In R. D. Sorkin’s framework for logic in physics a clear separation is made between the collection of unasserted propositions about the physical world and the affirmation or denial of these propositions by the physical world. The unasserted propositions form a Boolean algebra because they correspond to subsets of an underlying set of spacetime histories. Physical rules of inference apply not to the propositions in themselves but to the affirmation and denial of these propositions by the actual world. This physical logic may or may not respect the propositions’ underlying Boolean structure. We prove that this logic is Boolean if and only if the following three axioms hold: (i) The world is affirmed, (ii) Modus Ponens and (iii) If a proposition is denied then its negation, or complement, is affirmed. When a physical system is governed by a dynamical law in the form of a quantum measure with the rule that events of zero measure are denied, the axioms (i)–(iii) prove to be too rigid and need to be modified. One promising scheme for quantum mechanics as quantum measure theory corresponds to replacing axiom (iii) with axiom (iv). Nature is as fine grained as the dynamics allows.
We thank Rafael Sorkin for helpful discussions. Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI. FD and PW are supported in part by COST Action MP1006. PW was supported in part by EPSRC grant EP/K022717/1. PW acknowledges support from the University of Athens during this work.
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