In this paper we introduce a novel way of building arithmetics whose background logic is R. The purpose of doing this is to point in the direction of a novel family of systems that could be candidates for being the infamous R#1/2 that Meyer suggested we look for.

Weak enough relevant logics are often closed under depth substitutions. To determine the breadth of logics with this feature, we show there is a largest sublogic of R closed under depth substitutions and that this logic can be recursively axiomatized.

In a recent paper , Urbaniak and Cieśliński describe an analogue of the Yablo Paradox, in the domain of formal provability. Just as the infinite sequence of Yablo sentences inherit the paradoxical behavior of the liar sentence, an infinite sequence of sentences can be constructed that inherit the distinctive behavior of the Gödel sentence. This phenomenon—the transfer of the properties of self-referential sentences of formal mathematics to their “unwindings” into infinite sequences of sentences—suggests a number of interesting logical questions. The (...) purpose of this paper is to give a precise statement of a conjecture from Cieśliński and Urbaniak regarding the unwinding of the Rosser sentence, and to demonstrate that this precise statement is false. We begin with some preliminary motivation, introduce the conjecture against the background of some related results, and finally, in the last section, move on to the proof, which adapts a method used by Solovay and Guaspari. (shrink)

Categorial logic, as its name suggests, applies the techniques and machinery of category theory to topics traditionally classified as part of logic. We claim that these tools deserve attention from a greater range of philosophers than just the mathematical logicians. We support this claim with an example. In this paper we show how one particular tool from categorial logic---hyperdoctrines---suggests interesting metaphysics. Hyperdoctrines can provide semantics for quantified languages, but this account of quantification suggests a metaphysical picture quite different from the (...) one suggested by standard model-theoretic semantics. (shrink)

In this paper, I present a class of ‘structural’ abstraction principles, and describe how they are suggested by some features of Cantor's and Dedekind's approach to abstraction. Structural abstraction is a promising source of mathematically tractable new axioms for the neo-logicist. I illustrate this by showing, first, how a theorem of Shelah gives a sufficient condition for consistency in the structural setting, solving what neo-logicists call the ‘bad company’ problem for structural abstraction. Second, I show how, in the structural setting, (...) we can measure the logical strength of abstraction principles using categories of interpretations between theories. (shrink)

The paradox that appears under Burali-Forti’s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be—absurdly—an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of Burali-Forti is first and foremost a problem about concept formation by abstraction, not about sets. We contend, furthermore, that some (...) hundred years after its discovery the paradox is still without any fully satisfactory resolution. A survey of the current literature reveals one key assumption of the paradox that has gone unquestioned, namely the assumption that ordinals are objects. Taking the lead from Russell’s no class theory, we interpret talk of ordinals as an efficient way of conveying higher-order logical truths. The resulting theory of ordinals is formally adequate to standard intuitions about ordinals, expresses a conception of ordinal number capable of resolving Burali-Forti’s paradox, and offers a novel contribution to the longstanding program of reducing mathematics to higher-order logic. (shrink)

This paper presents an overview of Carnap, a free and open framework for the development of formal reasoning applications. Carnap’s design emphasizes flexibility, extensibility, and rapid prototyping. Carnap-based applications are written in Haskell, but can be compiled to JavaScript to run in standard web browsers. This combination of features makes Carnap ideally suited for educational applications, where ease-of-use is crucial for students and adaptability to different teaching strategies and classroom needs is crucial for instructors. The paper describes Carnap’s implementation, along (...) with its current and projected pedagogical applications. (shrink)

Russell’s paradox is purely logical in the following sense: a contradiction can be formally deduced from the proposition that there is a set of all non-self-membered sets, in pure first-order logic—the first-order logical form of this proposition is inconsistent. This explains why Russell’s paradox is portable—why versions of the paradox arise in contexts unrelated to set theory, from propositions with the same logical form as the claim that there is a set of all non-self-membered sets. Burali-Forti’s paradox, like Russell’s paradox, (...) is portable. I offer the following explanation for this fact: Burali-Forti’s paradox, like Russell’s, is purely logical. Concretely, I show that if we enrich the language \ of first-order logic with a well-foundedness quantifier W and adopt certain minimal inference rules for this quantifier, then a contradiction can be formally deduced from the proposition that there is a greatest ordinal. Moreover, a proposition with the same logical form as the claim that there is a greatest ordinal can be found at the heart of several other paradoxes that resemble Burali-Forti’s. The reductio of Burali-Forti can be repeated verbatim to establish the inconsistency of these other propositions. Hence, the portability of the Burali-Forti’s paradox is explained in the same way as the portability of Russell’s: both paradoxes involve an inconsistent logical form—Russell’s involves an inconsistent form expressible in \ and Burali-Forti’s involves an inconsistent form expressible in \. (shrink)

Frege’s Basic Law V and its successor, Boolos’s New V, are axioms postulating abstraction operators: mappings from the power set of the domain into the domain. Basic Law V proved inconsistent. New V, however, naturally interprets large parts of second-order ZFC via a construction discovered by Boolos in 1989. This paper situates these classic findings about abstraction operators within the general theory of F-algebras and coalgebras. In particular, we show how Boolos’s construction amounts to identifying an initial F-algebra in a (...) certain category, we identify a natural coalgebraic dual to Boolos’s axiom which naturally interprets large parts of Aczel’s non-well-founded set theory via the construction of a certain terminal F-coalgebra, and we suggest a coalgebraic way forward for an abstraction-theoretic axiomatization of the real numbers. (shrink)

It’s well known that it’s possible to extract, from Frege’s Grudgesetze, an interpretation of second-order Peano Arithmetic in the theory  HP2, whose sole axiom is Hume’s principle. What’s less well known is that, in Die Grundlagen Der Arithmetic §82–83 Boolos (2011), George Boolos provided a converse interpretation of HP2 in PA2 . Boolos’ interpretation can be used to show that the Frege’s construction allows for any model of PA2 to be recovered from some model of HP2. So the space (...) of possible arithmetical universes is precisely characterized by Hume’s principle. -/- In this paper, I show that a large class of second-order theories admit characterization by an abstraction principle in this sense. The proof makes use of structural abstraction principles, a class of abstraction principles of considerable intrinsic interest, and categories of interpretations in the sense of Visser (2003). (shrink)

The norm of assertion, to be in force, is a social norm. What is the content of our social norm of assertion? Various linguistic arguments purport to show that to assert is to represent oneself as knowing. But to represent oneself as knowing does not entail that assertion is governed by a knowledge norm. At best these linguistic arguments provide indirect support for a knowledge norm. Furthermore, there are alternative, non-normative explanations for the linguistic data (as in recent work from (...) Van Elswyk). Direct arguments would rely on normative judgments about the permissibility or impermissibility of assertions with and without the speaker knowing the content asserted. John Turri's work experimental results purport to show that our norm of assertion is factive, and probably knowledge. But as a number of recent experimentalists (Kneer, Reuter and Broessel, Marsili and Wiegmann) have shown, Turri's results rely on the problemmatic use of 'should' as in 'Maria should assert that she owns a 1990 watch.' Correcting for this, these experimentalists provide strong evidence that our norm of assertion is not factive. The standard reply to evidence like this from the assertion of unlucky falsehoods is the excuse maneuver. But Marsili and Wiegmann, following Turri's protocol for testing for excuse validation, show that participants who judge that assertions unlucky falsehoods are permissible are not guilty of excuse validation. Given our current understanding of the experimental evidence, knowledge is not our norm of assertion. (shrink)

The problem that Tallis attempts to address in Freedom: An Impossible Reality (2021) is that science appears to describe the entire world deterministically and that this seems to leave no room for free will. In the face of this threat, Tallis defends the existence of free will by arguing that science does not explain our intentional awareness of the world; and it is our intentional awareness that makes both science and free will possible. Against Tallis, it is here argued that (...) his argument is vulnerable to two criticisms. Firstly, his characterisation of science as apparently deterministic is inaccurate. Secondly, he has not solved the problem he has set himself but rather recast it, so that his conclusion leaves us having to account for free will, not in a deterministic universe, but either as a product of chance or as a miracle. It is here suggested that when we set aside the illusory threat of scientific determinism, we also set aside the temptation of free will (as its spurious answer). That done, we may better focus upon agent’s freedom of action – as discussed by philosophers such as Hobbes, Locke and Hume – the rational capability of an agent to act upon their wishes, given the constraints under which they find themselves. (shrink)

This book is a unique contribution to the philosophy of religion. It offers a comprehensive discussion of one of the most famous arguments for the existence of God: the ontological argument. The author provides and analyses a critical taxonomy of those versions of the argument that have been advanced in recent philosophical literature, as well as of those historically important versions found in the work of St Anselm, Descartes, Leibniz, Hegel and others. A central thesis of the book is that (...) ontological arguments have no value in the debate between theists and atheists. There is a detailed review of the literature on the topic (separated from the main body of the text) and a very substantial bibliography, making this volume an indispensable resource for philosophers of religion and others interested in religious studies. (shrink)

This article examines the distinction that Russell drew between his work as a philosopher and his work as a journalist. It explains why, when warning against the threat posed by a nuclear arms race, Russell thought it better to write as a journalist (speaking on behalf of common sense) rather than as a philosopher. It is argued that to put aside philosophy in favour of common sense is, in this instance, a mistake.

What is the biological function of perception? I hold perception, especially visual perception in humans, has the biological function of accurately representing the environment. Tyler Burge argues this cannot be so in Origins of Objectivity (Oxford, 2010), for accuracy is a semantical relationship and not, as such, a practical matter. Burge also provides a supporting example. I rebut the argument and the example. Accuracy is sometimes also a practical matter if accuracy partly explains how perception contributes to survival and reproduction.

In American universities, even Asian Philosophy is still often taught following methods adapted from European universities of the nineteenth century. Whether or not this approach is well-suited to philosophy as it was conceived in that era, it is inadequate if the aim is to develop a deep appreciation of Japanese philosophy. To limit what we consider Japanese philosophy to only what bears a distinct resemblance to academic Western philosophy, and accordingly to approach Japanese philosophy purely theoretically, is to risk missing (...) the greater part. Much of Japanese philosophy is applied philosophy, or in other words, what Pierre Hadot calls a “way of life,” and to appropriate it meaningfully requires practice rather than mere intellectual study alone. Thus, I contend that a proper means for introducing Western students is a more holistic method grounded in practicing traditional arts, such as composing haiku. I argue that the seventeenth century poet Matsuo Bashō can serve as a valuable resource in this process. I conclude with a description of the methods that I use in my efforts at teaching Japanese philosophy to undergraduate university students in South Texas. (shrink)

This is the first introductory textbook on non-classical propositional logics.

In this book, Graham Oppy examines arguments for and against the existence of God. He shows that none of these arguments is powerful enough to change the minds of reasonable participants in debates on the question of the existence of God. His conclusion is supported by detailed analyses of the arguments as well as by the development of a theory about the purpose of arguments and the criteria that should be used in judging whether or not arguments are successful. (...) Oppy discusses the work of a wide array of philosophers, including Anselm, Aquinas, Descartes, Locke, Leibniz, Kant, Hume and, more recently, Plantinga, Dembski, White, Dawkins, Bergman, Gale and Pruss. (shrink)

Genocide in Rwanda, multiple murder at Denver or Dunblane, the gruesome activities of serial killers - what makes these great evils, and why do they occur? In addressing such questions this book, unusually, interconnects contemporary moral philosophy with recent work in New Testament scholarship. The conclusions to emerge are surprising. Gordon Graham argues that the inability of modernist thought to account satisfactorily for evil and its occurrence should not lead us to embrace an eclectic postmodernism, but to take seriously (...) some unfashionable pre-modern conceptions - Satan, demonic possession, spiritual powers, cosmic battles. Precisely because it strives to observe the high standards of clarity and rigour that are the hallmarks of philosophy in the analytical tradition, the book makes a powerful case for the rejection of humanism and naturalism, and for explaining the moral obligation to struggle against evil by reference to the New Testament's cosmic narrative. (shrink)

Māori (Indigenous people of New Zealand (NZ)) suffer food insecurity disproportionately in New Zealand. Some research suggests that Māori value mātauranga Māori (traditional Māori knowledge) when it comes to the collection, preparation and eating of kai (food). This study explores the connections between mātauranga Māori and kai in regional NZ schools for potential pathways to impact food security for children. Semi-structured interviews were undertaken with five primary school principals in the Hawke’s Bay region. Principals were purposively selected on commitments to (...) proactively incorporating mātauranga Māori into their school environment. Reflective thematic analysis was used to analyse the data. Three main themes were identified: teaching and learning around mātauranga Māori and kai; environmental sustainability and the sustainability of kai initiatives; and school values. A strong emphasis was placed on the learning of traditional Māori values around kai and the whenua (land) and the sustainability of teaching programs. School values were invariably bound in Te Ao Māori (a Māori worldview) however, schools felt challenged in aligning their values and their mātauranga Māori-bound teaching practices with the current food provision programme in their schools (Ka Ora, Ka Ako). The programme’s strict nutritional guidelines appeared to challenge traditional Māori approaches to kai. Schools remain an ideal environment for the incorporation of mātauranga Māori to support food security and food education, and future work should explore a demonstration project that incorporates the knowledge gained in this study and ways to integrate mātauranga Māori into Ka Ora, Ka Ako. (shrink)

This volume presents a selection of the philosophical essays which Richard Rorty wrote during the first decade of his career, and complements four previous volumes of his papers published by Cambridge University Press. In this long neglected body of work, which many leading philosophers still consider to be his best, Rorty develops his views on the nature and scope of philosophy in a manner which supplements and elucidates his definitive statement on these matters in Philosophy and the Mirror of Nature. (...) He also develops his groundbreaking version of eliminative materialism, a label first coined to describe his position, and sets out original views on various central topics in the philosophy of language, concerning private language, indeterminacy, and verificationalism. A substantial introduction examines Rorty's philosophical development from 1961 to 1972. The volume completes our understanding of Rorty's intellectual trajectory and offers lucid statements of positions which retain their relevance to current debates. (shrink)

This revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, and (...) their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area. (shrink)

This book is an exploration of philosophical questions about infinity. Graham Oppy examines how the infinite lurks everywhere, both in science and in our ordinary thoughts about the world. He also analyses the many puzzles and paradoxes that follow in the train of the infinite. Even simple notions, such as counting, adding and maximising present serious difficulties. Other topics examined include the nature of space and time, infinities in physical science, infinities in theories of probability and decision, the nature (...) of part/whole relations, mathematical theories of the infinite, and infinite regression and principles of sufficient reason. (shrink)

Based on Michel Foucault's 1978 and 1979 lectures at the Collège de France on governmental rationalities and his 1977 interview regarding his work on imprisonment, this volume is the long-awaited sequel to Power/Knowledge.

This a general account of the school of Mo-tzu, its social basis as a movement of craftsmen, its isolated place in the Chinese tradition, and the nature of its later contributions to logic, ethics, and science. It assesses the relation of Mohist thinking to the structure of the Chinese language, and grapples with the textual dynamics of later Mohist writings, particularly in regard to grammar and style, technical terminology, the use and significance of stock examples, and overall organization. Includes edited (...) and annotated Chinese text with an English translation and commentary, a glossary, and a photographic reproduction of the unemended text from the Taoist Patrology. (shrink)

How have human cultures engaged with and thought about animals, plants, rocks, clouds, and other elements in their natural surroundings? Do animals and other natural objects have a spirit or soul? What is their relationship to humans? In this new study, Graham Harvey explores current and past animistic beliefs and practices of Native Americans, Maori, Aboriginal Australians, and eco-pagans. He considers the varieties of animism found in these cultures as well as their shared desire to live respectfully within larger (...) natural communities. Drawing on his extensive casework, Harvey also considers the linguistic, performative, ecological, and activist implications of these different animisms. (shrink)

This paper argues for the general proper functionalist view that epistemic warrant consists in the normal functioning of the belief-forming process when the process has forming true beliefs reliably as an etiological function. Such a process is reliable in normal conditions when functioning normally. This paper applies this view to so-called testimony-based beliefs. It argues that when a hearer forms a comprehension-based belief that P (a belief based on taking another to have asserted that P) through the exercise of a (...) reliable competence to comprehend and filter assertive speech acts, then the hearer's belief is prima facie warranted. The paper discusses the psychology of comprehension, the function of assertion, and the evolution of filtering mechanisms, especially coherence checking. (shrink)

This two-part volume collects the complete fragments and most important testimonies for the leading presocratic philosophers. The Greek and Latin texts are translated on facing pages and accompanied by a brief commentary for each philosopher.

The _Physics_ is a foundational work of western philosophy, and the crucial one for understanding Aristotle's views on matter, form, essence, causation, movement, space, and time. This richly annotated, scrupulously accurate, and consistent translation makes it available to a contemporary English reader as no other does—in part because it fits together seamlessly with other closely associated works in the New Hackett Aristotle series, such as the _Metaphysics_, _De Anima_, and forthcoming _De Caelo_ and _On Coming to Be and Passing Away_. (...) Eventually the series will include all of Aristotle's works. Sequentially numbered endnotes provide the information most needed at each juncture, while a detailed Index of Terms indicates places where focused discussion of key notions occurs. An illuminating general Introduction describes the book that lies ahead, explaining what sort of work it is and what sorts of evidence it relies on. (shrink)

Hylomorphism is the Aristotelian theory according to which substances are composites of matter and form. If my house is a substance, then its matter would be a collection of bricks and timbers and its form something like a structure that unites those bricks and timbers into a single substance. Contemporary hylomorphists are divided on how to understand forms best, but a prominent group of theorists argue that forms are emergent powers. According to such views, when material components are arranged appropriately, (...) a novel substance emerges with the power to impose unity on its components through time. I argue that these accounts of form fall prey to a bootstrapping problem, and so, suffer from issues of redundancy, given plausible assumptions about inherence. In their place, I suggest an ontologically minimalist conception of forms as collective manifestations of the powers of matter. (shrink)