The Incomputable pp 47-61 | Cite as

# Physical Logic

## 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.

## Keywords

Physical World Classical Logic Event Algebra Quantum Logic Classical Physic## Notes

### 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.

## References

- 1.R.P. Feynman, R.B. Leighton, M. Sands,
*Lectures on Physics*, vol. iii (Addison-Wesley, 1965)Google Scholar - 2.R. Sorkin, Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A
**9**(33), 3119–3127 (1994)MathSciNetCrossRefzbMATHGoogle Scholar - 3.R. Sorkin, Quantum measure theory and its interpretation, in
*Quantum Classical Correspondence: Proceedings of the 4th Drexel Symposium on Quantum Nonintegrability*, ed. by D. Feng, B.-L. Hu (International Press, Cambridge, MA, 1997), pp. 229–251. Preprint gr-qc/9507057v2Google Scholar - 4.R. Sorkin, Quantum dynamics without the wavefunction. J. Phys. A: Math. Theor.
**40**(12), 3207–3221 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 5.R. Sorkin, An exercise in “anhomomorphic logic”. J. Phys.: Conf. Ser.
**67**, 012018 (8 pp.) (2007)Google Scholar - 6.R. Sorkin, To what type of logic does the “Tetralemma Belong? (2010). Preprint 1003.5735Google Scholar
- 7.R. Sorkin, Logic is to the quantum as geometry is to gravity, in
*Foundations of Space and Time: Reflections on Quantum Gravity*, ed. by G. Ellis, J. Murugan, A. Weltman (Cambridge University Press, Cambridge, 2011). Preprint 1004.1226v1Google Scholar - 8.F. Dowker, Y. Ghazi-Tabatabai, The Kochen–Specker theorem revisited in quantum measure theory. J. Phys. A: Math. Theor.
**41**(10), 105301-1–105301-17 (2008)Google Scholar - 9.F. Dowker, G. Siret, M. Such, A Kochen-Specker system in histories form (2016, in preparation)Google Scholar
- 10.J. Henson, Causality, Bell’s theorem, and Ontic Definiteness (2011). Preprint 1102.2855v1Google Scholar
- 11.F. Dowker, D. Benincasa, M. Buck, A Greenberger-Horne-Zeilinger system in quantum measure theory (2016, in preparation)Google Scholar
- 12.L. Carroll, What the tortoise said to Achilles. Mind
**4**(14), 278–280 (1895)CrossRefGoogle Scholar - 13.N. Rescher, R. Brandom,
*The Logic of Inconsistency*(Basil Blackwell, Oxford, 1979)zbMATHGoogle Scholar - 14.S. Kochen, E. Specker, The problem of hidden variables in quantum mechanics. J. Math. Mech.
**17**(1), 59–87 (1967)MathSciNetzbMATHGoogle Scholar - 15.J. Bell, On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys.
**38**(3), 447–452 (1966)MathSciNetCrossRefzbMATHGoogle Scholar - 16.D. Greenberger, M. Horne, A. Zeilinger, Going beyond Bell’s theorem, in
*Bell’s Theorem, Quantum Theory and Conceptions of the Universe*, vol. 37, ed. by M. Kafatos. Fundamental Theories of Physics (Kluwer, Dordrecht, 1989), pp. 69–72Google Scholar - 17.D. Greenberger, M. Horne, A. Shimony, A. Zeilinger, Bell’s theorem without inequalities. Am. J. Phys.
**58**(12), 1131–1143 (1990)MathSciNetCrossRefzbMATHGoogle Scholar - 18.N. Mermin, Quantum mysteries revisited. Am. J. Phys.
**58**(8), 731–734 (1990)MathSciNetCrossRefGoogle Scholar - 19.N. Mermin, What’s wrong with these elements of reality? Phys. Today
**43**(6), 9–11 (1990)CrossRefGoogle Scholar - 20.J.B. Hartle, The spacetime approach to quantum mechanics. Vistas Astron.
**37**, 569–583 (1993)MathSciNetCrossRefGoogle Scholar - 21.J. Hartle, Spacetime quantum mechanics and the quantum mechanics of spacetime, in
*Gravitation and Quantizations: Proceedings of the 1992 Les Houches Summer School (Les Houches, France, 1992)*, vol. LVII, ed. by B. Julia, J. Zinn-Justin (Elsevier, Amsterdam, 1995). Preprint gr-qc/9304006v2Google Scholar