Clase DetectionActivity

To firmly fix how the probabilities that I posit operate and how they relate to more

familiar probabilities, let us look at a particular example. First, some terminology must be

introduced. I will refer to two different sorts of chance argued for above as “macrochance”

and “microchance”. The macrochance at a time of the macrostate that actually obtains at that

time is 1, and so the macrochance at a time of some other macrostate (or microstate)

obtaining (at any time) is implicitly conditional on the actual macrostate that obtains at that

time. Similarly for microchance, the microchance at a time of the microstate that obtains at

that time is 1, and so the microchance at a time of some other microstate (or macrostate)

obtaining is implicitly conditional on the actual microstate that obtains at that time. For

shorthand, I will refer to the macrochance of an event as ‘CH’ and the microchance of an

event as ‘ch’.

Now, for the example that follows, let ‘S’ stand for some macrostate instantiated at t1

and let ‘s’ stand for some microstate instantiated at t1. Let ‘R’ stand for some macrostate

instantiated at t2 and let ‘r’ stand for some microstate instantiated at t2. The following

diagram describes this situation, along with a specification of the various chances of each

         CHt1(St1) cht2(Rt2) S CHt1(Rt2|St1) R t1 t2        CH(st1|St1)      ch(Rt2|rt2) s cht1(rt2|st1) r cht1(st1) CHt1(st1) Figure 7

Let us consider first cht1(st1), which is of the general form cht(st`). This is the

microchance of s. As already mentioned, cht1(st1)=1, since the microchance at a time of the

conditional on the microstate at that time, the value of this chance varies over time. For

example, in the example depicted above, the microchance of r at t2 may be different at t1 than

it is at t2. Microchances need not always be extremal, and will not be in a fundamentally

indeterministic world. In the example above, for instance, cht1(rt2)=cht1(rt2|st1), which may

well be non-maximal.

Now, consider next CHt1(St1) and its general form CHt(St’). This is the macrochance

of S, and it behaves much like the microchance of s worked above. Again, CHt1(St1)=1 since

the macrochance at a time of the macrostate that obtains at that time is one. Since

this chance varies over time. For example, in the example above, CHt1(Rt2) might not equal

CHt2(Rt2). These chances, like their micro-counterparts above, need not be maximal, and will

not be in a macro-indeterministic world. In the above diagram, for instance,

CHt1(Rt2)=CHt1(Rt2|St1), which may well be non-maximal.

Next consider cht1(rt2|st1), which is of the general form cht(rt`|st``). This is the

microchance of s given r, and is perhaps the most familiar sort of chance. For example,

David Lewis’ history-to-chance conditionals express this sort of microchance17. The value

of this chance, like the chances above, varies over time, and these values may be non-

extremal, depending on whether or not the world is fundamentally deterministic. For

example, in the above diagram, cht1(rt2|st1) may well be non-maximal, but, if the chance is

indexed to t2 instead, its value is maximal. That is, cht2(rt2|st1)=1.

The macro-counterpart to cht1(rt2|st1) is CHt1(Rt2|St1), and its general form is

CHt(Rt’|St``). This is the macrochance of R given S, and is the sort of chance we typically

think of when we think of macrochances. For example, the chance that a coin will land

heads given that it is flipped is this sort of conditional chance. It operates analogously to the

above conditional microchances. For example, as the indexing to the t indicates, its value

changes over time, and if the world is macro-indeterministic, its value can be non-extremal.

In the example above, for instance, CHt1(Rt2|St1) might be non-maximal, but CHt2(Rt2|St1)=1.

So much for the relatively uncontroversial chances, and on now to the particular sorts

of chances I posit to handle the counterexamples to supervenience. CH(st1|St1) is the

macrochance of a microstate at a time given a macrostate at that time. Its general form is

CH(st|St). The first thing to notice about this macrochance is that it is not time indexed,

because its value does not change over time. The chance, for instance, that in the far away       

17

 Lewis (1980) 

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future some particular microstate will obtain given that some particular macrostate obtains, is

the same right now as it will be in that distant future. The second interesting feature of this

chance is that the macrostate and the microstate are indexed to the same time. This is

reflected in the intuition that, for example, that there is some chance that microstate a obtains

at a time, given that the roulette wheel is vigorously spinning at that time.

Finally, the value of these chances is typically non-maximal, even though the world

may be fundamentally deterministic, or macro-deterministic, or both. Again, this is reflected

in the intuition that the chance of, say, microstate a given that the macrostate is a vigorously

spun roulette wheel, is lower, but greater than zero, than the chance of microstate a given that

wheel’s spin is biased, which is higher, but less than, one. And, all of this is so regardless of

whether, say, the world is fundamentally deterministic.

Next up is CHt1(st1), which has the general form CHt(st’). This is the macrochance

that a particular microstate obtains. This chance is, like other macrochances, is implicitly

conditional on the macrostate that obtains at the time the chance is indexed to, so its value

changes over time. This implicit conditionalization, however, causes controversy for this

chance form. Consider a case where t and t’ have the same value, as in the example above.

Then, the value of CHt1(st1)=CH(st1|St1) if S obtains at t1. If s could realize some other

macrostate, say, T, then CHt1(st1)=CH(st1|Tt1) if T obtains. If no macrostate that s could

realize obtains at the time in question, CHt1(st1)=0. As a result of these equalities, CHt(st) is

typically non-maximal, even when s obtains.

Initially, this seems counter-intuitive. If s obtains, the chance of s seems like it

should be one. To sooth this intuition, it is important to keep in mind that, when s obtains,

work as well, and that is CHt(St). Further, if we are careful, perhaps this result is not really

all that counter-intuitive. Suppose I walk up to a casino dealer who holds in her hand a deck

of cards and I ask her to fan out the cards on the table to reveal their arrangement. I point to

that arrangement and I say “What is the chance of that arrangement?” My question is, I

submit, ambiguous. On one reading of my question, the appropriate answer is “One. After

all, it is the way that the cards are arranged.” This answer gives the microchance of the

arrangement at the time my question is asked, which is, I admit, one. On another, and

intuitively more helpful, reading of my question, the right answer is, “The chance is small,

and equal to the chance of any other arrangement. After all, it is a fair deck.” This answer

gives the macrochance of the arrangement, which, since it is a fair deck, is non-maximal. It

seems, then, that as long as we are careful to keep track of which probability we are

interested in, it is not that strange that the there are chances of certain microstates obtaining

that are non-maximal, even at the time that they obtain.

We end on a hopefully less controversial note with ch(Rt2|rt2), which is of the general

form ch(Rt|rt), and cht2(Rt2), which is of the general form cht(Rt`). The value of ch(Rt|rt) is

extremal, regardless of whether or not the world is deterministic, and is 1 if R obtains at 0

otherwise. The idea here is that the probability that a microstate realizes the macrostate that

it, in fact, does, is (as one would expect) one. So, in the above example, ch(Rt1|rt1) is one.

The value of cht(Rt`) changes over time, and need not be extremal. For example, the

value of cht1(Rt2) may be non-maximal if the world is fundamentally deterministic, but

cht2(Rt2) is extremal (and equal to ch(Rt2|rt2)), regardless of whether or not the world is

deterministic.

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