Monday, July 30, 2007

Notes Toward a Formal Typology of Argument III

In Part I I laid out, in a rough way, the rationale behind my notation for this typology. In Part II I identified the key intra-hierarchy relation of attenuation and noted how it structures the four hierarchies I had particularly identified: R1, R2, ~R1, ~R2. I left off noting that R1 and R2 on the one hand and ~R1 and ~R2 on the other are parallel, and that R1 and ~R1 on the one hand and R2 and ~R2 on the other are reciprocal. This leads us to the key inter-hierarchy relation of preclusion.

Like attenuation, preclusion can be seen as an a fortiori relation; but it is, unlike attenuation, a relation of opposition. The use of one argument often precludes, if we wish not to be self-defeating in our reasoning, another. Suppose you have a given bit of evidence, e.g., my seeing that the sky is blue. And I conclude from this that, given this, the sky must be blue. That is an argument of the type a : Ra(XLT). If a is a good reason for (XLT), however, that precludes its being a good reason for opposing conclusions, e.g., that it is necessary that it is not true that the sky is blue. This is what I mean by preclusion; and I think it's fairly clear that it's an important relation between types of arguments, one that is important for argumentation in general. Unlike attenuation, preclusion is symmetrical; every argument-type precludes every type that precludes it.

Preclusion shows a number of interesting patterns.

(1) Preclusion between R1 and R2. Whether a given type of argument in R1 precludes a given type of argument in R2 (or vice versa) depends crucially on how strong the conclusion is.

Every argument-type precludes its modal- & stance-reversed counterpart in the opposing hierarchy and every type stronger than it. Thus, RL(XT) precludes RM(X~T) because they have opposing modalities and stances; and since preclusion of the more attenuated conclusion is a fortiori preclusion of the less attenuated conclusion, RL(XT) precludes every argument-type in R2 stronger than RM(X~T). Since RM(X~T) is the attenuation of every other argument-type in R2, RL(XT) precludes every type in R2. R(X~F) has as its reverse counterpart R(XF); it precludes R(XF) and every type of which R(XF) is the attenuation.

(2) Preclusion between ~R1 and ~R2. There is no preclusion between ~R1 and ~R2. It's easy to see why; since the ~R hierarchies are types of arguments that remove relevance by indicating that the base is not a good reason for the conclusion, if the base is entirely irrelevant to the claim, then all the types of ~R arguments can be made with regard to it. So no argument-type in the ~R1 hierarchy precludes any argument-type in the ~R2 hierarchy, and vice versa.

(3) Preclusion between R1 and ~R1. RL(XT) certainly precludes the whole ~R1 hierarchy; RL(XT) precludes ~RL(XT) for obvious reasons, and what precludes the weaker a fortiori precludes the stronger. This pattern is repeated. Every R1 argument-type has a reciprocal in ~R1 that it precludes; since the hierarchies themselves are reciprocal, and what precludes the weaker precludes the stronger, every argument-type in R1 precludes its reciprocal in R2 and every type stronger than it; and, of course, vice versa.

(4) Preclusion between R2 and ~R2. Preclusion between R2 and ~R2 follows the same pattern as preclusion between R1 and ~R1.

If you think about it, we have some sort of at least vague intuitive feel for both attenuation and preclusion; this is why a fortiori reasoning is possible, since these are clearly the two primary foundations for it. And it is in seeing the interaction between attenuation and preclusion in the typology that we begin to see how useful such a typology has the potential for being.

It has recently become fashionable to talk about defeaters, which are usually taken to fall into two groups, namely, rebutting defeaters and undercutting defeaters. It's immediately clear that preclusion and defeat are somehow related. Preclusion itself can't be defeat, because preclusion obtains between types of arguments, and defeat between particular tokens of a type. But preclusion is that which makes defeat possible. There is some dispute about how precisely we should distinguish between rebutting defeaters and undercutting defeaters; but the usual way of doing so at least roughly is to say that rebutting defeaters give one a reason for thinking the original conclusion argued for is false; whereas undercutting defeaters give one a reason for thinking the premises for the conclusion do not actually yield it (at least in that given case). But given our typology we can give a simple account of both that is more rigorous than this. R-hierarchy (whether one-base or multiple base) argument-types are rebutting defeaters for those opposing R-hierarchy argument-types they preclude. And I think it can be argued fairly easily, although I will not do so here because it involves going beyond the one-base case I've been considering, that ~RR-hierarchy argument-types (that is, types that are not one-base) are undercutting defeaters for R-hierarchy argument-types (i.e., lower-level arguments) they preclude, and so forth. Self-defeat is also easily characterized; it occurs in one-base cases when, given the nature of the base, the argument does not rule out the tokens of precluding argument-types. A more rigorous account of preclusion than I've here provided would yield a more rigorous account of defeaters. And note several essential points shown by the typology:

(1) Because the attenuation relations group the argument-types into separate hierarchies, we can see the reason why people could make the rebutting/undercutting distinction in the first place, since they are, in fact, rather different in structure.
(2) However, the typology shows that there are forms of preclusion, the tokens of whose argument-types do not exhibit either rebutting defeat or undercutting defeat, namely, R preclusions of ~R arguments. Undercutting defeaters are always ~RR to R (or ~RRR to RR and R, etc.); rebutting defeaters R to R; so the oppositions of R to ~R and ~R to R, as well as oppositions to undercutting defeaters, do not fall under either the category of rebutting defeater or undercutting defeater. But they certainly are important for ruling out arguments.
(3) Moreover, it shows that while the distinction latches onto something, it also glosses over an immense amount of structure. There are many different kinds of rebutting defeaters, and what they can defeat is both precisely characterizable and very different in each case. Likewise with undercutting defeaters -- indeed, there appear to be infinitely many possible types of undercutting defeaters, each with their own place in the structure, although, no doubt, most of them are not practically interesting and, perhaps, they might under futher investigation fall into groups.

That's all I'll suggest on this point right now; it's sketchy, but I think it shows some of the promise of the typology for understanding the workings of arguments. In the next post I want to add a further level of sophistication to it, and probably will close in the post after that with some examples and, perhaps, other applications.

Continue to Part IV

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