Assertions, Hypotheses, Conjectures, Expectations: Rough-Sets Semantics and Proof Theory

  • Gianluigi  BellinEmail author
Part of the Trends in Logic book series (TREN, volume 39)


In this chapter bi-intuitionism is interpreted as an intensional logic which is about the justification conditions of assertions and hypotheses, extending C. Dalla Pozza and C. Garola’s pragmatic interpretation [18] of intuitionism, seen as a logic of assertions according to a suggestion by M. Dummett. Revising our previous work on this matter [5], we consider two additional illocutionary forces, \((i)\) conjecturing, seen as making the hypothesis that a proposition is epistemically necessary, and \((ii)\) expecting, regarded as asserting that a propostion is epistemically possible; we show that a logic of expectations justifies the double negation law. We formalize our logic in a calculus of sequents and study bimodal Kripke semantics of bi-intuitionism based on translations in S4. We look at rough set semantics following P. Pagliani’s analysis of “intrinsic co-Heyting boundaries” [40] (after Lawvere). A Natural Deduction system for co-intuitionistic logic is given where proofs are represented as upside down Prawitz trees. We give a computational interpretation of co-intuitionism, based on T. Crolard’s notion of coroutine [16] as the programming construction corresponding to subtraction introduction. Our typed calculus of co-routines is dual to the simply typed lambda calculus and shows features of concurrent and distributed computations.


Classical Logic Intuitionistic Logic Natural Deduction Sequent Calculus Epistemic Possibility 
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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversità di VeronaVeronaItaly

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