# Actual Futures in Peircean

Branching-Time Logic

## Bruno Barcellan and Alberto Zanardo

### Abstract

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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