This paper deals with the question of the logicality of modal logics from a proof-theoretic perspective. It is argued that if Dos̆en’s analysis of logical constants as punctuation marks is embraced, it is possible to show that all the modalities in the cube of normal modal logics are indeed logical constants. It will be proved that the display calculus for each displayable modality admits a purely structural presentation based on double-line rules which, following Dos̆en’s analysis, allows us to claim that (...) the corresponding modal operators are logical constants. (shrink)

Starting from a proof-theoretic perspective, where meaning is determined by the inference rules governing logical operators, in this paper we primarily aim at developing a proof-theoretic alternative to the model-theoretic meaning-invariant logical pluralism discussed in Beall and Restall. We will also outline how this framework can be easily extended to include a form of meaning-variant logical pluralism. In this respect, the framework developed in this paper—which we label two-level proof-theoretic pluralism—is much broader in scope than the one discussed in Beall (...) and Restall’s book. (shrink)

In the recent literature on proof-theoretic semantics, there is mention of a generality condition on defining rules. According to this condition, the schematic formulation of the defining rules must be maximally general, in the sense that no restrictions should be placed on the contexts of these rules. In particular, context variables must always be present in the schematic rules and they should range over arbitrary collections of formulae. I argue against imposing such a condition, by showing that it has undesirable (...) results and that it is ill-supported by the arguments brought in its favour. (shrink)

In philosophical logic and proof theory, we often find multiple-conclusion systems that induce a conjunctive reading of premises and a disjunctive reading of conclusions. In mathematical logic, in contrast, we often find multiple-conclusion systems that induce a conjunctive reading of both premises and conclusions. This paper studies some technical and philosophical aspects of this latter approach to multiple-conclusion consequence. The takeaway is that, while the importance of disjunctive multiple conclusions is beyond doubt, conjunctive multiple conclusions also have philosophical interest. First, (...) because there is some evidence that there are arguments with conjunctive multiple conclusions in natural language. Second, because conjunctive multiple conclusions are compatible with the reflexivity and transitivity of logical consequence, and this allows them to cohere better with some of our best accounts of what logical consequence is. (shrink)

Arguments, the story goes, have one or more premises and only one conclusion. A contentious generalisation allows arguments with several disjunctively connected conclusions. Contentious as this generalisation may be, I will argue nevertheless that it is justified. My main claim is that multiple conclusions are epiphenomena of the logical connectives: some connectives determine, in a certain sense, multiple-conclusion derivations. Therefore, such derivations are completely natural and can safely be used in proof-theoretic semantics.

This paper concentrates on how to capture harmony in sequent calculi. It starts by considering a proposal made by Tennant and some objections to it which have been presented by Steinberger. Then it proposes a different analysis which makes use of a double-line presentation of sequent calculi in the style of Dosen and it shows that this proposal is able to dismiss disharmonious operators without thereby adopting any global criterion.

Logical pluralists are committed to the idea of a neutral metalanguage, which serves as a framework for debates in logic. Two versions of this neutrality can be found in the literature: an agreed upon collection of inferences, and a metalanguage that is neutral as such. I discuss both versions and show that they are not immune to Quinean criticism, which builds on the notion of meaning. In particular, I show that the first version of neutrality is sub-optimal, and hard to (...) reconcile with the theories of meaning for logical constants, and the second version collapses mathematically, if rival logics, as object languages, are treated with charity in the metalanguage. I substantiate by proving a collapse theorem that generalizes familiar results. Thus, the existence of a neutral metalanguage cannot be taken for granted, and meaning-invariant logical pluralism might turn out to be dubious. (shrink)

The paper exposes the relevance of permuting conversions (in natural-deduction systems) to the role of such systems in the theory of meaning known as proof-theoretic semantics, by relating permuting conversion to harmony, hitherto related to normalisation only. This is achieved by showing the connection of permuting conversion to the general notion of canonicity, once applied to arbitrary derivations from open assumption. In the course of exposing the relationship of permuting conversions to harmony, a general definition of the former is proposed, (...) generalising the specific cases of disjunction and existential quantifiers considered in the literature. (shrink)