This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to (...) equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems. (shrink)

This book argues for a view in which processes of dialogue and interaction are taken to be foundational to reasoning, logic, and meaning. This is both a continuation, and a substantial modification, of an inferentialist approach to logic. As such, the book not only provides a critical introduction to the inferentialist view, but it also provides an argument that this shift in perspective has deep and foundational consequences for how we understand the nature of logic and its relationship with meaning (...) and reasoning. This has been upheld by several technical results, including, for example a novel approach to logical paradox and logical revision, and an account of the internal justification of logical rules. The book shows that inferentialism is greatly strengthened, such that it can answer the most stringent criticisms of the view. This leads to a view of logic that emphasizes the dynamics of reasoning, provides a novel account of the justification and normativity of logical rules, thus leading to a new, attractive approach to the foundations of logic. The book addresses readers interested in philosophy of language, philosophical and mathematical logic, theories of reasoning, and also those who actively engage in current debates involving, for example, logical revision, and the relationship between logic and reasoning, from advanced undergraduates, to professional philosophers, mathematicians, and linguists. (shrink)

One of the central issues facing contemporary political theory is the problem of difference. This problem is perhaps clearest in disagreements regarding the role of pluralism between advocates of deliberative, and agonistic, approaches to democracy. According to agonists, deliberative democracy has only paid lip-service to pluralism, emphasising agreement, consensus, and universalism. Instead, agonists argue that we should accommodate incommensurable difference as central to political organisation. But this shift threatens to emphasise particularity at the expense of commonality, so preventing the transformation (...) of social positions. These debates turn largely around the ways in which power is conceived in relation to democratic interaction, with deliberativists emphasising ideals of free and equal discourse, and agonists, the irreducible role of power. I begin by outlining a structural approach to social power, to show how we are all structurally implicated in power relationships. I go on to show how emphasising structural implication shifts us beyond debates that revolve around difference, drawing attention to the more foundational problem of how agents may come to understand these structures and each other, in ways that surpass or transcend the specific constraints acting upon them. Considering this, I develop a complex account of structural implication that emphasises the relationships between social norms, reasoning, and objective power relationships. This approach illuminates lacunae in similar approaches, particularly regarding the way in which structural power disenables us from often seeing structural imbalances of power, and clarifies the ways in which collective political transformation is possible. (shrink)

This paper considers a topos-theoretic structure for the interpretation of co-constructive logic for proofs and refutations following Trafford :22–40, 2015). It is notoriously tricky to define a proof-theoretic semantics for logics that adequately represent constructivity over proofs and refutations. By developing abstractions of elementary topoi, we consider an elementary topos as structure for proofs, and complement topos as structure for refutation. In doing so, it is possible to consider a dialogue structure between these topoi, and also control their relation such (...) that classical logic is simulated where proofs and refutations are conclusive. (shrink)

This paper discusses the philosophical and logical motivations for rejectivism, primarily by considering a dialogical approach to logic, which is formalized in a Question–Answer Semantics. We develop a generalized account of rejectivism through close consideration of Mark Textor's arguments against rejectivism that the negative expression ‘No’ is never used as an act of rejection and is equivalent with a negative sentence. In doing so, we also shed light upon well-known issues regarding the supposed non-embeddability and non-iterability of force indicators.

This peer-reviewed paper intervenes in debates relating to overarching themes that impact upon mass media studies, communication theory and theories of cognition more generally. In particular, the paper discusses issues involving how our ordinary psychological thinking relates to norms of rationality (and how these latter are conceived). In essence, I argue against a dominant approach taken by Christopher Peacocke, that rationality can be grounded in the possession of certain concepts. The article makes a new contribution to the field by arguing (...) against the dominant approach on two grounds: (a) it fails to distinguish between true and false normative commitments; (b) it is empirically unsound. In response, I briefly offer suggestions towards an alternative, and psychologically tractable, account of rational commitment. Presentations of earlier drafts of the paper were given at a seminar at the Centre for Study of Mind and Nature, Oslo, as well as the conference of the European Society for Philosophy and Psychology. (shrink)

It is well known that classical inferentialist semantics runs into problems regarding abnormal valuations. It is equally well known that the issues can be resolved if we construct the inference relation in a multiple-conclusion sequent calculus. The latter has been prominently developed in recent work by Restall, with the guiding interpretation that the valid sequent says that the simultaneous assertion of all of Γ with the denial of all of Δ is incoherent. However, such structures face significant interpretive challenges, and (...) they do not provide an adequate grasp on the machinery of the duality of assertions and denials that could provide an abstract account of inferential semantics; show why the dual treatment is semantically superior. This paper explores a slightly different tack by considering a dual-calculus framework consisting of two, single-conclusion, inference relations dealing with the preservation of assertion and the preservation of denial, respectively. In this context, I develop an abstract inferentialist semantics, before going on to show that the framework is equivalent to Restall’s, whilst providing a better grasp on the underlying proof-structure. (shrink)

This paper tackles foundational issues regarding the justification of logical rules. It is argued that standard accounts from both proof-theoretical and semantical points of view do not su ffi ce to account for the justification of basic logical rules. By way of response, an analysis of logical inference as acts taking place in dialogical situations is provided. In turn, this makes way for an internal justification of logical rules at the termination of dialogue, that can be formalized in terms of (...) co-constructive logic for proof and refutation [author reference omitted]. (shrink)

This peer-reviewed essay is an intervention into the emerging field of 'Speculative Realism', which has links to the field of Speculative Aesthetics. The work is essentially an attempt to develop a theory of perception (and more broadly consciousness) that is not at odds with the scientific worldview. In this respect, the dominant views of aesthetic perception (Kantian / neo-Kantian phenomenology) are critiqued in favour of neurophilosophical views stemming from Thomas Metzinger. In order to position myself, I go on to analyse (...) the fiction of Thomas Ligotti to develop the idea of a non-human-centric phenomenal experience. This work operates within several philosophical contexts: one is the debate between phenomenology and scientific realism, of which the developing field of Speculative Realism is a part, and upon which the journal Collapse has been incredibly influential; a further context is that of transcendental materialism; another is that of the relationship between literature and philosophy. The article marks a departure within these fields by suggesting that recent developments in cognitive science can be understood not through a naïve scientism or a kind of brute materialism, but only through understanding the transcendental ramifications to the phenomenal experience of the human. (shrink)

There are several issues with the standard approach to the relationship between conditionals and assertions, particularly when the antecedent of a conditional is false. One prominent alternative is to say that conditionals do not express propositions, but rather make conditional assertions that may generate categorical assertions of the consequent in certain circumstances. However, this view has consequences that jar with standard interpretations of the relationship between proofs and assertion. Here, I analyse this relationship, and say that, on at least one (...) understanding of proof, conditional assertions may reflect the dynamics of proving, which generate categorical assertions. In particular, when we think about the relationship between assertion and proof as rooted in a dialogical approach to both, the distinction between conditional and categorical assertions is quite natural. (shrink)

This article considers neoliberalism through the “peaceful violence” of its social spaces that are stratified and ordered around raciality whilst abjuring the explicit presence of racialised power. Many dominant analyses of neoliberalism in the social science have figured racial injustices as ideological fossils to be swept away by a fundamentally neutral political economy that has shaped all human activity according to market principles. As such, racial injustices are understood as material deviations from conditions of economic power on the one hand, (...) and cultural deviations in hegemonic common sense on the other. Against the grain of these approaches, I argue that racialised power is better understood as a productive technology fundamental to the structuring of our worlds, their neoliberal organisation just one iteration of ongoing coloniality. Contrary to popular analysis and this account of neoliberalism, contemporary racialised violence and nationalist discourse are, therefore, not only consistent with the power of neoliberal classes but also result from weaknesses in their hegemonic power. (shrink)

This article considers neoliberalism through the “peaceful violence” of its social spaces that are stratified and ordered around raciality whilst abjuring the explicit presence of racialised power. Many dominant analyses of neoliberalism in the social science have figured racial injustices as ideological fossils to be swept away by a fundamentally neutral political economy that has shaped all human activity according to market principles. As such, racial injustices are understood as material deviations from conditions of economic power on the one hand, (...) and cultural deviations in hegemonic common sense on the other. Against the grain of these approaches, I argue that racialised power is better understood as a productive technology fundamental to the structuring of our worlds, their neoliberal organisation just one iteration of ongoing coloniality. Contrary to popular analysis and this account of neoliberalism, contemporary racialised violence and nationalist discourse are, therefore, not only consistent with the power of neoliberal classes but also result from weaknesses in their hegemonic power. (shrink)

In (Béziau 2001), Béziau provides a means by which Gentzen’s sequent calculus can be combined with the general semantic theory of bivaluations. In doing so, according to Béziau, it is possible to construe the abstract "core" of logics in general, where logical syntax and semantics are "two sides of the same coin". Thecentral suggestion there is that, by way of a modification of the notion of maximal consistency, it is possible to prove the soundness and completeness for any normal logic (...) (without invoking the role of classical negation in the completeness proof). However, the reduction to bivaluation may be a side effect of the architecture of ordinary sequents, which is both overly restrictive, and entails certain expressive restrictions over the language. This paper provides an expansion of Béziau’s completeness results for logics, by showing that there is a natural extension of that line of thinking to n-sided sequent constructions. Through analogical techniques to Béziau’s construction, it is possible, in this setting, to construct abstract soundness and completeness results for n-valued logics.En (Béziau 2001), Béziau ofrece un recurso para combinar el cálculo de secuentes de Gentzen con la teoría semántica general de bivaluaciones. Al hacer esto, según Béziau, es posible construir el “núcleo” abstracto de la lógica en general, donde sintaxis y semántica son las dos caras de una misma moneda. La sugerencia clave es que, mediante una modificación de la noción de consistencia máxima, es posible probar la corrección y completud de cualquier lógica normal (sin invocar la función de la negación clásica en la prueba de completud). Sin embargo, la reducción a bivaluaciones puede ser un efecto colateral de la arquitectura de los secuentes ordinarios, que es abiertamente restrictiva y entraña determinadas restricciones expresivas sobre el lenguaje. Este artículo ofrece una expansión de los resultados de completud de Béziau para la lógica, mostrando que existe una extensión natural de esta línea de pensamiento a construcciones de secuentes de n lados. Mediante técnicas análogas a la construcción de Béziau, en este marco es posible construir resultados abstractos decorrección y completud para la lógica n-valuada. (shrink)

In, Béziau provides a means by which Gentzen’s sequent calculus can be combined with the general semantic theory of bivaluations. In do- ing so, according to Béziau, it is possible to construe the abstract “core” of logics in general, where logical syntax and semantics are “two sides of the same coin”. The central suggestion there is that, by way of a modification of the notion of maximal consistency, it is possible to prove the soundness and completeness for any normal logic. (...) However, the reduction to bivaluation may be a side e ff ect of the architecture of ordinary sequents, which is both overly restrictive, and entails certain expressive restrictions over the language. This paper provides an expansion of Béziau’s completeness results for logics, by showing that there is a natural extension of that line of thinking to n -sided sequent constructions. Through analogical techniques to Béziau’s construction, it is possible, in this setting, to construct abstract soundness and completeness results for n -valued logics. (shrink)

This article considers neoliberalism through the “peaceful violence” of its social spaces that are stratified and ordered around raciality whilst abjuring the explicit presence of racialised power. Many dominant analyses of neoliberalism in the social science have figured racial injustices as ideological fossils to be swept away by a fundamentally neutral political economy that has shaped all human activity according to market principles. As such, racial injustices are understood as material deviations from conditions of economic power on the one hand, (...) and cultural deviations in hegemonic common sense on the other. Against the grain of these approaches, I argue that racialised power is better understood as a productive technology fundamental to the structuring of our worlds, their neoliberal organisation just one iteration of ongoing coloniality. Contrary to popular analysis and this account of neoliberalism, contemporary racialised violence and nationalist discourse are, therefore, not only consistent with the power of neoliberal classes but also result from weaknesses in their hegemonic power. (shrink)

The lesson is familiar. Kripke’s arguments in favor of a posteriori necessary truths annul the idea that conceivability is a guide to metaphysical possibility because determining that which is a priori is a separate issue from determining that which is necessary. Modal rationalists do not completely agree with this conclusion. Following recent work on two-dimensional semantics, David Chalmers suggests that two distinct semantic values can be assigned to a statement, depending on whether we consider possible worlds as counterfactual or counteractual. (...) The idea is that counterfactual possibilities yield familiar Kripkean intuitions, but that counteractuals fulfill the desired link between a priori conceivability and metaphysical possibility. In this paper, I discuss a problem for modalrationalism that arises through the use of material conditionals, or conditionals in the indicative mood. I then turn to Chalmers’ response, and suggest reasons why it is inadequate. I turn to another response from Chalmers, and suggest that, whilst it solves the fi rst issue, it is incapable of grounding modal rationalism. In conclusion I will suggest a way in which a tempered version of modal rationalism can be salvaged. (shrink)