Abstract
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.
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Notes
- 1.
Lewis Carroll gives in [12] a witty account of the implications of a failure to take up modus ponens explicitly as a rule of inference.
- 2.
See [6] for a discussion of the ambiguity in the phrase “not A”.
- 3.
The notation \(\mathfrak{A}^{{\ast}}\) reflects the nature of the co-event space as dual to the event algebra.
- 4.
An alternative is to call any condition on the allowed co-events a dynamical law.
- 5.
Note also that at the level of the Boolean algebra of events we always have ¬ ¬ A = A and, moreover, if “the law of the excluded middle” is taken to mean that every event is either affirmed or denied then our framework respects it by fiat because that is just the statement that ϕ is a map to \(\mathbb{Z}_{2}\) [6]. This illustrates how careful one must be to be clear.
- 6.
We assume a filter is non-empty and not equal to the whole of \(\mathfrak{A}\).
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Acknowledgements
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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Clements, K., Dowker, F., Wallden, P. (2017). Physical Logic. In: Cooper, S., Soskova, M. (eds) The Incomputable. Theory and Applications of Computability. Springer, Cham. https://doi.org/10.1007/978-3-319-43669-2_3
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