Is it written into sport announcers’ contract that they shall speak in nothing but cliché?
Since there is always great confusion about why premises about “fairness” or “randomness” are not needed, we had better cover it in a main post.
(P1) I have a six-sided object, just one side of which is labeled ‘6’.
(P2) Upon tossing, only one side will show
(C) A ‘6’ will show.
The conclusion is not certain with respect to the premises. But we can say
Pr( C | P1 & P2 ) = 1/6.
Many would like to add this premise:
(P3) The die is “fair.”
But that is equivalent to
(P3) Each side of the die is equally likely to show,
which is the same as saying
(P3′) The probability of a ‘6’ is 1/6.
I can write
Pr( A ‘6’ will show | P1 & P2 & The probability of a ‘6’ is 1/6) = 1/6.
and since I do not now need (P1) or (P2), I can write
Pr( A ‘6’ will show | The probability of a ‘6’ is 1/6) = 1/6.
Or in plain English, “The probability a ‘6’ will show given the probability a ‘6’ will show is 1/6 is 1/6.”
This is a fine argument. It is valid. That means the conclusion has been deduced with certainty from the premises. Arguments which have conclusions that are deduced are the strongest there is, and isn’t that glorious.
But the argument is circular: its premises contain the conclusion, the thing we wanted to prove. Aristotle showed us that this argument is the same as saying, “A, therefore A,” where A is any statement.
So give up on (P3), because it is not needed. Instead, think carefully about (P1) and (P2); think hard about what they do not tell us. What words are there in (P1) that tell us that the sides are symmetric, that they are weighted equally, that they are of the same substance, that they have the same friction, or the same of any other probative factor?
Not one word. Too, there is not a shred of evidence that any of these things—and of an infinite number of other things—are such that we should consider them.
There isn’t any word in (P2) about the gravitational field in which the six-sided object—we do not know it is square, nor that it is a die—will be tossed, nor about the amount of spin or other force imparted to the object, nor about viscosity of the air in which it will be tossed, nor do we know that there will even be air! We do not know about the elasticity of the surface upon which the object will land, nor about that surface’s roughness. Nor do we know about an infinite number of other things, each of which could influence the object showing a ‘6’.
If there are 12 Johnstons and 18 Freihaufs in a room and you will select the one closest to the door, then, given this evidence and no other, there is a 60% chance that you will select a Freihauf. Just as above, we do not need to add any evidence that the people in the room are, perhaps by polkaing, distributed “randomly.”
Given only the evidence we have—and no other tacked on imaginatively —we have no idea where anybody in the room is. It is the lack of information that makes the outcome “random”, which is to say, unknown.
The problem we have with these types of arguments is that we cannot help but add premises: the stated ones are never enough; we are always greedy for more. Adding (unstated) premises is like a mathematician changing axioms mid-proof to suit him so that he gets the result he desires. This behavior might be because the above arguments sound like situations in which we have a lot of experience. But here is another example, adapted from Stove, which shows that we should consider arguments as given.
(P1) Just half of all winged horses are yellow.
(P2) Bob is a winged horses.
(C) Bob is yellow.
Here, everybody always agrees that the probability that “Bob is yellow” given just (P1) and (P2) is 50%. We don’t feel the need to talk about “opaque bags” of winged horses, with half of an infinite amount of them yellow, and with the other half some other color or colors. We don’t have the gut feeling that if we were to know that that probability is 50%, that we need to repeat a “random” experiment with winged horses an infinite number of times.
This, obviously, is because there are no winged horses that we know of, yellow or not. Thus, we are able to tackle the argument in its intended sense.
“OK, Briggs. Maybe. But what about real dice?” Well, as to that, stick around.