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.

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)

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)

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)

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)

We humans tend to believe that things are only real in as much as we perceive them, an idea reinforced by modern philosophy, which privileges us as special, radically different in kind from all other objects. But as Graham Harman, one of the theory's leading exponents, shows, Object-Oriented Ontology (OOO) rejects the idea of human specialness: the world, he states, is clearly not the world as manifest to humans. "To think a reality beyond our thinking is not nonsense, but (...) obligatory." At OOO's heart is the idea that objects--whether real, fictional, natural, artificial, human, or non-human--are mutually autonomous. This core idea has significance for nearly every field of inquiry which is concerned in some way with the systematic interaction of objects, and the degree to which individual objects resist full participation in such systems. In this brilliant new introduction, Graham Harman lays out OOO's history, ideas, and impact, taking in art and literature, politics and natural science along the way. From Sherlock Holmes, unicorns, and videogames to Dadaism, Voltaire, and string theory, this book will change the way you understand everything. (shrink)

Graham Priest presents a ground-breaking account of the semantics of intentional language--verbs such as "believes," "fears," "seeks," or "imagines." Towards Non-Being proceeds in terms of objects that may be either existent or non-existent, at worlds that may be either possible or impossible. The book will be of central interest to anyone who is concerned with intentionality in the philosophy of mind or philosophy of language, the metaphysics of existence and identity, the philosophy of fiction, the philosophy of mathematics, or (...) cognitive representation in AI. (shrink)

This chapter states the contrast between presumptivism about testimonial warrant (often called anti-reductionism) and strict reductionism (associated with Hume) about testimonial warrant. Presumptivism sees an analogy with modest foundationalism about perceptual warrant. Strict reductionism denies this analogy. Two theoretical frameworks for these positions are introduced to better formulate the most popular version of persumptivism, a competence reliabilist account. Seven arguments against presumptivism are then stated and critiqued: (1) The argument from reliability; (2) The argument from reasons; (3) the argument from (...) positive reasons; (4) the argument from negative reasons; (5) the argument from agency; (6) the argument from psychological force; and (7) the argument from gullibility. If presumptivm is false, it is not for any of these arguments. (shrink)

Epistemic defeat is standardly understood in either evidentialist or responsibilist terms. The seminal treatment of defeat is an evidentialist one, due to John Pollock, who famously distinguishes between undercutting and rebutting defeaters. More recently, an orthogonal distinction due to Jennifer Lackey has become widely endorsed, between so-called doxastic (or psychological) and normative defeaters. We think that neither doxastic nor normative defeaters, as Lackey understands them, exist. Both of Lackey’s categories of defeat derive from implausible assumptions about epistemic responsibility. Although Pollock’s (...) evidentialist view is superior, the evidentialism per se can be purged from it, leaving a general structure of defeat that can be incorporated in a reliabilist theory that is neither evidentialist nor responsibilist in any way. (shrink)

OOO and art: a first summary -- Formalism and its flaws -- Theatrical, not literal -- The canvas is the message -- After high modernism -- Dada, surrealism, and literalism -- Weird formalism.

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 case of very young children is a test case for the plausibility of reductionism about testimonial warrant. Reductionism requires reductive reasons, reductively justified and actively deployed for testimonial justification. Though nascent language-users enjoy warranted testimony based beliefs, they do not meet these three reductionist demands. This paper clearly formulates reductionism and the infant/child objection. Two rejoinders are discussed: an influential conceptual argument from Jennifer Lackey’s paper “Testimony and the Infant/Child Objection” and the growing empirical evidence from developmental psychology on (...) selective trust in children. Neither Lackey’s argument nor the empirical evidence vindicate reductionism. (shrink)

The problem of higher-order misrepresentation poses a dilemma for the higher-order theory of consciousness. The two ways of conceiving of the theory each run into a different difficulty raised by the problem of misrepresentation. If the theory is conceived relationally, i.e., conceived so as the higher-order state causes or makes a first-order state conscious, then the theory faces a problem raised by Block concerning the implausibility of non-existent conscious states. If conceived non-relationally, i.e., conceived in such a way as it (...) is the higher-order state itself which is the conscious state, then consideration of Neander’s original misrepresentation problem, that this seems to make the first-order state a spinning wheel in the explanation of consciousness, raises the conceptual problem that this version of the higher-order theory seems not to offer a philosophical theory of consciousness that is distinct from the first-order theory. (shrink)

Prometheanism -- Brassier at Goldsmiths -- Brassier's nihilism -- The path ahead -- Vitalist idealism -- Grant at Goldsmiths -- Philosophies of nature after Schelling -- A new sense of idealism -- Object-oriented ontology (OOO) -- OOO at Goldsmiths -- The withdrawn -- Objects and their qualities -- Vicarious causation -- The crucial place of aesthetics -- Speculative materialism -- Meillassoux at Goldsmiths -- After finitude -- Glimpses of the divine inexistence -- The two axes of speculative realism.

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)

ENGLISH POLITICAL PHILOSOPHY HOBBES I. ON MAN § In the year there was published in England a very remarkable book, one of England's Bibles, an original and ...

This biography illuminates the life and thought of Baroness Mary Warnock, whose active years spanned the second half of the twentieth century, a period during which opportunities for middle-class women rapidly and vastly improved.

Many technological advancements do not readily lend themselves to incorporation into a philosophy curriculum, but Wikipedia is an exception. Courses can be designed around implementing or improving Wikipedia pages, which will help students both learn technological skills and engage with the world beyond the classroom. In the fall of 2012 I led such a class, in which we created the Wikipedia page for (appropriately) Collective Intentionality. This essay recounts my experience leading this class, examines its pedagogical and philosophical import, and (...) provides a template for future classes. (shrink)

Social knowledge, for the most part, is knowledge through testimony. This essay is an overview of the epistemology of testimony. The essay separates knowledge from justification, characterizes testimony as a source of belief, explains why testimony is a source of knowledge, canvasses arguments for anti-reductionism and for reductionism in the reductionism vs. anti-reductionism debate, addresses counterexamples to knowledge transmission, defends a safe basis account of testimonial knowledge, and turns to social norms as a partial explanation for the reliability of testimony.

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)

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)

It is a very curious thing that proponents of Idealism have considered it to be a satisfactory counter to ‘scepticism’, ‘nihilism’, and the like. On the contrary, it seems to me that Idealism is a very close cousin to ‘brain-in-a-vat’ scepticism and other anti-naturalistic fantasies. Moreover, it seems to me that Idealism is inferior to Naturalism for much the same kinds of reasons that ‘brain-in-a-vat’ scepticism and other anti-naturalistic fantasies are inferior to Naturalism: a proper weighing of theoretical virtues discloses (...) that Naturalism is the most virtuous worldview. (shrink)

This introductory article outlines how this special issue contributes to existing scholarship that calls for a rethinking and re-evaluation of common assumptions about early modern philosophy. One way of challenging existing narratives is by questioning what role systems or systematicity play during this period. Another way of rethinking early modern philosophy is by considering assumptions about the role of philosophy itself and how philosophy can effect change in those who form philosophical beliefs or engage in philosophical argumentation. A further way (...) of advancing early modern scholarship is by examining the tight links between early modern views on metaphysics and epistemology, on the one hand, and moral, social, and political philosophy, on the other hand. Moreover, there are ongoing debates whether and how the traditional distinction between rationalism and empiricism, which has been questioned for several reasons, should be replaced. It has been proposed to replace it with a distinction between experimental and speculative philosophy, but this distinction cannot easily accommodate early modern debates concerning moral, social, and political philosophy. In addition to highlighting several ways how early modern philosophy has and can be rethought, we summarize how the papers in this special issue contribute to these ongoing efforts. (shrink)

This is the concluding chapter of a debate book about the existence of God: *Theism and Atheism: Opposing Arguments in Philosophy* (Gale, 2019). The book has a large number of contributors on both sides. My chapter suggests one way of unifying the contributions that are made on the atheistic side.

There is an important distinction between two different kinds of expressions of gratitude: propositional expressions of gratitude and prepositional expressions of gratitude. I argue that there is a corresponding distinction between two different kinds of expression of resentment: propositional expressions of resentment and prepositional expressions of resentment. I then argue that theists should suppose neither that propositional expressions of gratitude are prepositional expressions of gratitude to God, nor that propositional expressions of resentment are prepositional expressions of resentment of God.