other aspects, as well, AM3 combines what are, arguably, the strengths of both CC1 and C. It also allows (...) us an interesting look at how connexivity and the intuitionistic understanding of negation relate to each other. However, some problems remain, and we end by pointing to a large family of weaker logics that AM3 invites us to further explore. (shrink)

In this introduction, we offer an overview of main systems developed in the growing literature on connexive logic, and also point to a few topics that seem to be collecting attention of many of those interested in connexive logic. We will also make clear the context to which the papers in this special issue belong and contribute.

In this introduction to the special issue “40 years of FDE”, we offer an overview of the field and put the papers included in the special issue into perspective. More specifically, we first present various semantics and proof systems for FDE, and then survey some expansions of FDE by adding various operators starting with constants. We then turn to unary and binary connectives, which are classified in a systematic manner. First-order FDE is also briefly revisited, and we conclude by listing (...) some open problems for future research. (shrink)

This edited volume collects essays on the four-valued logic known as Belnap-Dunn logic, or first-degree entailment logic. It also looks at various formal systems closely related to it. These include the strong Kleene logic and the Logic of Paradox. Inside, readers will find reprints of seminal papers written by the fathers of the field: Nuel Belnap and Michael Dunn. In addition, the collection also features a well-known but previously unpublished manuscript of Dunn, an interview with Belnap, and a new essay (...) by Dunn. Besides the original, monumental papers, the book also includes research by leading scholars. They consider the extraordinary importance of Belnap-Dunn logic from several perspectives. They look at how, philosophically, it has served as a basic system of inconsistency-tolerant reasoning, as the core of underlying logics for theories based on dialetheism, and, more recently, for theories based on Buddhist philosophy. Coverage also explores its contributions to computer science, such as knowledge representation and information processing. This mix of seminal papers and insightful analysis by top scholars offers readers a comprehensive outlook on Belnap-Dunn logic and its related expansions, which have been agenda setting for the debate on philosophical logic as well as philosophy of logic. The book will also enhance further discussion on the philosophical issues related to nonclassical logics in general. (shrink)

One of the problems we face in many-valued logic is the difficulty of capturing the intuitive meaning of the connectives introduced through truth tables. At the same time, however, some logics have nice ways to capture the intended meaning of connectives easily, such as four-valued logic studied by Belnap and Dunn. Inspired by Dunn’s discovery, we first describe a mechanical procedure, in expansions of Belnap-Dunn logic, to obtain truth conditions in terms of the behavior of the Truth and the False, (...) which gives us intuitive readings of connectives, out of truth tables. Then, we revisit the notion of functional completeness, which is one of the key notions in many-valued logic, in view of Dunn’s idea. More concretely, we introduce a generalized notion of functional completeness which naturally arises in the spirit of Dunn’s idea, and prove some fundamental results corresponding to the classical results proved by Post and Słupecki. (shrink)

There is a relatively recent trend in treating negation as a modal operator. One such reason is that doing so provides a uniform semantics for the negations of a wide variety of logics and arguably speaks to a longstanding challenge of Quine put to non-classical logics. One might be tempted to draw the conclusion that negation is a modal operator, a claim Francesco Berto, 761–793, 2015) defends at length in a recent paper. According to one such modal account, the negation (...) of a sentence is true at a world x just in case all the worlds at which the sentence is true are incompatible with x. Incompatibility is taken to be the key notion in the account, and what minimal properties a negation has comes down to which minimal conditions incompatibility satisfies. Our aims in this paper are twofold. First, we wish to point out problems for the modal account that make us question its tenability on a fundamental level. Second, in its place we propose an alternative, non-modal, account of negation as a contradictory-forming operator that we argue is superior to, and more natural than, the modal account. (shrink)

Dialetheism is the metaphysical claim that there are true contradictions. And based on this view, Graham Priest and his collaborators have been suggesting solutions to a number of paradoxes. Those paradoxes include Russell’s paradox in naive set theory. For the purpose of dealing with this paradox, Priest is known to have argued against the presence of classical negation in the underlying logic of naive set theory. The aim of the present paper is to challenge this view by showing that there (...) is a way to handle classical negation. (shrink)

We observe that Jc Beall’s shrieking and shrugging strategy gives us an opportunity to reflect on the Australian plan for negation in FDE, a basic subclassical logic that is used in Beall’s argument for subclassical logics. An implication of our observation is applied to a recent defense of the Australian plan for negation by Francesco Berto and Greg Restall.

Change of logic is typically taken as requiring that the meanings of the connectives change too. As a result, it has been argued that legitimate rivalry between logics is under threat. This is, in a nutshell, the meaning‐variance argument, traditionally attributed to Quine. In this paper, we present a semantic framework that allows us to resist the meaning‐variance claim for an important class of systems: classical logic, the logic of paradox and strong Kleene logic. The major feature of the semantics (...) is that the connectives have the same meanings in these systems, so that the meaning‐variance argument is straightforwardly blocked. We discuss the effects of this semantics for two uses of the argument of meaning variance, and also consider its impact on related issues. (shrink)

Sören Halldén’s logic of nonsense is one of the most well-known many-valued logics available in the literature. In this paper, we discuss Peter Woodruff’s as yet rather unexplored attempt to advance a version of such a logic built on the top of a constructive logical basis. We start by recalling the basics of Woodruff’s system and by bringing to light some of its notable features. We then go on to elaborate on some of the difficulties attached to it; on our (...) way to offer a possible solution to such difficulties, we discuss the relation between Woodruff’s system and two-dimensional semantics for many-valued logics, as developed by Hans Herzberger. (shrink)

In a recent article, Mario Günther presented a conditional that is claimed to be connexive. The aim of this short discussion note is to show that Günther’s claim is not without problems.

We investigate the notion of classical negation from a non-classical perspective. In particular, one aim is to determine what classical negation amounts to in a paracomplete and paraconsistent four-valued setting. We first give a general semantic characterization of classical negation and then consider an axiomatic expansion BD+ of four-valued Belnap–Dunn logic by classical negation. We show the expansion complete and maximal. Finally, we compare BD+ to some related systems found in the literature, specifically a four-valued modal logic of Béziau and (...) the logic of classical implication and a paraconsistent de Morgan negation of Zaitsev. (shrink)

We outline the rather complicated history of attempts at axiomatizing Jaśkowski’s discussive logic $$\mathbf {D_2}$$ D2 and show that some clarity can be had by paying close attention to the language we work with. We then examine the problem of axiomatizing $$\mathbf {D_2}$$ D2 in languages involving discussive conjunctions. Specifically, we show that recent attempts by Ciuciura are mistaken. Finally, we present an axiomatization of $$\mathbf {D_2}$$ D2 in the language Jaśkowski suggested in his second paper on discussive logic, by (...) following a remark of da Costa and Dubikajtis. We also deal with an interesting variant of $$\mathbf {D_2}$$ D2, introduced by Ciuciura, in which negation is also taken to be discussive. (shrink)

In this paper, we generalize the ordered-pair semantics advanced by Matthew Clemens for the Logic of Paradox to n-tuple semantics, for each fixed n. Moreover, we show that the resulting semantics can accommodate not only LP, but also classical logic as well as strong Kleene logic depending on the set of designated values that one chooses. Building on the technical observations, we offer intuitively plausible readings for the semantics, and we also discuss some weaknesses of the original intuitive reading advanced (...) by Clemens. (shrink)

Intuitionism can be seen as a verificationism restricted to mathematical discourse. An attempt to generalize intuitionism to empirical discourse presents various challenges. One of those concerns the logical and semantical behavior of what has been called ' empirical negation'. An extension of intuitionistic logic with empirical negation was given by Michael De and a labelled tableaux system was there shown sound and complete. However, a Hilbert-style axiom system that is sound and complete was missing. In this paper we provide the (...) missing axiom system which is shown sound and complete with respect to its intended semantics. Along the way we consider some further applications of empirical negation. (shrink)

In his “The simple argument for subclassical logic,” Jc Beall advances an argument that led him to take FDE as the one true logic (the latter point is explicitly made clear in his “FDE as the One True Logic”). The aim of this article is to point out that if we follow Beall’s line of reasoning for endorsing FDE, there are at least two additional reasons to consider that FDE is too strong for Beall’s purposes. In fact, we claim that (...) Beall should consider another weaker subclassical logic as the logic adequate for his project. To this end, we first briefly present Beall’s argument for FDE. Then, we discuss two specific topics that seem to motivate us to weaken FDE. We then introduce a subsystem that will enjoy all the benefits of Beall’s suggestion. (shrink)

The present note revisits the joint work of Leonard Goddard and Richard Routley on significance logics with the aim of shedding new light on their understanding by studying them under the lens of recent semantic developments, such as the plurivalent semantics developed by Graham Priest. These semantics allow sentences to receive one, more than one, or no truth-value at all from a given carrier set. Since nonsignificant sentences are taken to be neither true nor false, i.e. truth-value gaps, in this (...) essay we show that with the aid of plurivalent semantics it is possible to straightforwardly instantiate Goddard and Routley’s understanding of how the connectives should work within significance logics. (shrink)

Over the past ten years, the community researching connexive logics is rapidly growing and a number of papers have been published. However, when it comes to the terminology used in connexive logic, it seems to be not without problems. In this introduction, we aim at making a contribution towards both unifying and reducing the terminology. We hope that this can help making it easier to survey and access the field from outside the community of connexive logicians. Along the way, we (...) will make clear the context to which the papers in this special issue on Frontiers of Connexive Logic belong and contribute. (shrink)

A world is trivial if it makes every proposition true all at once. Such a world is impossible, an absurdity. Our world, we hope, is not an absurdity. It is important, nevertheless, for semantic and metaphysical theories that we be able to reason cogently about absurdities—if only to see that they are absurd. In this note we describe methods for ‘observing’ absurd objects like the trivial world without falling in to incoherence, using some basic techniques from modal logic. The goal (...) is to begin to locate the trivial world’s relative position in modal space; the outcome is that the less we assume about relative possibility, the more detail we can discern at the edge of reason. (shrink)

In one of his papers, JC Beall advanced a new and interesting interpretation of Weak Kleene logic, in terms of on-topic/off-topic. In brief, Beall suggests to read the third value as off-topic, whereas the two classical values are read as true and on-topic and false and on-topic. Building on Beall’s new interpretation, the aim of this paper is threefold. First, we discuss two motivations to enrich Beall’s interpretation, and offer an alternative semantic framework that reflects our motivations. Second, by making (...) use of our new framework, we will offer a new interpretation of the logic of Catuskoti which combines Beall’s proposal of having FDE as the correct logic with the on-topic/off-topic divide. Finally, we will offer a general result that will allow us to make sense of a family of infectious logics in terms of Beall’s on-topic/off-topic reading. (shrink)

The present note offers a proof that systems developed by Majkić are actually extensions of intuitionistic logic, and therefore not paraconsistent.

This note offers a non-deterministic semantics for mbC1, introduced by Janusz Ciuciura, and establishes soundness and completeness results with respect to the Hilbert-style proof system. Moreover, based on the new semantics, we briefly discuss an unexplored variant of mbC1 which has a contra-classical flavor.

Metabolic dysfunction is a ubiquitous underlying feature of many neurological conditions including acute traumatic brain injuries and chronic neurodegenerative conditions. A central problem in neurological patients, in particular those with traumatic brain injuries, is an impairment in the utilization of glucose, which is the predominant metabolic substrate in a normally functioning brain. In such patients, alternative substrates including ketone bodies and lactate become important metabolic candidates for maintaining brain function. While the potential neuroprotective benefits of ketosis have been recognized for (...) up to almost a century, the majority of work has focused on the use of ketogenic diets to induce such a state, which is inappropriate in cases of acute disease due to the prolonged periods of time (i.e., weeks to months) required for the effects of a ketogenic diet to be seen. The following review seeks to explore the neuroprotective effects of exogenous ketone and lactate preparations, which have more recently become commercially available and are able to induce a deep ketogenic response in a fraction of the time. The rapid response of exogenous preparations makes their use as a therapeutic adjunct more feasible from a clinical perspective in both acute and chronic neurological conditions. Potentially, their ability to globally moderate long-term, occult brain dysfunction may also be relevant in reducing lifetime risks of certain neurodegenerative conditions. In particular, this review explores the association between traumatic brain injury and contusion-related dementia, assessing metabolic parallels and highlighting the potential role of exogenous ketone and lactate therapies. (shrink)

One of the oldest systems of paraconsistent logic is the set of so-called C-systems of Newton da Costa, and this has been generalized into a family of systems now known as logics of formal inconsistencies by Walter Carnielli, Marcelo Coniglio and João Marcos. The characteristic notion in these systems is the so-called consistency operator which, roughly speaking, indicates how gluts are behaving. One natural question then is to ask if we can let not only gluts but also gaps be around (...) and generalize the notion of consistency into classicality. This is already considered by Andréa Loparić and da Costa in the style of C-systems. The aim of this paper is to develop a family of systems that generalizes the system of Loparić and da Costa which may be called logics of formal classicality. (shrink)

In [1] Béziau developed the paraconsistent logic Z, which is definitionally equivalent to the modal logic S5, and gave an axiomatization of the logic Z: the system HZ. In the present paper, we prove that some axioms of HZ are not independent and then propose another axiomatization of Z. We also discuss a new perspective on the relation between S5 and classical propositional logic with the help of the new axiomatization of Z. Then we conclude the paper by making a (...) remark on the paraconsistency of HZ. (shrink)

In one of their papers, Michael De and Hitoshi Omori observed that the notion of classical negation is not uniquely determined in the context of so-called Belnap-Dunn logic, and in fact there are 16 unary operations that qualify to be called classical negation. These varieties are due to different falsity conditions one may assume for classical negation. The aim of this paper is to observe that there is an interesting way to make sense of classical negation independent of falsity (...) conditions. We discuss two equivalent semantics, and offer a Hilbert-style system that is sound and complete with respect to the semantics. (shrink)

Subintuitionistic (propositional) logics are those in a standard intuitionistic language that result by weakening the frame conditions of the Kripke semantics for intuitionistic logic. In this paper we consider two negation expansions of subintuitionistic logic, one by classical negation and the other by what has been dubbed “empirical” negation. We provide an axiomatization of each expansion and show them sound and strongly complete. We conclude with some final remarks, including avenues for future research.

In [Waragai & Shidori, 2007], a system of paraconsistent logic called PCL1, which takes a similar approach to that of da Costa, is proposed. The present paper gives further results on this system and its related systems. Those results include the concrete condition to enrich the system PCL1 with the classical negation, a comparison of the concrete notion of “behaving classically” given by da Costa and by Waragai and Shidori, and a characterisation of the notion of “behaving classically” given by (...) Waragai and Shidori. (shrink)

Connectives such as Tonk have posed a significant challenge to the inferentialist. It has been recently argued that the classical semanticist faces an analogous problem due to the definability of “nasty connectives” under non-standard interpretations of the classical propositional vocabulary. In this paper, we defend the classical semanticist from this alleged problem.

Some philosophers have thought that life could only be meaningful if there is no God. For Sartre and Nagel, for example, a God of the traditional classical theistic sort would constrain our powers of self-creative autonomy in ways that would severely detract from the meaning of our lives, possibly even evacuate our lives of all meaning. Some philosophers, by contrast, have thought that life could only be meaningful if there is a God. God and the Meanings of Life is interested (...) in exploring the truth in both these schools of thought, seeking to discover what God could and couldn't do to make life meaningful (as well as what he would and wouldn't do). Mawson espouses a version of the 'amalgam' or 'pluralism' thesis about the issue of life's meaning – in essence, that there are a number of different legitimate meanings of 'meaning' (and indeed 'life') in the question of life's meaning. According to Mawson, God, were he to exist, would help make life meaningful in some of these senses and hinder in some others. He argues that whilst there could be meaning in a Godless universe, there could be other sorts of meaning in a Godly one and that these would be deeper. (shrink)

T. A. Cavanaugh's Hippocrates' Oath and Asclepius' Snake: The Birth of the Medical Profession articulates the Oath as establishing the medical profession's unique internal medical ethic - in its most basic and least controvertible form, this ethic mandates that physicians help and not harm the sick. Relying on Greek myth, drama, and medical experience (e.g., homeopathy), the book shows how this medical ethic arose from reflection on the most vexing medical-ethical problem -- injury caused by a physician -- and argues (...) that deliberate iatrogenic harm, especially the harm of a doctor choosing to kill (physician assisted suicide, euthanasia, abortion, and involvement in capital punishment), amounts to an abandonment of medicine as an exclusively therapeutic profession. The book argues that medicine as a profession necessarily involves stating before others what one stands for: the good one seeks and the bad one seeks to avoid on behalf of the sick, and rejects the view that medicine is purely a technique lacking its own unique internal ethic. It concludes noting that medical promising (as found in the White Coat Ceremony through which U. S. medical students matriculate) implicates medical autonomy which in turn merits respect, including honoring professional conscientious objections. (shrink)

This text is designed to help students construct and evaluate arguments.