Roger White has proven, to my satisfaction, that perfectly rational agents could not have imprecise credences. In this post, I want to explain away a potential source of contrary intuitions. It seems perfectly reasonable -- and perhaps even rationally required -- to refrain from accepting bets on issues you are completely ignorant about. But if I were required to always have precise credence, then there are many (intuitively dubious) bets that I would have to accept. For example, suppose I have no evidence for or against a proposition P. If I must give it a precise credence of 0.5, then I must accept any better-than-even bet (e.g. costing $10 if P is false, and paying $15 if P is true).
The worry is that this leaves me very vulnerable to exploitation from more knowledgeable dealers. They might offer me deals that seem tempting in my ignorance, but which they have carefully set up so that the option I'm expected to choose is in fact a loser. The offering of a bet is itself a piece of new evidence: since the bookie is out to rip me off, an offer that looks "too good to be true" probably is. So it's a good practical rule of thumb to be disposed not to accept bets that others are keen to offer you.
Of course, we want to bracket these practical considerations for philosophy's sake. To avoid any fear of shady dealings or manipulation, we may suppose the details of the bet were determined by some completely random process. In that case, my intuitions sharpen up considerably. It no longer seems permissible to reject a better-than-even bet. Let me now offer an argument to back this up.
We saw in the previous post that a better-than-even Bet A can be combined with its converse Bet B (i.e. offering the same, better-than-even payoffs for the opposite result) to yield a sure win ($5 in the case of $15 vs. -$10 payoffs). So, even if you don't know anything else, you at least know that it's more desirable to take both bets than neither. The expected payoff is (say) $5 rather than zero.
But now suppose you are offered the following: we flip a coin, and if it lands heads you're committed to Bet A only, and if tails you get Bet B only. Is this a game you should accept? It seems so. If you play it over and over again, you can expect to make ever increasing winnings ($5 for every two games, on average). And it's perfectly random, so your one-shot expected utility must be positive too.
So, if you're offered Bet A alone (assuming the background conditions are such that the details of this bet were selected via some random, non-shady process), you should take it. A sharp credence of 0.5 won't leave you vulnerable in these properly sanitized conditions.
Monday, March 10, 2008
Shady Gambles
Sunday, March 09, 2008
Combining Bets
Suppose you have no idea whether P is true. Is it rationally permissible for you to reject the following bet?
(A) You win $15 if P is true, and lose $10 if P is false.
If so, the following bet is presumably also permissible to turn down:
(B) You win $15 if P is false, and lose $10 if P is true.
But if someone offers you both bets at once, it would be crazy to turn them down: you can net $5 no matter the outcome, it's a guaranteed win! We may take this to show that the rational status of bets is not closed under conjunction. It can be rationally permissible to reject A, and permissible to reject B, but impermissible to reject (A and B).
What if you do not know that both bets will be offered? Suppose you are offered A, and permissibly reject it. Then, to your surprise, you are offered B. Are you now rationally required to accept bet B, based on the principle that it would be irrational to reject both? That would be bizarre. Instead, I'd suggest that the rational principle in play is the following:
(Sure Win) It is irrational to knowingly turn down a sure win (unless it comes with opportunity costs, etc.).
It is irrational to reject A and B together, for together they offer a sure win. But the person who rejects A, and is only later offered B, was never offered a sure win. Bet A by itself (with no guarantee that B will be offered too) is not a sure win. So it may be permissibly rejected. And once you've rejected A, bet B by itself is not a sure win either. So it too may now be rejected without violating the Sure Win principle.
Why am I going on about this, you ask? Adam Elga uses this combination of bets to argue that a rational agent must be disposed to accept at least one of them (and so have precise credences). But it seems like a bad argument to me, since the defender of imprecise credence can appeal to the Sure Win principle - as I did above - to explain why the irrationality of the combined bet rejection does not imply the irrationality of rejecting either bet alone. Right?
[See also: Is Imprecise Credence Rational?]
Friday, February 15, 2008
Is Imprecise Credence Rational?
If you haven't the faintest clue whether some proposition p is true or false, what subjective probability (credence) should you give it? 1/2? A common answer these days is that you shouldn't give it a precise credence at all. Instead, your credence should be spread over an interval, such as [0,1]. Greater precision than that ought to be based on real knowledge, e.g. of objective probabilities. Mere ignorance doesn't qualify one to make such claims.
Roger White, in his talk 'Evidential Symmetry and Mushy Credence', offers a neat argument for the old-fashioned answer of 1/2, which goes roughly as follows:
Coin Game: Suppose you're given a fair coin which has 'p' plastered on one side, and '~p' on the other. Moreover, you know that whichever one is true was plastered over the Heads side. You toss the coin and it happens to land on 'p'.
(1) It's a fair coin, so you should initially give P(heads) = 1/2.
(2) This should not changed upon seeing the coin land on 'p' -- you have no idea whether p is the true one or not, so there is no new evidence for you here. So your updated P+(heads) = initial P(heads) = 1/2.
(3) Since the coin landed on 'p', this will be heads-up iff p is true. Hence P+(p) = P+(heads) = 1/2.
(4) But the coin landing on 'p' doesn't tell you anything new about the proposition's truth value. So your prior credence P(p) should also have been 1/2.
Convinced?
Sunday, January 22, 2006
Misleading statistics: second-order ratios
I saw an ad on TV the other day, promising that its product would "kill 99% of most household germs", or something like that. Of course, "most" could mean anything over 50%, and to take a proportion (even 99%) of that will make it even less, so the promise doesn't really say much. It's almost as bad as the sales where everything is "up to 50% off" (which of course is consistent with most of the discounts actually being far less). Damn advertisers.
More generally though, it can be difficult to report second-order percentages in a non-misleading way. Suppose a political candidate starts with 10% of the vote. If someone reported that his popularity had "increased by 50%", you might be startled into thinking that he was now winning the race, with 60% of the vote. But they might instead merely mean to say that he increased his vote share by five percentage points, to a total of 15% -- which is, after all, half again as good as (hence, 50% improvement on) his original position of 10%.
Anyway, it would be nice if people took more care to resolve this ambiguity when reporting these sorts of statistics. Clarity might be achieved by describing the first sort of increase as "an improvement of 50 percentage points", which I think is more clearly talking about the first-order units. To report second-order percentages, one might specify the background set, e.g. "50% of his original 10 per cent share".
Such care might significantly improve 99% of some of the world.
Sunday, September 11, 2005
Abstract and Concrete Probabilities
There seems to be an important distinction between two types of probability. Let's say there's some object O with a property P1, but you don't know what other properties it has. Further suppose that what you're really interested in is whether O has some other property P2. What's the probability that O has P2? We could interpret this as an epistemic question, whereby we abstract away all the unknown details of O and just ask what proportion of P1-objects also have P2. Here we aren't really talking about the specific object O at all. We treat it merely as a token of type "P1-object", without regard for the multiplicity of other (unknown) ways O might be categorized. That's one option. Alternatively, we might focus on the concrete object O, and ask: of this specific object, including its many other properties that we are not yet aware of, how likely is O to have P2? This is a question about the real or metaphysical modal properties of O, rather than being a purely epistemic question. Let's illustrate the distinction with a couple of real-world examples.
The first example came up in the comments to my recent post on ad hominems. Suppose a notoriously unreliable person makes an argument. You are wondering whether the argument (O) is likely to be a sound one (P2), given that it is being made by this notoriously unreliable person (P1). One way to answer this would be to abstract away from the details of the argument itself, and just use your background knowledge of the advocate's unreliability - in particular, that most of his arguments are unsound - to conclude that this argument is therefore likely unsound. This is a classic ad hominem fallacy. Although such abstraction may be rational in a sense, it isn't really fair on the argument in question. After all, you've completely ignored the argument itself, when the dialectical norms of civil discourse recommend that we consider each argument on its own merits.
This latter suggestion is taken seriously by the concrete probabilist. He focuses on the argument (object O) rather than who makes it (property P1). In particular, he recognizes that the validity of the argument is metaphysically independent of the person making it. So he ignores the person, and instead assesses the argument on its own merits, trying to determine whether it is logically valid, and how plausible the premises are. It is from these considerations that he estimates the soundness of the argument.
The second example is provided by stereotyping individuals (e.g. racial profiling). Say you want to know whether Jack (O) committed the crime (P2), when the only other information you have about him is that he is a black male (P1). You can probably tell what comes next. The abstract probabilist ignores all the complexities of Jack the individual, and makes his judgment in light of how many other black males have been known to commit crimes. As with the ad hominems, we can see that this prejudice is rational in a sense, but also incredibly unfair to Jack as an individual. Your judgment of him is based on what you know of others who share the same property as him (i.e. of being a black male), rather than anything concrete and intrinsic to Jack himself. I argue elsewhere that this is morally problematic.
Consider a pair of cross-racial "twins", Alan and Bill, who are identical in all respects except that Alan is white and Bill is black. Clearly this is a difference that makes no difference. As a matter of fact, each is equally likely to make any particular decision, or have any other property or characteristic. These facts are what determine concrete probabilities, and they are unaffected by whether we know about them. But abstract probability is an epistemic notion. If all you knew was that Alan was white and Bill black, then you would judge their abstract probabilities very differently. You might think Alan is likely to be wealthier, better educated, and so forth. But these "probabilities" are metaphysically superficial -- merely reflecting facts about group frequencies and correlations. Concrete probability is much deeper, being grounded in causal and explanatory facts about the concrete individuals themselves.
So, like I said, this strikes me as an important distinction. (If someone with more formal training in statistics can go beyond the intuitive graspings I've offered here, please do leave a comment.) I think the close analogy between ad hominems and sterotyping is also interesting to note.
Sunday, September 04, 2005
Raven Paradox Essay
Suppose we are wondering whether the hypothesis (S1) All ravens are black is true. Intuitively, it seems that observing black ravens ought to boost the credence we give to S1. For similar reasons, observing a non-black non-raven presumably provides confirming evidence for the statement (S2): Whatever is not black is not a raven. But note that S1 and S2 are logically equivalent. Each is true if and only if there are no non-black ravens.[1] So whatever confirms the one statement must also confirm the other. In particular, a red herring is a non-black non-raven, and thus confirms S2, and thus confirms S1. But it seems incredible that observing a red herring could provide any evidence whatsoever for the claim that all ravens are black. I will argue that this conclusion is no ‘paradox’. On the contrary, it is demonstrably true. Building on the insights this yields into the nature of confirmation, I will further argue that black ravens are not always confirming evidence that all ravens are black. Common sense is wrong on both counts.
There are two general approaches that one might take in seeking evidence related to a universal statement like S1. One can try to find “confirming instances” of the hypothesis, i.e. objects that it is true of. Alternatively, one can seek disconfirming instances that would falsify the hypothesis.[2] Discussions of the raven paradox have traditionally focused on the first approach. I will argue that this is a mistake, and that there is no such thing as direct confirmation of a universal statement. Instead, the way to confirm S1 is, roughly, to try as hard as one can to falsify it. If it survives all such attempts, then this is evidence that there is no falsity to be had, i.e. that the hypothesis is true.
On this approach, the way to confirm that all ravens are black is to look for a raven that isn’t. More precisely, we employ a method that might reasonably be expected to turn up a non-black (henceforth, ‘coloured’) raven, were such to exist. When it fails to do so, this provides some evidence that there are no such coloured ravens to be found.[3] The strength of this evidence will depend upon how likely the method would be to expose coloured ravens were there any. Clearly, if the method is guaranteed to expose any coloured ravens that exist, then, when it fails to do so, this would conclusively prove that none do. By contrast, if the method has only a very faint chance of picking up a coloured raven, then its failure to do so provides correspondingly faint evidence against them.
The perfect test of S1 would be to exhaustively check every object in existence to see whether any are coloured ravens. If we found one, S1 would be shown to be false, and otherwise we would have established its truth. But we can also do imperfect tests, which merely sample the population of worldly objects, rather than doing an exhaustive census. So let’s consider the method of randomly sampling objects from the world. Every sampled object that is not a coloured raven would count as (weak) evidence that there are no such creatures. For if there were coloured ravens then we should expect them to turn up occasionally in random samples. The fact that they never do counts against them. Put another way, an object of any given type is more likely to turn up in a random sample if there are comparatively few objects of other types in the sample population. In particular, red herrings are slightly more like to be observed if there are no coloured ravens in the population than if there are some, all else being equal. This explains why it is that red herrings can be confirming evidence for S1.
The intuitive oddness of this conclusion can be partly dissolved when we recall that the strength of confirmation depends upon the proportion of coloured ravens we should expect to find in the sample population, were any to exist. If our sample population comprises all worldly objects, then this proportion is vanishingly small. We can improve it by shrinking the size of the sample population without excluding any possible coloured ravens from consideration. One way to do this would be to sample from the population of coloured objects. But of course there are still an impractically large number of those. Better yet, we could sample from the population of ravens. This would provide much stronger results. Hence our intuition that the way to confirm S1 is to observe black ravens, not red herrings. For all practical purposes, our intuition is quite right. But the difference is only one of degree, not kind. To illustrate: imagine a world bursting full of ravens, but with extremely few coloured (non-black) objects.[4] In such a world, the best way to confirm that all ravens are black would not be to look first for ravens and then check their colour. It would be much more efficient to instead search through the coloured objects for one that is a raven.
Despite the above arguments, one might remain suspicious of the ‘paradoxical conclusion’. Intuitively, it seems that when faced with a bucket of herrings, going through it and identifying each red herring would do nothing at all to confirm that all ravens are black. In fact, I will argue that this intuition is correct. Although it would confirm S1 to observe of coloured things that they are non-ravens, this is different from observing of non-ravens that they are coloured.
The distinction can be clarified in terms of the ‘sample population’. If we sample objects from the population of all coloured objects, then any coloured ravens that exist will be included in this population, and so stand a chance of being sampled. However, this is not true of the population of non-ravens, nor, for that matter, the population of black things. Even if coloured ravens existed, they would stand no chance of being sampled from such a restricted population. So our failure to find them in such a sample provides no evidence whatsoever about their existential status. Thus, whether any given object counts as confirming a universal statement like S1 will depend upon the circumstances of the observation. This shows that the commonly assumed principle of Direct Confirmation is false:
(DC) “Whenever an object has two attributes C1, C2, it constitutes confirming evidence for the hypothesis that every object which has the attribute C1 also has the attribute C2.”[5]
To illustrate: Suppose I am wondering whether the hypothesis (B) Nothing exists beyond the boundaries of my backyard is true. DC implies that I could obtain evidence for this hypothesis without having to look beyond my backyard at all. Exploring my lawn, I notice plenty of objects that both (1) exist, and (2) are in my backyard; hence, by DC, each such object serves as evidence for B, the claim that all existing objects are in my backyard. But this is clearly absurd. Exploring the confines of my backyard tells me nothing about the existential status of the world beyond it. For the same reason, exploring a bucket of herrings cannot tell us whether coloured ravens exist elsewhere.
This shows that DC is a mistaken account of confirmation. Hypotheses are not directly confirmed by their instances. Instead, they are confirmed indirectly, when a potentially falsifying test returns a negative result. The reasoning behind indirect confirmation was explained earlier in this essay. Let us call a population ‘CR-open’ if it would contain coloured ravens were any to exist, and ‘CR-closed’ if it would not. So, for example, the populations of ravens, of coloured objects, and of everything that exists, are all CR-open, whereas populations of black objects, or of herrings, are CR-closed. The upshot of our earlier discussion is that evidence concerning the existence of coloured ravens can be obtained by randomly sampling CR-open populations, but not CR-closed ones. Thus finding a red herring in the population of coloured objects is confirming evidence for S1, whereas finding a black raven by exploring the population of black objects is not. Sometimes red herrings really do confirm S1, and sometimes black ravens really don’t. Insofar as our intuitions rebel against this ‘paradoxical’ result, they are mistaken to do so, as my account of the nature of confirmation makes clear.
Hempel defends DC against the above objections by arguing that CR-closed populations are corrupted by background knowledge: we “know anyhow” that they contain no coloured ravens, so further testing is simply unnecessary.[6] The suggestion here is that the bucket of herrings does provide evidence for S1, but it does this as soon as you find out that it is a bucket of herrings. You learn nothing new from searching through it. But this is insufficient to explain our intuitions here. DC implies that the number of herrings in the bucket makes a difference. Each one provides an additional evidential boost to the hypothesis S1. But this is not plausible. Whether the bucket contains two herrings or two thousand can make no difference to the likelihood of a coloured raven existing elsewhere. This is something that my theory of indirect confirmation can clearly explain. DC appears quite unmotivated in comparison. Indeed, it doesn’t really explain why confirmation ever occurs at all.
One possible foundation for direct confirmation comes from what I will call ‘the argument from exclusion’.[7] We begin, a priori, with various rival hypotheses about the existential status of coloured ravens, and assign each some non-zero probability. We are primarily interested in the claim (S3) There are no coloured ravens; but one rival to this might[8] be the hypothesis (EC) Everything is a coloured raven. When we observe a bucket of red herrings, in any circumstances whatsoever, this allows us to rule out EC, thus boosting the a posteriori probability of all its rivals, including S3.
However, this is only part of the story. When we observe the bucket of red herrings, that allows us to rule out any hypothesis about the world which is inconsistent with the existence of such a bucket. Some of these, like EC, may be rivals to S3. But others – such as (ED) Everything is a dog – will be consistent with S3, and so our credence in the latter will be harmed by their exclusion. We have no reason to think that the sum effect of these exclusions will be favourable towards S3. So the argument from exclusion fails.
To recap: I have argued that the standard account of direct confirmation is mistaken. This helps to explain some of our intuitive resistance to the ‘paradoxical conclusion’. We are right to think that there is no evidence to be found from observing objects that are already known not to be ravens. It is only through sampling objects that might be coloured ravens that the possibility of falsification arises, and this possibility is a prerequisite for indirect confirmation. The raven paradox can survive this modification, however, since we can still establish the surprisingly conclusion that a red herring can, in the right circumstances, constitute confirming evidence that all ravens are black.
There is another source of intuitive resistance worth exploring. The red herring confirms that all ravens are black because it is evidence that there are no coloured ravens. But we could just as well say that the herring is evidence that there are no non-green ravens, and hence that all ravens are green. However, it seems decidedly odd to say that one piece of evidence can confirm both that all ravens are black and that all ravens are green. The problem is that the sampled red herring, on its own, serves as evidence against the existence of all other object types. In particular, it is evidence that there are no ravens at all. It is only by factoring in our background knowledge that there are some black ravens that we get to rule out the more general hypothesis that there are no ravens at all, and hence that all zero of them are green.
How we interpret this result will depend upon whether we take universal statements to have existential import – that is, whether “all ravens are black” entails that “some ravens exist”. Modern logicians deny this, holding that empty universal statements are vacuously true. That is, “there are no ravens” entails “all ravens are X” for any X whatsoever. Since the red herring confirms (in isolation) that there are no ravens at all, so it confirms the logical entailment that “all ravens are X” for every possible X. It is important to note that on this view, “all ravens are black” and “all ravens are green” are not inconsistent. Both will be true if there are no ravens at all. So this is no problem for the herring’s confirmation of S1.
Let us now consider how affirming existential import would affect this. It would imply that S1 and S2 are no longer logically equivalent,[9] so the standard raven argument would fail. After all, observing the herring does nothing at all to confirm the existence of a black raven. Quite the opposite, in fact: in isolation from our background knowledge, it provides evidence that there are no ravens at all, and hence no black ones. So, if S1 requires that some black ravens exist, then observing a red herring does not in isolation confirm S1. What it does support – even in isolation – is the claim (S3) there are no coloured ravens. When we combine this with our background knowledge, we obtain support for S1. But again, the core of the raven paradox survives these minor alterations. It is intuitively surprising that a red herring could confirm S3, or that in conjunction with our background knowledge of black ravens it could confirm S1. But we might consider these results to be less surprising than the original claim that a red herring alone is evidence that all ravens are black. This reaction would show that we were (mis)understanding S1 as having existential import. Our intuitions are quite right to reject the view that red herrings can confirm the existence of black ravens. But the raven argument, properly understood, makes no such claim.
We are now in a position to agree with Hempel that the appearance of ‘paradox’ in the raven argument is misleading. I have shown how a red herring can, in appropriate circumstances, constitute probabilistic evidence that all ravens are black. Further, three potential causes of intuitive resistance to this conclusion can be safely dispelled. Firstly, differences in population size mean that much stronger confirmation is to be had by finding a raven to be black than by finding a coloured object to be other than a raven. The degree of confirmation from the latter is so weak as to be unworthy of notice for all practical purposes. Second, there is no evidence to be found by exploring a population already known to not contain coloured ravens. This intuitively pleasing result has some surprising implications, however, including (i) that traditional accounts of ‘direct confirmation’ are mistaken; and (ii) that black ravens are not always evidence that all ravens are black. Finally, our intuitions might be misled by interpreting the universal claim to have existential import. We are right to think that the herring cannot serve as evidence for the existence of black ravens; all it suggests is that there are no non-black ravens.
Despite this agreement, my disagreements with Hempel may be more significant. I have disputed the traditional principle of Direct Confirmation, and proposed an alternative theory of indirect confirmation, which clearly explains how the ‘paradoxical conclusion’ arises and why it is true. It also deepens the ‘paradox’ by implying that in some circumstances, black ravens do not confirm that all ravens are black. The advocated theory of indirect confirmation does more than just solve Hempel’s raven paradox, it also expands and illuminates it.
Bibliography
Fitelson, B. & Hawthorne, J. (forthcoming) ‘How Bayesian Confirmation Theory Handles the Paradox of the Ravens’ http://fitelson.org/ravens.pdf
Hempel, C. (1945) ‘Studies in the Logic of Confirmation I’ Mind, 54 (213): 1-26.
Hempel, C. (1946) ‘A Note on the Paradoxes of Confirmation’ Mind, 55 (217): 79-82.
Musgrave, A. (2004) ‘How Popper [Might Have] Solved the Problem of Induction’ Philosophy, 79: 19-31.
Whiteley, C. (1945) ‘Hempel’s Paradoxes of Confirmation’ Mind, 54 (214): 156-158.
[1] I will discuss the implications of denying this equivalence later in the essay.
[2] Cf. Musgrave, p.24, though I depart from the Popperians in holding that resistance to falsification is itself evidence for the truth of the hypothesis in question, at least in the circumstances I describe below.
[3] Whiteley, p.157, hints at this line of argument, though he fails to follow through on it. In particular, he makes the mistake of thinking that if an event is “not improbable even if S1 is false” then it “provides no evidence” for S1. But it is relative, not absolute, probabilities that matter here. If an event is more probable given S1’s truth than its falsity (even if this latter probability is itself high in absolute terms) then the event boosts the a posteriori probability of S1.
[4] I owe this thought experiment to Doug Campbell.
[5] Hempel (1946), p.79. There he calls it the “R1” requirement, and elsewhere in the literature it is referred to as “Nicod’s Condition”.
[6] Hempel (1945), pp.19-20. Though Fitelson & Hawthorne, p.7, argue that this contradicts Hempel’s own theory of confirmation.
[7] Thanks to Doug Campbell for suggesting this argument to me.
[8] To ensure that the two hypotheses really are mutually inconsistent, we must assume that the universe is not empty, i.e. that at least one object exists.
[9] In particular, S1 would entail the existence of a raven, whereas S2 would instead entail the existence of a non-black object. So, in a possible world containing nothing but red herrings, S2 would be true but S1 false.
Saturday, August 27, 2005
The Argument from Exclusion
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?
