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Explaining probability

Simon Saunders

The explanans:

….tracked by credence (the ‘principal principle’)

… unobservable (no ‘probability meter’)

….unaffected by lucky charms

….manifested, but only with high credence, and only in

statistics of repeated trials

….bound up (or credence is bound up) with uncertainty

Physical probability is….

The explanandum:

Probabilities are relative weights of branches in EQM

There is no unitary evolution



i.e. there is no ‘absolute probability’ meter.




i.e. there is no ‘relative probability’ meter either.

But is there a unitary evolution [???]

where the absolute value of c (respectively d) is a function of the absolute value of a or b?



(That is, using the principal principle: can relative probabilities at least be measured with high credence?)



(That is, using the principal principle: can relative probabilities at least be measured with high credence?)

Still not Possible!

There is however a unitary evolution

There is however a unitary evolution:

There is a unitary evolution:

i.e. relative probabilities can be manifested (with high credence) in the statistical structure of repeated trials, or of a single trial with a large number of systems prepared in the same state.

But why should credence track the numbers |c|2, etc?

Answer: the Deutsch-Wallace representation theorem(derivation of the principal-principle when ‘objective chance’is amplitude squared).

Or rather (because it is a theorem) it’s premise, ‘Equivalence’.

The explanans:

….tracked by credence (the ‘principal principle’)… unobservable (no ‘probability meter’)….unaffected by lucky charms….manifested, but only with high credence, and only in statistics of repeated trials ….bound (or credence is bound up) up with uncertainty


The explanans:

….tracked by credence (the ‘principal principle’)… unobservable (no ‘probability meter’)….unaffected by lucky charms….manifested, but only with high credence, and only in statistics of repeated trials ….bound up (or credence is bound up) with uncertainty


The explanans:

….tracked by credence (the ‘principal principle’)… unobservable (no ‘probability meter’)….unaffected by lucky charms….manifested, but only with high credence, and only in statistics of repeated trials ….bound up (or credence is bound up) with uncertainty


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The world is, it does not happen. Only to the gaze of consciousness, crawling up the life-line of your body, does it fleetingly come to life (Weyl)

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The world is, it does not happen. Only to the gaze of consciousness, crawling up the life-line of your body, does it fleetingly come to life (Weyl)

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In a block universe…..

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There is no changeIn a block universe…..

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There is no change

•Time does not flow


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There is no change


•There is no becoming


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There is no change


•There is no becoming

•There is no free-will


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Change resides in the relations among events throughout your life-line W


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•The flow of time is….

•Becoming is…

•Free-will is…


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•Becoming is…

•Free-will is…

•‘I’ uttered at S refers to an ordered pair <S,W>

Utterance of ‘I’ at S


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•Becoming is…

•Free-will is…

•‘I’ uttered at S refers to an ordered pair <S,W>

Utterance of ‘I’ at S

Truth condition for ‘E will happen to me’ uttered at S is ‘W contains E after S’


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In a branching universe….

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There is no probability


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There is no probability •There is no such thing as uncertainty


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There is no probability •There is no such thing as uncertainty•There is nothing about which to be ignorant


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Probabilities are relative weights of branches


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Probabilities are relative weights of branches• There is uncertainty about…


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Probabilities are relative weights of branches• There is uncertainty about…• There is ignorance of….


Probabilities are relative weights of branches• There is uncertainty about…• There is Ignorance of…..


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Probabilities are relative weights of branches• There is uncertainty about….• There is ignorance of ….• ‘I’ uttered at S refers to ordered pair <S,Entire>


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S

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Justificatory condition for ‘I’m uncertain about E’ uttered at S is ‘E lies in some branch in <S,Entire> and not in others’

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E F G



Justificatory condition for ‘I’m uncertain about E’ uttered at S is ‘E lies in some branch in <S,Entire> and not in others’

But: ‘E will happen’ had better not have a truth value – failure of bivalence

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Probabilities are relative weights of branches• There is uncertainty about…• There is Ignorance of…..• ‘I’ refers to….


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Probabilities are relative weights of branches• There is uncertainty about….• There is ignorance of ….• ‘I’ refers to a unique continuant

In a branching universe…

…attribute utterances to unique continuants

.

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Probabilities are relative weights of branches• There is uncertainty about….• There is ignorance of ….• ‘I’ refers to a unique continuant

…..attribute utterances to unique continuants

.

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S

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W1 W2 W3

Probabilities are relative weights of branches• There is uncertainty about….• There is ignorance of ….• ‘I’ is uttered at S by each unique continuant overlapping at S

….attribute utterances to unique continuants

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W1 W2 W3

Probabilities are relative weights of branches• There is uncertainty about …• There is ignorance of …. • ‘I’ is uttered at S by each unique continuant overlapping at S


Justificatory condition for ‘I am uncertain about E’ uttered by Wk

at S is that Wk at S does not the know he isWk’

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W1 W2 W3

Probabilities are relative weights of branches• There is uncertainty about E• There is ignorance of who I am • ‘I’ is uttered at S by each unique continuant overlapping at S




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S

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W1 W2 W3

Probabilities are relative weights of branches• There is uncertainty about E• There is ignorance of who I am • ‘I’ is uttered at S by each unique continuant overlapping at S

Alternatively: attribute utterances to unique continuants



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S

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W1 W2 W3

‘E will happen’ uttered by <Wk,S> is true iff and only if E happens to Wk - Bivalence

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W1 W2 W3

Transition amplitudes, branch weights

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Measure over histories

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W1 W2 W3

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S S S

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<S,W1> <S,W2> <S,W3>

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W1 W2 W3

<E,W1><F,W2> <G,W3>

<S,W1> <S,W2> <S,W3>

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<E,W1><F,W2> <G,W3>

<S,W1> <S,W2> <S,W3>

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Transition amplitudes

Overlap Divergence

E.g. McDaniel (in Lewisian Themes) writes:

Despite David Lewis’s impressive arguments for modal realism, it is safe to say that it has won few adherents. Moreover, modal realism with overlap – the view that objects are literally present at more than one world – is probably the least popular position to occupy in modal metaphysics. Consequently, the possibility that a version of modal realism with overlap might be a serious contender has been virtually ignored. (p.140).

Lewis has exactly two arguments against a metaphysics of overlapping worlds (in Plurality of Worlds):

1. It faces ‘the problem of accidental intrinsics’

2. It renders incoherent our ordinary thinking of the future

Lewis has exactly two arguments against a metaphysics of overlapping worlds (in Plurality of Worlds):

1. It faces ‘the problem of accidental intrinsics’

2. It renders incoherent our ordinary thinking of the future

What does it mean to say the future holds a sea fight? …we have three options, (1) It is false that the future holds a sea flight; because ‘the future’ is a denotationless improper description. (2) It is true that the future holds a sea fight; because ‘the future’ denotes neither of the two partial futures but rather their disunited sum, which does hold a sea fight. (3) It is neither true nor false that the future holds a sea fight; because ‘the future; has indeterminate denotation, and we get different truth values on different resolutions of the indeterminacy.

Respect for common sense gives us reason to reject any theory that says that we ourselves are involved in branching….. But we needn’t reject the very possibility that a world branches. The unfortunate inhabitants of such a world, if they think of ‘the future’ as we do, are of course sorely deceived, and their peculiar circumstances do make nonsense of how they ordinarily think. But that is their problem, not ours….. (Plurality of Worlds p.209).

In divergence also there are many futures; that is, there are many later segments of worlds that begin by duplicating initial segments of our world. But in divergence, only one of those futures is truly ours. The rest belong not to us but to our other-worldly counterparts. Our future is the one that is part of the same world as ourselves. It alone is connected to us by the relations – the (strictly or analogically) spatiotemporal relations, or perhaps natural external relations generally – that unify a world. It alone is influenced causally by what we do and how we are in the present. We wonder which of one is the future that has the special relation to ourselves. We care about it in a way that we do not care about all the other-worldly futures. Branching, and the limited overlap it requires, are to be rejeced as making nonsense of the way we take ourselves to be related to our futures; and divergence without overlap is to be preferred. (op cit. p.208).

In divergence also there are many futures; that is, there are many later segments of worlds that begin by duplicating initial segments of our world. But in divergence, only one of those futures is truly ours. The rest belong not to us but to our other-worldly counterparts. Our future is the one that is part of the same world as ourselves. It alone is connected to us by the relations – the (strictly or analogically) spatiotemporal relations, or perhaps natural external relations generally – that unify a world. It alone is influenced causally by what we do and how we are in the present. We wonder which of one is the future that has the special relation to ourselves. We care about it in a way that we do not care about all the other-worldly futures. Branching, and the limited overlap it requires, are to be rejected as making nonsense of the way we take ourselves to be related to our futures; and divergence without overlap is to be preferred. (op cit. p.208).

The shared stage S does the thinking for both of the continuants to which it belongs. Any thought it has must be shared. It cannot desire one thing on behalf of C1 and another on behalf of C2. If it has an urgent, self-interested desire for survival on the part of C1, that very thought must also be an urgent, self-interested (and not merely benevolent) desire for survival on the part of C2 . It is not possible that one thought should be both. So it is not possible for S to have such a desire on behalf of C1. So it is not possible for C2 at t to have the straight-forward commonsensical desire that he himself survive. (‘Survival and Identity, p.74)

Against the proposed semantics, directly:

The ‘me’ in their shared thought (unless it refers to the thinking stage) has the status of an improper description. It cannot refer to C1 in C1 ‘s thought and to C2 in C2 ‘s thought, for these thoughts are one and the same. (p.75)

Against the proposed semantics, directly:

Two tales of Schrodinger’s cat………

A shot rang out in the dark.

The cat hissed. Peter turned to Susan and said




‘It’s the end’





‘No’ said Susan, grabbing the cat, ‘there must be a way out’.


The vet looked apologetic. The operation had taken much longer than expected.







he said, ‘he’ll be right as rain now’. Susan was delighted.


‘

‘




‘

‘











‘No’ said Susan, grabbing the cat, ‘there must be a way out’.

The vet looked apologetic. The operation has taken much longer than expected.

he said, ‘he’ll be right as rain now’. Susan was delighted.



Two reasons this semantics may have been ignored:

1. In the context of modal realism, it does not provide de re identities

2. In the context of Parfitian fision, ‘tensed identity’ is artificial.


What is wanted in de re identities? McDaniel:

“It is possible that Al Gore won the 2000 U.S. presidential election” must include, in some intimate sense of ‘include’, the object that the claim is about.

But

Al Gore could have won the 200 U.S. presidential election because a part of him is a part of an Al Gore who did win the 2000 U.S. presidential election.


What is wanted in de re identities? McDaniel:

“It is possible that Al Gore won the 2000 U.S. presidential election” must include, in some intimate sense of ‘include’, the object that the claim is about.

But

Al Gore could have won the 200 U.S. presidential election because a part of him is a part of an Al Gore who did win the 2000 U.S. presidential election.

Is that intimate enough?


Thus Lewis:

Then what do we say when a stage shared between two (or more) people is present? Strictly speaking, two people are present there by way of that one stage, but the fact that there are two is extrinsic to the time in question. It seems for all the world that there is only one. We will have to say something quite counter-intuitive, but we get a choice of evils.


We could say that there are two people; or that there is one, but really we’re counting stages rather than people; or that there is one, and we’re counting people, but we’re not counting all the people who are present; or that there is one, and we’re counting people, but we’re not counting them by identity…..it really isn’t nice to have to say any of these things – but after all, we’re talking about something that doesn’t really ever happen to people except in science fiction stories and philosophy examples, so is it really so very bad that peculiar cases have to get described in peculiar ways? We get by because ordinary cases are not pathological. But modality is different: pathology is everywhere...(Plurality of Worlds, p.218)

The special conditions that have to be in place if this semantics is to be serviceable to metaphysicians:

1. Branching is massive (yielding de re identities for modal realists)



2. All objects in the environment branch as well (yielding identity conditions similar to tensed identity).



2. All objects in the environment branch as well (yielding identity conditions similar to tensed identity).

3. There had better be an equivariant probability measure over all branches at each time.

Conclusion:

Given these three conditions, the natural semantics for branching is the same as that for divergence – overlapping worlds differ little from diverging worlds.

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