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arXiv:1409.0755 · Synthese

The logic of the future in quantum theory

Anthony Sudbery

Published Jul 8, 2016• 11 citations
arXivHTMLPDF

Abstract

According to quantum mechanics, statements about the future made by sentient beings like us are, in general, neither true nor false; they must satisfy a many-valued logic. I propose that the truth value of such a statement should be identified with the probability that the event it describes will occur. After reviewing the history of related ideas in logic, I argue that it gives an understanding of probability which is particularly satisfactory for use in quantum mechanics. I construct a lattice of future-tense propositions, with truth values in the interval [0, 1], and derive logical properties of these truth values given by the usual quantum-mechanical formula for the probability of a history.

DOI
10.1007/s11229-016-1142-9

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According to quantum mechanics, statements about the future made by sentient beings like us are, in general, neither true nor false; they must satisfy a many-valued logic. I propose that the truth value of such a statement should be identified with the probability that the event it describes will occur. After reviewing the history of related ideas in logic, I argue that it gives an understanding of probability which is particularly satisfactory for use in quantum mechanics. I construct a lattice of future-tense propositions, with truth values in the interval [0, 1], and derive logical properties of these truth values given by the usual quantum-mechanical formula for the probability of a history.
I discuss the problems of probability and the future in the Everett-Wheeler understanding of quantum theory. To resolve these, I propose an understanding of probability arising from a form of temporal logic: the probability of a future-tense proposition is identified with its truth value in a many-valued and context-dependent logic. I construct a lattice of tensed propositions, with truth values in the interval [0,1][0,1][0,1], and derive logical properties of the truth values given by the usual quantum-mechanical formula for the probability of histories.
I discuss the meaning of probability in the Everett‐Wheeler interpretation of quantum mechanics, together with the problem of defining histories. To resolve these, I propose an understanding of probability arising from a form of temporal logic: the probability of a future‐tense proposition is identified with its truth value in a many‐valued and context‐dependent logic. In short, probability is degree of truth. These ideas relate to traditional naive ideas of time and chance. Indeed, I argue that Everettian quantum mechanics is the only form of scientific theory that truly incorporates the perception that the future is open.
I discuss the meaning of probability in the Everett‐Wheeler interpretation of quantum mechanics, together with the problem of defining histories. To resolve these, I propose an understanding of probability arising from a form of temporal logic: the probability of a future‐tense proposition is identified with its truth value in a many‐valued and context‐dependent logic. In short, probability is degree of truth. These ideas relate to traditional naive ideas of time and chance. Indeed, I argue that Everettian quantum mechanics is the only form of scientific theory that truly incorporates the perception that the future is open.
The quantum-mechanical description of the world, including human observers, makes substantial use of entanglement. In order to understand this, we need to adopt concepts of truth, probability and time which are unfamiliar in modern scientific thought. There are two kinds of statements about the world: those made from inside the world, and those from outside. The conflict between contradictory statements which both appear to be true can be resolved by recognising that they are made in different perspectives. Probability, in an objective sense, belongs in the internal perspective, and to statements in the future tense. Such statements obey a many-valued logic, in which the truth values are identified as probabilities.
In quantum logic, i.e., within the structure of the Hilbert lattice imposed on all closed linear subspaces of a Hilbert space, the assignment of truth values to quantum propositions (i.e., experimentally verifiable propositions relating to a quantum system) is unambiguously determined by the state of the system. So, if only pure states of the system are considered, can a probability measure mapping the probability space for truth values to the unit interval be assigned to quantum propositions? In other words, is a probability concept contingent or emergent in the logic of quantum propositions? Until this question is answered, the cause of probabilities in quantum theory cannot be completely understood. In the present paper it is shown that the interaction of the quantum system with its environment causes the irreducible randomness in the relation between quantum propositions and truth values.