Paraconsistency and its Philosophical Interpretations

Many authors have considered that the notions of paraconsistency and dialetheism are intrinsically connected, in many cases, to the extent of confusing both phenomena. However, paraconsistency is a formal feature of some logics that consists in invalidating the rule of explosion, whereas dialetheism is a semantical/ontological position consisting in accepting true contradictions. In this paper, we argue against this connection and show that it is perfectly possible to adopt a paraconsistent logic and reject dialetheism, and, moreover, that there are examples of non-paraconsistent logics that can be interpreted in a dialetheic way

Logic as a Puzzle-Solving Activity

Some authors have recently argued in favor of anti-exceptionalism about logic. The general idea is that logic is not different from the other sciences, and its principles are as revisable as scientific principles. This paper has three sections. In section 1, I discuss the meaning of anti-exceptionalism and its place in contemporary logic. In section 2, I analyze some recent developments on this topic by Williamson (2017) and Hjortland (2017), which will motivate my view. In section 3, I propose a puzzle-solving perspective on logical practice. According to my view, there is a common methodology, in which scientists may use non-classical in order to solve some specific puzzles, but classical logic stays in a privileged position, as a common language and as a general theory of reasoning. This role cannot be fulfilled by other logics, and therefore the comparison between classical and non-classical logic is not like a regular comparison between competing hypotheses in science. The methodology of logical practice is therefore not abductive, at least in many important cases. Classical logic is not the “best available theory”, but the fundamental piece of our scientific methodology. My position is still anti-exceptionalist: logic is like any other science, or at least like any other science which can be characterized by a puzzle-solving methodology

Against Classical Paraconsistent Metatheory

There was a time when 'logic' just meant classical logic. The climate is slowly changing and non-classical logic cannot be dismissed off-hand. However, a metatheory used to study the properties of non-classical logic is often classical. In this paper, we will argue that this practice of relying on classical metatheories is problematic. In particular, we will show that it is a bad practice because the metatheory that is used to study a non-classical logic often rules out the very logic it is designed to study

There is no Logical Negation: True, False, Both, and Neither

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

There is no Logical Negation: True, False, Both, and Neither

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

On Williamson's new Quinean argument against nonclassical logic

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

Rethinking inconsistent mathematics

This dissertation has two main goals. The first is to provide a practice-based analysis of the field of inconsistent mathematics: what motivates it? what role does logic have in it? what distinguishes it from classical mathematics? is it alternative or revolutionary? The second goal is to introduce and defend a new conception of inconsistent mathematics - queer incomaths - as a particularly effective answer to feminist critiques of classical logic and mathematics. This sets the stage for a genuine revolution in mathematics, insofar as it suggests the need for a shift in mainstream attitudes about the rolee of logic and ethics in the practice of mathematics

Dialetheism in Action: A New Strategy for Solving the Equal Validity Paradox

This paper starts from the Equal Validity Paradox, a paradoxical argument connected to the so-called phenomenon of faultless disagreement. It is argued that there are at least six strategies for solving the paradox. After presenting the first five strategies and their main problems, the paper focuses on the sixth strategy which rejects the assumption that every proposition cannot be both true a false. Dialetheism is the natural candidate for developing strategy six. After presenting strategy six in detail, we formulate a normative problem for the dialetheist and offer a tentative solution to it. We then elaborate further considerations connecting strategy six to pluralism about truth and logic. Even if strategy six is a hard path to take, its scrutiny highlights some important points on truth, logic and the norms for acceptance and rejection

Higher-Order Logical Pluralism as Metaphysics

Higher-order metaphysics is in full swing. One of its principle aims is to show that higher-order logic can be our foundational metaphysical theory. A foundational metaphysical theory would be a simple, powerful, systematic theory which would ground all of our metaphysical theories from modality, to grounding, to essence, and so on. A satisfactory account of its epistemology would in turn yield a satisfactory epistemology of these theories. And it would function as the final court of appeals for metaphysical questions. It would play the role for our metaphysical community that ZFC plays for the mathematical community. I think there is much promise in this project. There is clear value in having a shared foundational theory to which metaphysicians can appeal. And there is reason to think that higher-order logic can play this role. After all, it has long been known that one can do math in higher-order logic. And there is growing reason to think that one can do metaphysics in higher-order logic in much the same way. However, most of the research approaches higher-order logic from a monist perspective, according to which there is 'one true' higher-order logic. And in the midst of the enthusiasm, metaphysicians seem to have overlooked that this approach leaves the program susceptible to epistemological problems that plague monism about other areas, like set theory. The most significant of these is the Benacerraf Problem. This is the problem of explaining the reliability of our higher-order-logical beliefs. The problem is sufficiently serious that, in the set-theoretic case, it has led to a reconception of the foundations of mathematics, known as pluralism. In this dissertation I investigate a pluralist approach to higher-order metaphysics. The basic idea is that any higher-order logic which can play the role of our foundational metaphysical theory correctly describes the metaphysical structure of the world, in much the way that the set-theoretic pluralist maintains that any set theory which can play the role of our foundational mathematical theory is true of a mind-independent platonic universe of sets. I outline my view about what it takes for a higher-order logic to play this role, what it means for such a logic to correctly describe the metaphysical structure of the world, and how it is that different higher-order logics which seem to disagree with each other can meet both of these conditions. I conclude that higher-order logical pluralism is the most tenable version of the higher-order logic as metaphysics program. Higher-order logical pluralism constitutes a radical departure from conventional wisdom, requiring a significant reconception of the nature of validity, modality, and metaphysics in general. It renders moot some of the most central questions in these domains, such as: Is the law of excluded middle valid? Is it the case that necessarily everything is necessarily something? Is the grounding relation transitive? On this picture, these questions no longer have objective answers. They become like the question of whether the Continuum Hypothesis is true, according to the set-theoretic pluralist. The only significant question in the neighborhood of the aforementioned questions is: which metaphysical principles are best suited to the task at hand