We made rather merry at the Chapter House last night, an annual tradition. Class begins in two hours. I can still hear the vuvus. Sheesh.
Statements about the unknown made with reference to the known are common enough. The example David Hume gave us is, since all the many flames I have observed before have been hot, I expect the next flame will be hot.
That kind of inference is inductive. The outcome itself—the future flame being hot—is not guaranteed by the universe to be true. By which I mean, there are no set of true premises known to us that allow us to deduce the statement “the next flame will be hot.” It could be, for example, that there exists some mystery physics which will come into play with the next flame, causing it not to be hot.
A deductive inference known to everybody (from yesterday) starts with the premises, “All men are mortal, and Socrates is a man.” We supply the conclusion of interest. If we supply, “Socrates is mortal”, then our intuitions and the given premises tell us the probability of the conclusion is one.
But if we supplied “Socrates the immortal man has the flu” then given just our stated premises we cannot say anything about the probability of this conclusion, because, based on our knowledge of common English words and names, the conclusion is not related to the premises.
In this situation, it is easy to see that the conclusion and premises are unrelated; but of course it isn’t always so easy to gauge the interrelatedness of premises and conclusions. Just think of politics to know what I mean.
Supplying reasonable premises—which are nothing but data, models, and judgments by another name—and relevant conclusions is the basis of all probability modeling. Two people with different premises (models, data) but the same conclusion can come to different probabilities of the truth of that conclusion.
Both of those probabilities are correct, given the premises. It is always the premises—the evidence—that are in dispute.
The important thing to take away is that the truth, falsity, or probability of any statement cannot be made without reference to something. We need some evidence, premises, information upon which to condition any statement.
The probability of a die showing a ‘6’ is not “1/6″l; but the probability of a die showing a ‘6’ given that the die is a six-sided object with just one side labeled ‘6’ and that only one side will show, is 1/6.
This probability, given these premises, has been deduced. If you want to bring up questions about this or that real, physical die, well, then you are adding premises, different evidence. If all—which means everything—you know is that you have a real, physical die with six sides, just one of which is painted with six dots, then the probability of the conclusion is 1 in 6.
If you’re interested whether this next toss will result in a six, well, that’s the same thing. If you’re interested in how many times a six will show in the next dozen tosses—its relative frequency—then that is easy to compute, too (we’ll learn how tomorrow).
If you add the premise that “Something might be wrong with this die; something which subtracts from its symmetry”, then this is equivalent to “The die is asymmetric or it is not.” That is a tautology and is always true, no matter what the state of the universe and no matter whether your real die is symmetric or not.
Tautologies add no information, they cannot change the probability of any conclusion. Pause and think hard about this, because this important fact confuses. If we want to learn about this die we have to change the argument, especially the conclusion. We also need to add premises in the form of observations of actual die tosses. We’ll learn how to do that later, too.
As we have long discussed, we do not need to add any premises about “randomness” or “fair dies” to our premises. Random only means “unknown” and nothing more. Of course the conclusion is not certain: we cannot deduce a ‘6’ will show (without adding premises). The outcome is uncertain, which means it has a probability greater than zero but less than one.
Adding words about “fair” die is equivalent to saying “the probability of a ‘6’ is 1/6” to the list of premises. This is not wrong, but it makes the argument circular. It puts the conclusion in the premises, from which we can deduce the conclusion.