Forthcoming in Saunders, Barrett, Kent, and Wallace (ed.),

I develop the decision-theoretic approach to quantum probability, originally
proposed by David Deutsch, into a mathematically rigorous proof
of the Born rule in (Everett-interpreted) quantum mechanics. I sketch the
argument informally, then prove it formally, and lastly consider a number
of proposed "counter-examples" to show exactly which premises of the
argument they violate.

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Simon Saunders and DW,

A reply to comments on Saunders and Wallace (below) by Paul Tappenden (this paper)

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Simon Saunders and DW,

Following Lewis, it is widely held that branching worlds differ in important ways from
diverging worlds. There is, however, a simple and natural semantics under which ordinary
sentences uttered in branching worlds have much the same truth values as they
conventionally have in diverging worlds. Under this semantics, whether branching or
diverging, speakers cannot say in advance which branch or world is theirs. They are uncertain
as to the outcome. This same semantics ensures the truth of utterances typically
made about quantum mechanical contingencies, including statements of uncertainty, if
the Everett interpretation of quantum mechanics is true. The ‘incoherence problem’ of
the Everett interpretation, that it can give no meaning to the notion of uncertainty, is
thereby solved.

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I consider exactly what is involved in a solution to the probability
problem of the Everett interpretation, in the light of recent work on
applying considerations from decision theory to that problem. I suggest
an overall framework for understanding probability in a physical theory,
and conclude that this framework, when applied to the Everett
interpretation, yields the result that that interpretation
satisfactorily solves the measurement problem.

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In submission

I investigate the consequences for semantics, and in particular for the
semantics of tense, if time is assumed to have a branching structure not
out of metaphysical necessity (to solve some philosophical problem) but
just as a contingent physical fact, as is suggested by a
currently-popular approach to the interpretation of quantum mechanics.

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In the light of recent work suggesting that the quantum probability rule can be derived in the Everett interpretation via decision theory, I consider what physical features of quantum mechanics make this possible. I analyse the status of the probabiliy rule in three different models of branching universes, each somewhat more complicated than the last, and conclude that only in the last model --- in which the branching structure, as in quantum mechanics, emerges in a somewhat imprecise way from the underlying physical reality --- is it possible to derive a probability rule, or indeed to behave in any rational way at all.

**NOTE: I wrote this four years ago intending to submit it fairly shortly afterwards, but for various reasons I still haven't got around to it. For that reason, I don't mind people citing or quoting it now.**

(PostScript PDF)

I present a proof of the quantum probability rule from decision-theoretic assumptions, in the context of the
Everett interpretation. The basic ideas behind the proof are those presented in Deutsch's recent proof of the
probability rule, but the proof is simpler and proceeds from weaker decision-theoretic assumptions. This makes
it easier to discuss the conceptual ideas involved in the proof, and to show that they are defensible.

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An analysis is made of Deutsch's recent claim to have derived the Born rule
from decision-theoretic assumptions. It is argued that Deutsch's proof
must be understood in the explicit context of the Everett
interpretation, and that in this context, it essentially succeeds. Some
comments are made about the criticism of Deutsch's proof by Barnum, Caves,
Finkelstein, Fuchs, and Schack; it is argued that the flaw which they
point out in the proof does not apply if the Everett interpretation is
assumed.

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An extended analysis is given of the program, originally suggested by Deutsch, of solving the probability problem in the Everett interpretation by means of decision theory. Deutsch's own proof is discussed, and alternatives are presented which are based upon different decision theories and upon Gleason's Theorem. It is argued that decision theory gives Everettians most or all of what they need from `probability'. Some consequences of (Everettian) quantum mechanics for decision theory itself are also discussed.

**NOTE: this long (70 pages) and occasionally rambling paper has been almost entirely superseded by
material in the above papers; if something is not included in them it usually means that I have had second thoughts.
I include it for completeness only.**
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