In my previous post, I laid down the R1, R2, ~R1, and ~R2 hierarchies without much comment. Each of these hierarchies are arguments that might be made given a single base. I will here use the R1 hierarchy to begin looking at the patterns of relations among arguments exhibited in these groupings. The R1 hierarchy, you will recall, is:
The reason for the hierarchy being structured as it is, is simply that the higher you go in the table, the stronger the argument, or, to be more exact, the stronger the claim made by the argument. Thus, a : RaL(XT) is the strongest in the schedule, and a : RaM(X~F) is the weakest. As we go from top to bottom the conclusions become attenuated; types of argument on the same line (e.g., a : RaL(X~F) and a : Ra(XLT)) are not attenuated with respect to each other. Thus the most important, and most interesting relation among these different types of argument, when we consider only a single hierarchy, is attenuation. When we make explicit the attenuation relations among these arguments, we get the following (sorry for the crudeness of this and all other graphics; they were thrown together as drafts):
The way to think of attenuation is to think of it as an a fortiori relation. If we have a good argument of type a : Ra(XL~F), then a fortiori an argument of type a : Ra(X~F) is a good argument. In other words, if some evidence is good reason to think that the sky is necessarily not false, a fortiori that evidence is good reason to that the sky is not false. The relation is transitive. If a is a good reason to conclude that it is a necessary truth that the whole cake is greater than any of its slices, a fortiori it is a good reason to conclude that this claim's being not false is a possible truth.
I mentioned before that some people might not like the idea of taking truth and false as contraries. What happens if you only allow two values, where T is taken as equivalent to ~F and F is taken as equivalent to ~T? You get the same basic hierarchy, but it collapses into a single line, since every argument-type on the left collapses into the argument-type immediately above it on the right.
When we recognize this it turns out to be easy enough to identify the attenuation relations for the other hierarchies. This is R2:
This is ~R1:
And this is ~R2:
Sharp eyes will recognize that R1 and R2, and ~R1 and ~R2, are parallel, whereas R1 and ~R1, and R2 and ~R2, are reciprocal. This brings us to the most important inter-hierarchy relation, preclusion. We'll discuss that in the next post on this subject.
Continue to Part III.