PyramidHead
Contributor
Suppose I am an omniscient being who knows the position and trajectory of all particles in the universe at any given time, like Laplace's demon.
I am assigned a lottery number by a computer when I buy a ticket (I do not pick the number myself). Looking at the number, and comparing it to what I know will be the result of the lottery tomorrow night based on my perfect knowledge of the future, I see that the number I was just assigned is the winning number.
It seems that I can now simultaneously hold two true beliefs about the situation:
1. There is a 100% probability that the number I was assigned will be the winning lottery number tomorrow.
(I believe this because I know the exact way the lady on TV will reach into the little container, the exact position of all the numbered balls, etc.)
2. The fact that I was randomly assigned the winning lottery number is an improbable coincidence.
(I believe this because there were many more trajectories the future could have taken that did not result in my number being picked, compared to the restricted number of trajectories in which it was picked.)
Even in a fully determined universe, the proportion of ways in which a fair lottery drawing could follow a path determined by initial conditions and governing laws and result in my independently selected number being picked is necessarily smaller than the ways it could have resulted in a number other than mine being picked, if the initial conditions and governing laws were slightly different than they were. Thus, I am presented with what seems like a paradox: tomorrow's lottery drawing will unfold in a completely expected, unsurprising manner from my omniscient vantage point, but its outcome will nonetheless be improbable from that same vantage point.
Clearly there are two forces at play here. One is my available knowledge. The more knowledge I have, the more confident I can be of some future event happening or not; the probability I assign to it will approach 0 or 1 in light of each new piece of data. Since my knowledge is infinite in this example, I assign a probability of 1 to the event.
The other is the coincidental relationship between an event and an independent designation, which in this example is the coincidence between the computer assigning me a number and the lady drawing that number from the container tomorrow. Regardless of how much knowledge I possess, as long as these two events were not connected in any way, there will have been many more ways for the two numbers to be different than ways for them to be the same.
One way to resolve this tension is to say that the first statement is a misuse of the word 'probability'. Confidence, expectation, and surprise are not measurements of actual probability, even though they may accompany some assignments of probability (because we are humans and not omniscient beings). It would be more accurate to say:
1. I know that the number I was assigned will be the winning lottery number tomorrow.
Which has nothing to do with improbability, and so doesn't say anything about improbability. This separates the epistemological aspect from the statistical aspect of the event. Actual improbability would depend on the intersection between an event--no matter how predictable--and an unrelated designation of the same event, at the same level of description. So, while we may use probability inferences as a substitute for epistemic confidence when we lack complete knowledge, strictly speaking they are not saying the same things.
For example, when CNN reports the projected winner of an election based on the votes so far, the number they display is not really a probability (which in a two-person race is always 50/50) but a measurement of how justified a certain expectation is in light of the evidence on hand.
Does this make sense?
I am assigned a lottery number by a computer when I buy a ticket (I do not pick the number myself). Looking at the number, and comparing it to what I know will be the result of the lottery tomorrow night based on my perfect knowledge of the future, I see that the number I was just assigned is the winning number.
It seems that I can now simultaneously hold two true beliefs about the situation:
1. There is a 100% probability that the number I was assigned will be the winning lottery number tomorrow.
(I believe this because I know the exact way the lady on TV will reach into the little container, the exact position of all the numbered balls, etc.)
2. The fact that I was randomly assigned the winning lottery number is an improbable coincidence.
(I believe this because there were many more trajectories the future could have taken that did not result in my number being picked, compared to the restricted number of trajectories in which it was picked.)
Even in a fully determined universe, the proportion of ways in which a fair lottery drawing could follow a path determined by initial conditions and governing laws and result in my independently selected number being picked is necessarily smaller than the ways it could have resulted in a number other than mine being picked, if the initial conditions and governing laws were slightly different than they were. Thus, I am presented with what seems like a paradox: tomorrow's lottery drawing will unfold in a completely expected, unsurprising manner from my omniscient vantage point, but its outcome will nonetheless be improbable from that same vantage point.
Clearly there are two forces at play here. One is my available knowledge. The more knowledge I have, the more confident I can be of some future event happening or not; the probability I assign to it will approach 0 or 1 in light of each new piece of data. Since my knowledge is infinite in this example, I assign a probability of 1 to the event.
The other is the coincidental relationship between an event and an independent designation, which in this example is the coincidence between the computer assigning me a number and the lady drawing that number from the container tomorrow. Regardless of how much knowledge I possess, as long as these two events were not connected in any way, there will have been many more ways for the two numbers to be different than ways for them to be the same.
One way to resolve this tension is to say that the first statement is a misuse of the word 'probability'. Confidence, expectation, and surprise are not measurements of actual probability, even though they may accompany some assignments of probability (because we are humans and not omniscient beings). It would be more accurate to say:
1. I know that the number I was assigned will be the winning lottery number tomorrow.
Which has nothing to do with improbability, and so doesn't say anything about improbability. This separates the epistemological aspect from the statistical aspect of the event. Actual improbability would depend on the intersection between an event--no matter how predictable--and an unrelated designation of the same event, at the same level of description. So, while we may use probability inferences as a substitute for epistemic confidence when we lack complete knowledge, strictly speaking they are not saying the same things.
For example, when CNN reports the projected winner of an election based on the votes so far, the number they display is not really a probability (which in a two-person race is always 50/50) but a measurement of how justified a certain expectation is in light of the evidence on hand.
Does this make sense?
