James Trafford

In [5], 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”. 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 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.

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.

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

This peer-reviewed paper investigates the dominant underlying approach to aesthetic experience and conscious experience more generally – that is, a neo-Kantian phenomenological approach. In essence, I argue that such approaches are based on a petitio principii in relation to what I call the 'principle of appearing qua appearing' – a principle that, I suggest, underlies the dominant approach to aesthetic perception. So, the ramifications of this argument are that we ought to question the dominance of the phenomenological approach to experience, and reassess our concepts of the aesthetic mode of perception in this light. The paper makes a specific contribution to knowledge in this field in that it identifies a weakness in the dominant arguments relevant to our understanding of conscious experience, and clears the path for views of conscious experience that emerge in cognitive science. The paper has been delivered at a number of major conferences, and is used as a course text at the American University of Berlin.

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.

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.

This paper aims to develop a unified account of communication, competence and reference fixing that surpasses problems with two of the most influential views on the philosophical market, neodescriptivism and the 'locking' theory. Our charge is that the conditions upon communication are less substantive than the neo-descriptivist account requires and the conditions upon reference-fixing are more substantive than those provided by the locking-view. In order to avoid the problems that neodescriptivist views face (e.g. holism), we suggest that the shareability of a specific set of inferences, or dispositions to infer, is not a prerequisite for conceptual shareability. In order to avoid the infamous ‘which-properties-speakers-lock-on’ problem of the locking-view, we establish a more robust causal relation between concepts and their referents.

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