Hypothetical Logic of Proofs

The Logic of Proofs (LP) [2–5] is a refinement of modal logic introduced by Artemov in 1995 which has recently been proposed for explaining well-known paradoxes arising in the formalization of Epistemic Logic. Assertions of knowledge and belief are accompanied by justifications: the formula [[t]]A states that proof witness t is “reason” for knowing/believing A. Also, LP is capable of reflecting its proofs in the object-logic: ` A implies ` [[t]]A, for some t. This suggests that computational interpretations in terms of the Curry-de Bruijn-Howard isomorphism could yield programming languages supporting a uniform treatment of programs and type derivations. Although some avenues in this direction have been explored (eg. certifying mobile computation [10] and history-aware computation [9]) they have been centred on intuitionistic fragments of LP. This work proposes to extend this analysis to full LP, which is based on classical logic. To this effect, we define the Hypothetical Logic of Proofs (HLP). The set of formulas and proof witnesses of HLP is defined by the following syntax: