Consistent Sets and Contrary Inferences: Reply to Griffiths and Hartle

TL;DR

This paper discusses the issue of contrary inferences in the consistent histories approach to quantum mechanics, defending the formalism against criticisms and clarifying the distinction between contradictory and contrary inferences.

Contribution

It clarifies the difference between contradictory and contrary inferences in the consistent histories formalism and defends the approach against recent criticisms.

Findings

Consistent histories can admit contrary inferences without contradiction.

Formalism based on ordered consistent sets excludes both contradictory and contrary inferences.

The paper critiques Griffiths and Hartle's response to the criticisms of the consistent histories approach.

Abstract

It was pointed out recently [A. Kent, Phys. Rev. Lett. 78 (1997) 2874] that the consistent histories approach allows contrary inferences to be made from the same data. These inferences correspond to projections $P$ and $Q$, belonging to different consistent sets, with the properties that $PQ = QP = 0$ and $P \neq 1 -Q$. To many, this seems undesirable in a theory of physical inferences. It also raises a specific problem for the consistent histories formalism, since that formalism is set up so as to eliminate contradictory inferences, i.e. inferences $P$ and $Q$ where $P = 1 - Q$. Yet there seems to be no sensible physical distinction between contradictory and contrary inferences. It seems particularly hard to defend the asymmetry, since (i) there is a well-defined quantum histories formalisms which admits both contradictory and contrary inferences, and (ii) there is also a well-defined formalism, based on ordered consistent sets of histories, which excludes both. In a recent comment, Griffiths and Hartle, while accepting the validity of the examples given in the above paper, restate their own preference for the consistent histories formalism. As this brief reply explains, in so doing, they fail to address the arguments made against their approach to quantum theory.