The true futures

Abstract

In this paper various branching time semantics are compared with the aim of clarifying the role of true futures of counterfactual moments, that is, true futures of moments outside the true chronicle. First we give an account of Arthur Prior’s Ockhamistic semantics where truth of a formula is relative to a moment and a chronicle. We prove that this is equivalent to a version of a semantics put forward by Thomason and Gupta where truth is relative to a moment and what is called a chronicle function which assigns a chronicle to each moment. Later we discuss how a semantic theory considered by Belnap and Green may be formalised. It comes about by assuming a chronicle function to be given once and for all. However, this semantics invalidates an intuitively valid formula, so we present an alternative semantics where the formula in question is valid. Furthermore, we shall exhibit an intuitively invalid formula which is invalid in our alternative semantics, but which is valid in Prior’s Ockhamistic semantics. So we can conclude that Prior’s Ockhamistic validity does not imply validity in the alternative semantics. On the other hand, the converse implication does hold, as we shall prove. Summary of mathematical results: We have proved that Prior’s Ockhamistic semantics has the same valid formulas as Thomason and Gupta’s semantics, and we have proved that Prior’s Ockhamistic semantics has strictly more valid formulas than the alternative semantics.