To continue on from my previous raven post, I want to further discuss what I will call "the argument from exclusion" for Hempel's R1 account of confirmation.
As previously explained, I hold that one's sampling method is of crucial importance to determining whether an observed instance of a G-ish F is evidence that all Fs are G. To borrow Blar's example, suppose we're wondering whether there are any extraterrestrials. If God provided me with a random sample of living creatures, and all of them were from Earth, then that would provide me with evidence that all living creatures are from Earth. But I get no such evidence from a sample that is restricted to Earthly creatures to begin with. If I see my neighbour's dog, it is both alive and on Earth, but - contrary to Hempel's R1 principle - this does not provide any evidence whatsoever that all life is on Earth (i.e. there are no aliens).
The argument from exclusion denies this. It begins by noting that there are various rival hypotheses about extraterrestrial life. On one, let's call it H(0), there are no aliens: all living creatures are on Earth. At the other extreme, we have H(1), the claim that everything is an alien, and nothing else but aliens exists. There are infinitely many such hypotheses H(p/q): that the proportion of aliens to objects in the universe is p/q. [We will see later that this strict definition fails to yield the rough description of H(0) given above.]
We can begin by assigning each of these hypotheses a non-zero probability. Now, when we observe my neighbour's dog (or indeed anything at all that isn't an alien), that allows us to rule out H(1). Consequently, the non-zero probability previously attached to this will get reassigned around all the other H's, including H(0). Each will receive some miniscule, but non-zero, boost. Hence, the argument goes, observing my neighbour's dog does (just slightly) confirm that there are no aliens.
I want to show that this argument fails. I previously showed that it yields a paradox of interpretation, because what conclusions we draw are radically relative upon how we choose to carve up the space of hypotheses. Suppose that, instead of appealing to the proportionally-focused 'H' hypotheses, we instead carved up the problem space in terms of cardinality. That is, we consider the range of hypotheses C(n): that there are exactly n aliens in existence.
Now, each partition method is exhaustive. It must be the case that exactly one of the H(p/q) hypotheses is correct, and exactly one of the C(n) hypotheses is correct. Each partitions logical space in such a way as to exhaust the possibilities. There is no possible state of affairs that is not covered by exactly one H hypothesis and exactly one C hypothesis. Okay. Now, it seems that H(0) and C(0) are the same hypothesis. The proportion of aliens to other existing objects is zero iff there are zero aliens (but see below). But while observing my neighbour's dog allows us to rule out H(1), and thus slightly confirm H(0), the very same observation does not falsify any of the C(n) hypotheses. For any n, it is possible for there to exist n aliens plus my neighbour's dog. So the C hypotheses are unaffected by the observation. In particular, C(0) is not confirmed.
Thus we have the following inconsistent triad:
1) C(0) = H(0)
2) H(0) is confirmed by evidence E
3) C(0) is not confirmed by evidence E
Perhaps we should reject (1). If it's possible for there to be infinitely many objects, then H(0) could be true when C(0) is false. The proportion of aliens to other objects could be zero even if there are some finite number of aliens. This suggests to me that H's focus on proportions or ratios is misplaced. We're not interested in the relational question of what proportion of objects are aliens. Rather, we're interested in the absolute number of aliens (and, particularly, whether it is non-zero).
This is further highlighted by considering repeated applications of Hempels R1 principle. It claims that each observed co-instantiation of F-properties with G-properties confirms that all F's are G. So if, after observing the dog, I also observe a rabbit, the rabbit provides further evidence that all living creatures are Earthly. How can the argument from exclusion deal with this? Well, it must say that we rule out the hypothesis E(-1) that everything is an alien except for one thing.
There are two things to note here. Firstly, E(-1) is not part of our previous partitions of the problem space. It does not correspond to any C or H hypothesis. Granted, an exhaustive partition of logical space is provided by the hypotheses E(-n): that everything is an alien except for n things. But this is a bizarre and unhelpful partition. None of the E hypotheses correspond to the desired hypothesis C(0) that there are no aliens. C(0) will be true iff, for some n, E(-n) is true and there are exactly n objects. Depending on how many objects there are, this could potentially end up being any of the E hypotheses. None of them are inconsistent with C(0). So, because they are not rival hypotheses, ruling out E(-1), say, does nothing to confirm C(0).
Are there any other exhaustive partitions of logical space possible that would support the argument from exclusion? I can't think of any, but it isn't obvious how to prove this. I guess I need to show that the C partition is the appropriate one to make, such that any other partition that yields contradictory results (e.g. the H partition) must be mistaken.
How can I show this? I'm not too sure. It seems fairly self-evident though, really. If we're wondering whether there are no aliens, this corresponds to the claim that the number of aliens is zero, and this contrasts with all the rival C(n) claims that the number of aliens is n. This partition appeals only to the absolute number of aliens -- it is in this sense the most basic and fundamental partition we could make. The R partition appeals to the relative proportion of aliens to other objects, and so is more complex, depending upon both the absolute number of aliens and the absolute number of everything else. Plus it misbehaves terribly if one allows for the possibility of infinitely many objects. Similarly the E partition appeals to both 'everything' and the absolute number of aliens. So both R and E can be redefined in terms of C plus some extra variables. It seems fairly clear that there is no simpler possibility than C. Any other such partition is going to build on it in much the fashion that R and E do.
But that's a fairly rough intuitive account, so if anyone can see how to turn this into a rigorous proof, please let me know!
P.S. Put another way, I need to show that there is no possible partition of logical space such that "there are no aliens" [i.e. C(0)] and "everything is an alien" [i.e. E(0), and perhaps R(1)] are rival hypotheses within this partition. But perhaps this follows simply from the fact that C is an exhaustive partition which contains C(0) but does not contain anything corresponding to E(0)? I don't think it is quite this simple though -- one might always respond that it simply requires a finer-grained partition. Perhaps the better answer is to simply point out that E(0) and C(0) are not mutually exclusive: if no objects existed at all, then it would be true both that there are no aliens and that everything that exists is an alien! Perhaps this suffices for my proof?