# Non-monotonic Probability Theory and Photon Polarization

- 47 Downloads
- 5 Citations

## Abstract

A non-monotonic theory of probability is put forward and shown to have applicability in the quantum domain. It is obtained simply by replacing Kolmogorov’s positivity axiom, which places the lower bound for probabilities at zero, with an axiom that reduces that lower bound to minus one. Kolmogorov’s theory of probability is monotonic, meaning that the probability of *A* is less then or equal to that of *B* whenever *A* entails *B*. The new theory violates monotonicity, as its name suggests; yet, many standard theorems are also theorems of the new theory since Kolmogorov’s other axioms are retained. What is of particular interest is that the new theory can accommodate quantum phenomena (photon polarization experiments) while preserving Boolean operations, unlike Kolmogorov’s theory. Although non-standard notions of probability have been discussed extensively in the physics literature, they have received very little attention in the philosophical literature. One likely explanation for that difference is that their applicability is typically demonstrated in esoteric settings that involve technical complications. That barrier is effectively removed for non-monotonic probability theory by providing it with a homely setting in the quantum domain. Although the initial steps taken in this paper are quite substantial, there is much else to be done, such as demonstrating the applicability of non-monotonic probability theory to other quantum systems and elaborating the interpretive framework that is provisionally put forward here. Such matters will be developed in other works.

## Key words

Boolean operations probability quantum domain## Preview

Unable to display preview. Download preview PDF.

## References

- Allen, E. H.: 1976, Negative probabilities and the uses of signed probability theory,
*Philosophy of Science***43**, 53–70.CrossRefGoogle Scholar - Araki, G.: 1948, Theory of elliptically polarized photons,
*Physical Review***74**, 472–479.CrossRefGoogle Scholar - Birkhoff, G. and Von Neumann, J.: 1936, The logic of quantum mechanics,
*Annals of Mathematics***37**(2nd Series), 823–843.CrossRefGoogle Scholar - Feynman, R.: 1987, Negative probabilities, in B. J. Hiley and F. D. Peat (eds.),
*Quantum Implications*, Routledge, New York.Google Scholar - Foulis, D.: 1999, A half-century of quantum logic: What have we learned? in D. Aerts and J. Pykacz (eds.),
*Quantum Structures and the Nature of Reality*, Kluwer, Dordrecht.Google Scholar - French, A. P. and Taylor, E. F.: 1978,
*An Introduction to Quantum Physics*, Norton, New York.Google Scholar - Gudder, S.: 1988,
*Quantum Probability*, Academic, New York.Google Scholar - Howson, K. and Urbach, P.: 1993,
*Scientific Reasoning: The Bayesian Approach*, Open Court, Chicago.Google Scholar - Khrennikov, A.: 1999,
*Interpretations of Probability*, Brill Academic Publishers, Leiden, The Netherlands.Google Scholar - Kolmogorov, A. N.: 1950,
*Foundations of Probability Theory*, Chelsea, New York. (First published in German in 1933).Google Scholar - Loomis, L.: 1955, The lattice-theoretic background of the dimension theory of operator algebras,
*Memoirs of the American Mathematical Society***18**, 1–36.Google Scholar - Margenau, H.: 1963, Measurements and quantum states: Part II,
*Philosophy of Science***30**, 138–157.CrossRefGoogle Scholar - Mellor, H.: 1971,
*The Matter of Chance*, Cambridge University Press, Cambridge, UK.Google Scholar - Muckenheim, W.: 1986, A review of extended probabilities,
*Physics Reports***133**, 337–401.CrossRefGoogle Scholar - Popper, K.: 1968,
*The Logic of Scientific Discovery*, Harper and Row, New York.Google Scholar - Reichenbach, H.: 1944,
*Philosophic Foundations of Quantum Mechanics*, University of California Press, Berkeley.Google Scholar - Roberts, J.: (in progress), Non-monotonic probabilities as a guide to life? For more information please contact the author.Google Scholar
- Suppes, P. and Zanotti, M.: 1991, Existence of hidden variables having only upper probabilities,
*Foundations of Physics***21**, 1479–1499.CrossRefGoogle Scholar