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This paper presents rules of inference for a binary quantifier I for the formalisation of sentences containing definite descriptions within intuitionist positive free logic. I binds one variable and forms a formula from two formulas. Ix[F, G] means ‘The F is G’. The system is shown to have desirable prooftheoretic properties: it is proved that deductions in it can be brought into normal form. The discussion is rounded up by comparisons between the approach to the formalisation of definite descriptions recommended (...) 

This paper considers prooftheoretic semantics for necessity within Dummett's and Prawitz's framework. Inspired by a system of Pfenning's and Davies's, the language of intuitionist logic is extended by a higher order operator which captures a notion of validity. A notion of relative necessary is defined in terms of it, which expresses a necessary connection between the assumptions and the conclusion of a deduction. 

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, multipleconclusion derivations. Therefore, such derivations are completely natural and can safely be used in prooftheoretic semantics. 

The problem of negative truth is the problem of how, if everything in the world is positive, we can speak truly about the world using negative propositions. A prominent solution is to explain negation in terms of a primitive notion of metaphysical incompatibility. I argue that if this account is correct, then minimal logic is the correct logic. The negation of a proposition A is characterised as the minimal incompatible of A composed of it and the logical constant ¬. A (...) 

This paper deals with the question of the logicality of modal logics from a prooftheoretic 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 doubleline rules which, following Dos̆en’s analysis, allows us to claim that (...) 

Generalelimination harmony articulates Gentzen’s idea that the eliminationrules are justified if they infer from an assertion no more than can already be inferred from the grounds for making it. Dummett described the rules as not only harmonious but stable if the Erules allow one to infer no more and no less than the Irules justify. Pfenning and Davies call the rules locally complete if the Erules are strong enough to allow one to infer the original judgement. A method is given (...) 