In the second post I introduced the notion of attenuation and used it to establish hierarchies of arguments.
In the third post I introduced the notion of preclusion and used it to show how distinct hierarchies of arguments are interrelated.
In the fourth post I introduced the notion of candidacy in order to show one way the typology could be extended (and corrected an inaccuracy in earlier posts).
Here I'd just like to give an example. But first it may be useful to note a few points that help when applying the typology to real-world examples.
The first is that the only difference in bases that is important is a difference that affects the structure of the argument. Thus, one can have as one's bases a thousand separate pieces of evidence, and if they all together only form one reason for a conclusion, they still can be treated as a simple, one-base type of argument. It's only when one base is put forward as a reason for thinking that another base is a reason for drawing a conclusion that we get into multiple-base hierarchies.
The second is that in a one-base case, multiplying R's doesn't really affect anything. a : Ra(Ra(XT)T) is just a more verbose a : Ra(XT).
In Hume's Enquiry Concerning Human Understanding there is a famous passage in which Hume presents a counterexample to his own position, the missing shade of blue:
There is, however, one contradictory phenomenon, which may prove that it is not absolutely impossible for ideas to arise, independent of their correspondent impressions. I believe it will readily be allowed, that the several distinct ideas of colour, which enter by the eye, or those of sound, which are conveyed by the ear, are really different from each other; though, at the same time, resembling. Now if this be true of different colours, it must be no less so of the different shades of the same colour; and each shade produces a distinct idea, independent of the rest. For if this should be denied, it is possible, by the continual gradation of shades, to run a colour insensibly into what is most remote from it; and if you will not allow any of the means to be different, you cannot, without absurdity, deny the extremes to be the same. Suppose, therefore, a person to have enjoyed his sight for thirty years, and to have become perfectly acquainted with colours of all kinds except one particular shade of blue, for instance, which it never has been his fortune to meet with. Let all the different shades of that colour, except that single one, be placed before him, descending gradually from the deepest to the lightest; it is plain that he will perceive a blank, where that shade is wanting, and will be sensible that there is a greater distance in that place between the contiguous colour than in any other. Now I ask, whether it be possible for him, from his own imagination, to supply this deficiency, and raise up to himself the idea of that particular shade, though it had never been conveyed to him by his senses? I believe there are few but will be of opinion that he can: and this may serve as a proof that the simple ideas are not always, in every instance, derived from the correspondent impressions; though this instance is so singular, that it is scarcely worth our observing, and does not merit that for it alone we should alter our general maxim.
So let's take the background argument, which is a set of reasons for the claim that every simple idea has a corresponding impression whence it derives. This is an argument of type a : Ra(IT), where I is the claim about the relation between ideas and impressions. He then presents a contradictory phenomenon, the fact that we can conceive of the scenario of the missing shade of blue (mb), which "may prove that it is not absolutely impossible for ideas to arise, independent of their correspondent impressions," in other words, that it is possible for I to be false. There is some ambiguity here about whether the argument is supposed to be of type mb : Rmb(IMF) or the slightly weaker mb : RmbM(IF). The phrase "not absolutely impossible" suggests to me that the intent is the weaker one; the idea appears to be that it is not in the strictest sense impossible for simple ideas to arise without simple impressions, not that it is possible given actual conditions. The type would then be mb : RmbM(IF). But this is controversial; one might choose to emphasize the terms in which Hume ends his description of the scenario: "this may serve as a proof that the simple ideas are not always, in every instance, derived from the correspondent impression". This might well be taken to suggest that the argument is of type mb : Rmb(IMF). However, if we might also wish to emphasize the 'may' in "this may serve", in which case we have a different sort of argument altogether, namely, mb : MRmb(IF). In this interpretation Hume is only admitting to the scenario's possibly being a reason for thinking that I is false. Thus we have three superficially similar but in fact rather different interpretations of the missing shade of blue:
mb : RmbM(IF)
mb : Rmb(IMF)
mb : MRmb(IF)
This sort of ambiguity is, in fact, very common in arguments made in modal terms that are not made very, very carefully. If the argument is of the first type, it is an argument that the missing shade of blue suggests that it is possible for I to be false (under some circumstances or other, which may or may not actually occur). If of the second, it says that I can be false under some conditions that might actually occur. And if of the third, it says that the missing shade of blue might be a reason for thinking that I is false. Which one we choose affects how we understand Hume's counter-reasoning, but it will be convenient to use the term Bmb to mean the exclusive disjunction: either the first, or the second, or the third.
After giving the scenario, and saying that it may be taken as proof that I is false, Hume then goes on to say the words that have so puzzled commentators, namely, "this instance is so singular, that it is scarcely worth our observing, and does not merit that for it alone we should alter our general maxim." This reasoning is, at least at first glance, of type s, mb : Rs(~Bmb(IF)T). That is, it's an argument that s is a reason for thinking the missing shade of blue is not a reason for thinking I false. Now, how does our understanding of Bmb affect our understanding of this line of reasoning?
If Bmb should be understood as mb : RmbM(IF), then the idea would appear to be this: the missing shade of blue really is a legitimate reason for thinking that it is possible for I to be false. But the singularity of the case could mean, among other things, that it is unlikely, and perhaps not even possible, for I to be false under conditions that would actually obtain in the real world. Thus I can be taken to be true as a "general maxim" on the original argument Hume made.
If Bmb should be understood as mb : Rmb(IMF), then the point of mentioning the singularity of the case is not to say that it's so unusual that it might not ever occur, but that it's so unusual that even if it does occur it's not a good reason for taking I as Hume actually understands it to be false. For instance, the idea might be that Hume is only interested in what generally occurs "in the wild" (in a restricted application of the principle), not under highly artificial circumstances like that which the missing shade of blue would have to be.
If Bmb should be understood as mb : MRmb(IF), then the singularity of the case would indicate that the scenario is not even a good candidate for an argument that I is false. This would seem to be contradicted by what Hume says immediately before this ("this may serve as a proof"), although, of course, that carries a typical Humean ambiguity: it may serve as a proof, but does Hume think it really does serve as one, or is he just observing that other reasonable persons might think it does? But it does seem less likely, so we can perhaps set aside the mb : MRmb(IF) as an unlikely possibility.
Now, Hume is very often criticized for his response, but it is clear that many commentators take the argument to be of type s, mb : Rs(~Rmb(IMF)T) -- that is, they take it as a straightforward denial that the missing shade of blue is a counterexample to I, where it is taken as actually being able to occur. But it's important to note that we can take the conclusion of the argument to be not (IMF) but M(IF), in which case almost all the criticisms of Hume's treatment of the missing shade of blue miss the point, for they take it as saying that it can really occur, whereas Hume might not even be committing himself to it's being able to occur at all -- he might just be saying that there is some conceivable scenario (and for him conceivability is possibility) in which it might, without committing himself to the claim that this scenario is at all consistent with other truths about the world. That is, all he would be conceding is that I is not a necessary truth; and he is right that singularity can show this to be irrelevant to actually taking I to be true, i.e., can be an undercutting defeater for I.
This is all very crude, but I think you can get the idea. As I said in the first post, this whole thing is just an idea that came to me one Monday in late July and was worked out over a few days of sporadic thought. It's likely to need fine-tuning in a great many spots. Comments are very welcome.