As might have been obvious from yesterday, the truth, falsity, or the somewhere-in-betweenness of any conclusion-hypothesis-proposition can only be assessed with reference to a list of assumptions, premises, data. That is, you cannot know the status of any proposition without some list of premises. Different premises lead to different statuses.
In particular, this means that you cannot ask, “What is the probability of X?”, where X is some proposition. For example, you cannot ask, X = “What is the probability that I roll a six on a die?” This probability does not exist. Similarly, you cannot ask, Y = “What is the probability that Socrates is mortal?” This probability also does not exist. There are no unconditional probabilities, no unconditional arguments of any kind.
If I assume that E = “All men are mortal and Socrates is a man” then I can claim that “It is certain, given E, that Y”, or that “If I assume it is true, regardless whether or not it is or that I can know it is, that all men are mortal and Socrates is a man, then the probability that Socrates is mortal is 1.” Or I can write:
Pr( Y | E ) = 1.
But I cannot write:
Pr( Y ) = something,
for that is forever unknown. There just is no such thing as an unconditional probability, just as there are no such things as unconditional logical arguments, just as there are no such things as unconditional mathematical theorems, and so on. If you find yourself disagreeing, have a go at creating the probability of some hypothesis that does not make reference to any assumptions/premises/data.
(It’s only true that in most textbooks probability is written as if it were unconditional. While this makes life for the author and for typesetters, it ends up producing confusion about the nature of probability.)
It’s important to understand that E is only that which we assume is true. It matters not one whit whether E—with respect to some other set of premises—really is true, or false, or somewhere in between. Logic concerns itself only with the connections between premises and conclusions. The premises and conclusions are something exterior, something given to us.
Another hoary example. Let Ed = “A six-sided object, just one side of which is labeled 6, will be tossed and only one side can show.” Then if X = “A 6 shows”,
Pr( X | Ed ) = 1/6.
We have deduced—just as we do with all probabilities—the probability that X will be true. Notice that this says nothing about real dice in any real situation. This is just a logical argument, no different in nature from the premise “All Martians wear hats and George is a Martian” which lets us deduce that “George wears a hat.” This conclusion with respect to this evidence is true, it’s probability is 1; and this is so even though we know, with respect to observational evidence, that there are no Martians.
Now if we write X = “A Buick shows”, we can write
0 < Pr( X | Ed ) < 1
We are stuck because our evidence says nothing about a Buick. There may be a Buick on one of the other five sides, there may not. The evidence is mostly mute on this subject. Except if we suppose there is an implicit call to the contingent nature of this object being tossed. If we assume that, then we can at least say the probability is not 0 and not 1, but it may be anywhere in between. But we can also make the argument that Ed should be interpreted more strictly. If it is the case, then the best we can do is this:
Pr( X | Ed ) = unknown.
Probability cannot be relative frequency. For example, given “Half of all Martians wear hats and George is a Martian” which lets us deduce that the probability “George wears a hat” is 0.5. But there is no relative frequency of this “experiment.” This one counter-example is enough to show that the relative frequency interpretation of probability is false (it doesn’t show it has things backwards; for that, read the book, paying attention to the references).
Probability cannot be subjective in the following sense. If we accept that “All men are mortal and Socrates is a man” then the probability that “Socrates is mortal” is 1. Even if we don’t want it to be. And above the probability that “George wears a hat” cannot be anything but 0.5. Probability only appears to be subjective in some instances because we often are bad at listing the premises we hold when assessing probabilities. However, if we agree on the exact list of premises (and on the rules of logic) then we must agree on the probabilities deduced.
Probability cannot always be quantified. If we accept that “Some men are mortal and Socrates is a man” then the probability that “Socrates is mortal” is something between 0 and 1, but the exact number cannot be deduced.
Why doesn’t adding “the six-sided object is weighted” or “the six-sided object is fair” to Ed change the probability that Pr( X | Ed ) = 1/6?
List the exact premises you hold in your Pr (“Barack Obama will be re-elected” | your premises).
Change the evidence “All men are mortal and Socrates is a man” so that the probability of “Socrates is mortal” is bounded between two fixed numbers.
In a tearing hurry today. Did not have time to check for typos!