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)

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)

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)

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)

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)

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)

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)

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)

Arguing for a new direction in postmodern theological thinking, away from the liberalism and nihilism of those who name themselves postmodern theologians, the ...

This paper is my second contribution to the Szatkowski volume. In the first paper, I provide a critical discussion of Bob Maydole's ontological arguments. In this second paper, I respond to Maydole's critical response to my first paper. My overall verdict is that Maydole does not successfully defend his arguments against my critical attack.

"Here is an excerpt. If you like where Collier goes with this you'll like the rest of the book: "I believe we make a mistake if we think that modern man is a rational creature. While it is a mark of primitive man to respond directly to the non logical and less rationally defensible images projected by the psyche, similar primitive or elemental responses lurk behind the civilized faced of which we are so proud. For example, we might be somewhat (...) amused by the cave dweller's belief that because he possessed the image of an animal he had gained some power capable of controlling the creature's life. But do we not believe something of the sort when we cherish the photograph of someone important to us and think carefully as to where it should be placed - where is the place we can contemplate it the most effectively? Can we honestly say that in possessing this image of a person we do not feel that some intangible link exists between us and them? And could we willfully and with passion deface the photograph without the irrational thought overtaking us that we had done some harm to the relationship and to the person concerned?"" -- customer review, Amazon.com. (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 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.

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 essay argues for a two-part thesis concerning the deflationist theories of truth and reference. First, I identify two points of contrast between the deflationist theories and their traditional inflationary opponents: (1) they each employ different orders of explanation for the variety of semantic phenomena, and (2) the inflationist is typically taken to be beholden to a reductive explanation of reference, whereas the deflationist is doubtful of this project. Secondly, I argue that these two points of contrast need not come (...) together to exhaust the space of possible views. There is room for a plausible middle ground: a moderate version of inflationism. My moderate inflationism will reject the deflationist's order of explanation, so it counts as genuinely inflationary. However, it also rejects the reductionist ambitions of the earlier inflationists, so it isn’t an apt target for deflationary scepticism. (shrink)

This brief paper builds on my recently published work on financial capitalism. My objective here is (a) to sketch an account of the primary social features of post-1970s financialised capitalism, (b) to identify select dynamics or mechanisms that have resulted in these features, (c) to outline a programme of research to enhance our explanatory understanding of the ‘fractured society’ via the sociological concepts of structure, culture and agency, and (d) to broach, characterize and assess likely prospects and triggers for the (...) accomplishment of a credible, defensible post-capitalist societal formation. (shrink)

Most phobias are focussed on a small number of fear-inducing stimuli (e.g., snakes, spiders). A review of the evidence supporting biological and cognitive explanations of this uneven distribution of phobias suggests that the readiness with which such stimuli become associated with aversive outcomes arises from biases in the processing of information about threatening stimuli rather than from phylogenetically based associative predispositions or “biological preparedness.” This cognitive bias, consisting of a heightened expectation of aversive outcomes following fear-relevant stimuli, generates and maintains (...) robust learned associations between them. Some of the features of such stimuli which determine this expectancy bias are estimates of how dangerous they are, the semiotic similarity between them and their aversive outcomes, and the degree of prior fear they elicit. Ontogenetic and cultural factors influence these features of fear-relevant stimuli and are hence important in determining expectancy bias. The available evidence does not exclude the possibility that both expectancy biases and specific evolved predispositions coexist, but the former can explain a number of important findings that the latter cannot. (shrink)

This paper considers how parents engage with a large, internationally supported childhood pneumococcal vaccine trial in The Gambia. Current analysis and professional reflection on public engagement is strongly shaped by the imperatives of public health and research institutions, and is thus couched in terms of acceptance and refusal, and . In contrast Gambian parents in the extreme, of free medical treatment, versus onepublic engagement with science’ in a globalized context might be recast, with implications for debates in biomedical ethics, and (...) the sustainability of public participation in medical research. (shrink)

Is your mind the same thing as your brain, or are there aspects of mind beyond the brain's biology? This is the mind-body problem, and it has captivated curious minds since the dawn of human contemplation. Today many insist that the mind is completely reducible to the brain. But is that claim justified? In this stimulating anthology, twenty-five philosophers and scientists offer fresh insights into the mind-brain debate, drawing on psychology, neurology, philosophy, computer science, and neurosurgery. Their provocative conclusion? The (...) mind is indeed more than the brain. (shrink)

This book reconceives disability as a set of social relations and practices, as experienced embodiment, and as an emancipatory movement, as well as a biomedical phenomenon. The author brings new attention to complex ethical questions surrounding disability, looking at not only the biomedical understanding of impairment, but also its cultural representations and social organization.

Altered state theories of hypnosis posit that a qualitatively distinct state of mental processing, which emerges in those with high hypnotic susceptibility following a hypnotic induction, enables the generation of anomalous experiences in response to specific hypnotic suggestions. If so then such a state should be observable as a discrete pattern of changes to functional connectivity (shared information) between brain regions following a hypnotic induction in high but not low hypnotically susceptible participants. Twenty-eight channel EEG was recorded from 12 high (...) susceptible (highs) and 11 low susceptible (lows) participants with their eyes closed prior to and following a standard hypnotic induction. The EEG was used to provide a measure of functional connectivity using both coherence (COH) and the imaginary component of coherence (iCOH), which is insensitive to the effects of volume conduction. COH and iCOH were calculated between all electrode pairs for the frequency bands: delta (0.1–3.9Hz), theta (4–7.9Hz) alpha (8–12.9Hz), beta1 (13–19.9Hz), beta2 (20–29.9Hz) and gamma (30–45Hz). The results showed that there was an increase in theta iCOH from the pre-hypnosis to hypnosis condition in highs but not lows with a large proportion of significant links being focused on a central-parietal hub. There was also a decrease in beta1 iCOH from the pre-hypnosis to hypnosis condition with a focus on a fronto-central and an occipital hub that was greater in high compared to low susceptibles. There were no significant differences for COH or for spectral band amplitude in any frequency band. The results are interpreted as indicating that the hypnotic induction elicited a qualitative change in the organization of specific control systems within the brain for high as compared to low susceptible participants. This change in the functional organization of neural networks is a plausible indicator of the much theorized “hypnotic-state”. (shrink)

Examines the influence of the physical and biological sciences on society, ethics, and philosophy during the twentieth century.

This article critically examines the structure and implications of the argument in ST 1, Q2, A3, associated with Aquinas’ First Way. Our central endeavor is to discern whether a certain disambiguation of point 6 (“There is something that is not moving/changing that moves/changes other things”) can be logically inferred from points 1-5. Through a three-part proof, the article establishes that under specific conditions, it can indeed be inferred. However, this interpretation notably diverges from Aquinas’ intended conclusion and subsequent stronger interpretations (...) of the claim. The paper also engages in discussions surrounding the soundness of the argument, assessing the plausibility of its premises in light of physics and logical analysis. The contribution does not seek to speculate on Aquinas’ understanding or to dismiss his perspective but rather to delineate what follows from his First Way premises, highlighting discrepancies with contemporary views on the nature of existence, particularly the notion of God. (shrink)

This monograph reappraises the role of Bertrand Russell's philosophical works in establishing the analytical tradition in philosophy. It's main aims are to: * improve our understanding of the history of analytical philosophy * engage in the important disputes surrounding the interpretation of Russell's philosophy * make a contribution to central issues in current analytical philosophy. Drawing extensively from Russell's less well known and unpublished works, this book is a welcome addition to the literature and will undoubtedly find a place on (...) the bookshelves of philosophers around the world. (shrink)

Even though disquotationalism is not correct as it is usually formulated, a deep insight lies behind it. Specifically, it can be argued that, modulo implicit commitment to reflection principles, all there is to the notion of truth is given by a simple, natural collection of truth-biconditionals.