On Fri, May 29, 1998 at 04:03:00AM +0000, Nick Bostrom wrote: > It seems we can interpret "I will observe X" as meaning: "There is a > future brain-state B2, similar in certain respects to the brain-state > B1 which instanciates this present cognition C1, such that B2 > instanciates C2, and C2 includes an observation of X.".

That is not going to give you nice results. For example if there is no wavefunction collapse, all possible brain-states exist in the future and "I will observe X" would have probability 1 for all X under your interpretation. I think I've found a way to define these kinds of probabilities that is consistent (it satisfies the axioms of probability theory) and still somewhat intuitive. Suppose we have a set I of observer-instants, a mapping G from observer-instants to their current perceptions and memories, a theory of identity, i.e., a function F that maps each observer-instant to the set of observer-instants that are identified with its future, and a probability measure Q on the observer-instants. Define: P(I currently observe X)=Q({i|X is in G(i)}) P(I did observe X)=Q({i|X is in G(j) for some j such that i is in F(j)}) P(I will observe X)=Q({i|there exist some j in F(i) such that X is in G(j)}) P(I will observe X and I currently observe Y)=Q({i|there exist some j in F(i) such that X is in G(j)} intersect {i|Y is in G(i)} (and similarly for other kinds of conjunctions) The basic idea is that "I will observe X" is interpreted as "A randomly distributed observer-instant has the property that some future version of it observes X", which seems obvious now that I think about it. For an example, consider the setup in the single-coin copying paradox. We have one observer-instant (a) at time 0 with measure 2/6, two observer-instants (b,c) at time 1 with measure 1/6 each, three observer-instants (d,e,f) at time 2 with measure 1/6 each. Also, F(a)={b,c,d,e,f}, F(b)={d,e}, and F(c)={f}. G(a)={clock reading 0} G(b)={clock reading 1, coin landing heads, memories of being a} G(c)={clock reading 1, coin landing tails, memories of being a} G(d)={clock reading 2, heads-up coin, memories of being a and b} G(e)={clock reading 2, heads-up coin, memories of being a and b} G(f)={clock reading 2, tails-up coin, memories of being a and c} According to the above definitions, P(I will see a heads-up coin and clock reading 2|I currently see clock reading 0) = 1/2, whereas P(I see a heads-up coin|I see clock reading 2) = 2/3. In general under this definition P(I currently observe X|I observed Y) <> P(I will observe X|I currently observe Y) whenever there is some possibility of copying or death between observations X and Y.