Actual Futures in Peircean
Branching-Time Logic

Bruno Barcellan and Alberto Zanardo


In the paper Indeterminsm and the Thin Red Line, Belnap and Green show that the notion of actual future seriously clashes with Objective Indeterminism. However, there are some Computer Science and Artificial Intelligence applications of the formal structure which arises from the actual future point of view. Of course, these applications have nothing to do with the ontology of objective indeterminism, and so the objections by Belnap and Green do not apply to them. We consider an extension of Peircean branching-time logic, which contains a new future operator to be read as `it will happen, in the future that will actually take place'. According to this reading of the new operator, Time is pictured as a tree in which each moment has one marked possible future that represents the actual future of that moment. This allows one to define a semantics for the extended language. We will provide a finite deductive system which is sound and strongly complete for the corresponding notion of validity.

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