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From intuitionistic justification logic to explicit modal type theory with diamond

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I am currently elaborating a grant project proposing to develop (explicit) modal type theory with diamond. As far as I can see, the state-of-the-art in modal type theories is focused on using generalisations of box modalities (right adjoints). From my non-expert view—and I may very well be wrong! it seems to stem from (i) the technical difficulty of using diamonds in type systems/natural deduction, and (ii) the apparent lack of application cases for these modalities.

My proposal aims at investigating (i) whether ‘recent’ proof theoretic tools to treat intuitionistic modal logics can help alleviate this technical difficulty, (ii) whether potential applications become clearer when we have the tools to integrate diamond modalities to type systems, and ‘cherry on top’ (iii) whether diamonds end up being the missing link to relate systems for modal logic and for arithmetics in the intuitionistic case in the same nice way as in the classical case.

This third point can actually be seen as the cornerstone of the project and relies on the development of intuitionistic justification logic. With R.Kuznets and L.Stra├čburger, we recently provided a treatment of the intuitionistic diamond modality in the style of justification logic. I will give an overview of this preliminary work and present my ideas for the proposal. I am hoping for generous feedback!

=== Zoom details === Meeting ID: 818 7333 5084 Passcode: 217

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