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Modal logic as dialogical logic. (English) Zbl 1108.03309

Summary: The title reflects my conviction that, viewed semantically, modal logic is fundamentally dialogical; this conviction is based on the key role played by the notion of bisimulation in modal model theory. But this dialogical conception of modal logic does not seem to apply to modal proof theory, which is notoriously messy. Nonetheless, by making use of ideas which trace back to Arthur Prior (notably the use of nominals, special proposition symbols which ‘name’ worlds), I show how to lift the dialogical conception to modal proof theory. I argue that this shift to hybrid logic has consequences for both modal and dialogical logic, and I discuss these in detail.

MSC:

03B45 Modal logic (including the logic of norms)
03B60 Other nonclassical logic
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