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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 (...) 