Nils Kürbis

Without directly addressing the Demarcation Problem for logic—the problem of distinguishing logical vocabulary from others—we focus on distinctive aspects of logical vocabulary in pursuit of a second goal in the philosophy of logic, namely, proposing criteria for the justification of logical rules. Our preferred approach has three components. Two of these are effectively Belnap’s, but with a twist. We agree with Belnap’s response to Prior’s challenge to inferentialist characterisations of the meanings of logical constants. Belnap argued that for a logical constant to exist, its rules must be conservative over a previously given consequence relation and guarantee the uniqueness of the constant. The twist is that we require logical vocabulary to be provably conservative, not over a previously given formal consequence relation, but over a previously given meaningful vocabulary of a language. Uniqueness is also a feature defined in terms of provability: if two syntactically distinguished expressions are governed by the same rules, then formulas with them as main operators are interderivable. Belnap’s criteria are not only those for the existence of a logical constant, but more: they are what distinguishes logical vocabulary from all other expressions. It is the defining mark of a logical constant that it is provably conservative over the fragment of the language which excludes it and that its rules guarantee its uniqueness. The third component is the topic neutrality of logic. The provable conservativeness of logic over a previously given vocabulary of a language is motivated, in part, by appeal to molecularism in the theory of meaning. Molecularity is a feature endorsed by the theory of meaning as a whole, so it does not distinguish logical constants from other expressions. The same is true for conservativeness. We argue that molecularity, and hence conservativeness, are implicit presuppositions of speakers’ use of language: the addition of new vocabulary to a language is presupposed to be conservative. But this presupposition is one that we should be able reflectively to endorse (such as when attempting a rational reconstruction of the use of the vocabulary). Thus, where conservativeness is found to fail, this defect should be remedied and the use of the vocabulary corrected (as in classical negation) or excised altogether (as in pejoratives). We go on to characterise a notion of topic neutrality, which we argue applies to logical vocabulary. We then note that reflective endorsement of the conservativeness of a topic neutral vocabulary requires a proof that the vocabulary is conservative relative to any base vocabulary. Thus we require that logical vocabulary be demonstrably conservative. Allying this requirement, distinctive of logic, to a general consideration about the commitments of assertion yields a mode of justification for logical constants akin to some conceptions of Harmony, i.e., to the idea that the consequences of assertion of a logical complex need to be warranted by the grounds and ought to be the strongest consequences warranted by the grounds. Intuitionistic logic acquires a somewhat familiar justification but emanating from a new motivation.

Throughout his career, Keith Hossack has made outstanding contributions to the theory of knowledge, metaphysics and the philosophy of mathematics. This collection of previously unpublished papers begins with a focus on Hossack's conception of the nature of knowledge, his metaphysics of facts and his account of the relations between knowledge, agents and facts. Attention moves to Hossack's philosophy of mind and the nature of consciousness, before turning to the notion of necessity and its interaction with a priori knowledge. Hossack's views on the nature of proof, logical truth, conditionals and generality are discussed in depth. In the final chapters, questions about the identity of mathematical objects and our knowledge of them take centre stage, together with questions about the necessity and generality of mathematical and logical truths.

Rumfitt has given two arguments that in unilateralist verificationist theories of meaning, truth collapses into correct assertibility. In the present paper I give similar arguments that show that in unilateral falsificationist theories of meaning, falsehood collapses into correct deniability. According to bilateralism, meanings are determined by assertion and denial conditions, so the question arises whether it succumbs to similar arguments. I show that this is not the case. The final section considers the question whether a principle central to Rumfitt's first argument, ‘It is assertible that if and only if it is assertible that it is assertible that ’, is one that bilateralists can reject, and concludes that they cannot. It follows that the logic of assertibility and deniability, according to a result by Williamson, is the little known modal logic K4 studied by Sobociński. The paper ends with a plaidoyer for bilateralists to adopt this logic.

This book argues that the meaning of negation, perhaps the most important logical constant, cannot be defined within the framework of the most comprehensive theory of proof-theoretic semantics, as formulated in the influential work of Michael Dummett and Dag Prawitz. Nils Kürbis examines three approaches that have attempted to solve the problem - defining negation in terms of metaphysical incompatibility; treating negation as an undefinable primitive; and defining negation in terms of a speech act of denial - and concludes that they cannot adequately do so. He argues that whereas proof-theoretic semantics usually only appeals to a notion of truth, it also needs to appeal to a notion of falsity, and proposes a system of natural deduction in which both are incorporated. Offering new perspectives on negation, denial and falsity, his book will be important for readers working on logic, metaphysics and the philosophy of language.