A few somewhat rambling musings.
Something we often have to deal with in history of philosophy -- whether we put it in precisely these terms or not -- is what might be called (and sometimes is called) the 'neighborhood of an argument'. Due to difficulties in interpretation, we can't always be certain that an interpretation I is the accurate interpretation of argument A. What we can do is determine if it is in the neighborhood of A, insofar as the evidence indicates it.
Now, this notion of a neighborhood for an argument is an interesting one, because it's more difficult to have a clear notion of what it means than one might think. It's generally a notion of the 'I know it when I see it' variety. But there would be advantages to making it more clear than this, particularly with regard to objections, since one thing that would be very nice is to be able to determine accurately whether an objection problematizes all arguments in the neighborhood of A, or just A. If, for instance, there is something wrong with A, it would be nice to know if there is something in the neighborhood of A that is more promising, or if the flaw infects all arguments in the neighborhood.
One tempting way to characterize the neighborhood is by commonality. An argument consists of a conclusion, premises, and a form or procedure according to which the premises are transformed into the conclusion. Let's take the following argument:
All men are mortal.
All politicians are men.
Therefore, all politicians are mortal.
We might be inclined to identify several neighborhoods for the argument by identifying them according to premises, conclusions, and forms. This would yield the following neighborhoods:
[arguments using the premise 'All men are mortal']
[arguments using the premise 'All politicians are men']
[arguments concluding to 'All politicians are mortal']
[arguments using the premise 'All men are mortal' and concluding to 'All politicians are mortal']
[arguments using the premise 'All politicians are men' and concluding to 'All politicians are mortal']
[arguments of Barbara syllogistic form]
[Barbara arguments using the premise 'All men are mortal']
[Barbara arguments using the premise 'All politicians are men']
[Barbara arguments concluding to 'All politicians are mortal']
[Barbara arguments using the premise 'All men are mortal' and concluding to 'All politicians are mortal']
[Barbara arguments using the premise 'All politicians are men' and concluding to 'All politicians are mortal']
and so forth.
Some of these are very small neighborhoods (the last two listed, for instance, which seem to be neighborhoods of one) while others are extraordinarily large (there are infinitely many Barbara arguments). And clearly not every objection will affect all the neighborhoods in the same way. Suppose the objection is:
At least one man is not mortal.
This objection, if true, would refute all arguments in any neighborhood in which all the arguments have the premise 'All men are mortal'. (In the above list, four neighborhoods would be eliminated by this objection.) A formal reductio ad absurdum of the argument, however, if it existed, would refute all Barbara arguments; that's one way of determining valid and invalid figures. All this seems fairly basic.
But I wonder if the premises, conclusions, forms approach (PCF) is really quite adequate. Consider arguments that are in some sense 'instantiations' of other arguments. Aquinas's Fifth Way is not a design argument; if all design arguments were unsound, this would not entail that the Fifth Way is unsound. Likewise, if Aquinas's Fifth Way is sound, this does not entail that any design arguments are. Thus, the Fifth Way and design arguments are not in the neighborhood of each other in terms of form or conclusion; nor do the design arguments in the neighborhood of, say, Paley's argument, share any premises with the Fifth Way. But it seems wrong to say that the refutation or confirmation of Paley's argument is irrelevant to our evaluation of Thomas's Fifth Way. And the reason for this does not seem hard to find. The Fifth Way presents a particular view of the relation of final causes to intelligence. The sort of design suggested by Paley's argument is one particular way in which certain kinds of final causes might be related to intelligence. If it turns out that those final causes are indeed related to intelligence in that way, then even if the Fifth Way fails in general, it will succeed for the domain of causes considered by Paley's argument. Thus, to that extent, Paley's argument and the Fifth Way might be said to share a neighborhood without sharing premises, conclusions, or form. An objection that final causes can't possibly have any relation to intelligence would refute this whole neighborhood.
One might modify PCF so that not only premises, conclusions, and forms would identify neighborhoods but also generalizations of them. But there are also reasons to include premises, conclusions and forms that are different but, under certain conditions, can be translated into each other (or in one direction). And so forth. And thus we seem to be in a state in which just about any argument can be in the neighborhood of any other argument under some condition or other. This actually seems right to me, but one wonders if there is an easier way to handle neighborhoods, one that's more clear than our usual handling, but is itself more easily done in an orderly way.
For practical purposes we might think in this way. What we actually need when talking about neighborhoods is something to reason on that helps clarify arguments and their relations to each other. So let's take descriptions; to neologize I'll simply call them 'approximativements'. What we actually are interested in when we talk about the neighborhood of an argument (or a kind of argument, or a class of argument) is a feature, identifiable by description, that has further properties relevant to evaluating the arguments in that neighborhood. So, for instance, if someone wants to argue for the conclusion 'No design arguments for God's existence are good', he's really interested in the properties of arguments in the neighborhood established by the approximativement, argument using the concept of design and concluding with 'There is a God'; what he'll do is argue that the concept of design is not well-formed, or not relevant to the existence of God.
Thus we'll tend to prefer neighborhoods that have well-behaved approximativements. The approximativement argument that can be formulated with at least one word that uses the letter e might be said to describe a neighborhood of arguments, but not a natural one; it's not a neighborhood we can do much with, because the use of the letter e is not relevant to either the truth or the validity or the plausibility of an argument. This makes sense; what you want are neighborhoods that are capable of refutation as neighborhoods, and the means of refutation are fairly standard: showing the premise false, showing the conclusion false, showing the form to be invalid, showing that all premises of a certain type are false, etc. Thus the whole conclusion here is unsurprising; but it is interesting to this extent, that in talking about neighborhoods of arguments, or kinds of arguments, we are often very fuzzy about what the relevant approximativement is, and this leads us to make mistakes about whether a given objection that refutes argument A also refutes argument B -- i.e., refutes some neighborhood of arguments shared by A and B. Thus, it's a common mistake to assume that Kant in refuting a certain type of (broadly Cartesian) ontological argument for God's existence thereby also refuted Anselm's argument in the Proslogion. This, as people have noted, is an assumption rather difficult to prove, even if we grant that Kant rather refuted the Cartesian argument. And the reason is that Kant's refutation can only apply to Anselm's argument if Anselm's argument falls within a neighborhood described by a particular approximativement, namely, argument for God's existence in which a premise uses existence as a predicate. When the relative approximativement is made explicit, it naturally raises the question of whether Anselm's argument -- or, indeed, the Cartesian argument -- really does use existence as a predicate. And one way in which one might defend Anselm is to show that, in fact, his argument does not fall within the neighborhood characterized by the approximativement. In other words, we show that Anselm's argument is not described by the approximativement.
So we see that whenever we talk about the neighborhood of an argument, or about kinds of arguments, is a way of talking about the relevance of one argument to another; clarifying it would give us a better means of evaluating arguments both as objections and as defensible from objections. But the question is still how to do this; the above gives one way in which one might make what we are doing more explicit, but it doesn't really clarify what we are doing in the first place. It's possible, too, that this whole notion of a 'neighborhood' is ill-formed; that, for instance, there are several wholly separate things that are jumbled together in it, which are better kept apart. And I can think of arguments that would suggest this: for instance, one could argue that the notion of a 'neighborhood' for an argument actually arises from conflating two different domains, like logic and rhetoric, or form and function. I don't know if any of these arguments would work. It's possible also, and this seems plausible to me, that we actually need to distinguish more carefully between conceptual neighborhoods and argumentative neighborhoods. But, in any case, if we could give a clear account, or at least a clearer account, of 'neighborhood', it would be immensely valuable for evaluating arguments -- particularly arguments that are objections to other arguments. It would also be useful for discovery, since it's clear that (to take an example) if we learned that a purported objection against arguments with the conclusion 'There are x's' did not actually refute the entire neighborhood of that argument, then we would be better equipped for formulating an argument for x's that works (or a better refutation of such arguments). Right now it seems very hit and miss, with a lot of guesswork.
