An extraordinary number of arguments, even those not originally in this form, can be summarized by a standard Aristotelian syllogism. Here would be a typical design argument summarized in this way, for instance:
What exhibits signs of design has a designer.
The world as a whole exhibits signs of design.
Therefore the world as a whole has a designer.
In practice, of course, you'd probably have all sorts of subordinate arguments attempting to support the premises, but we can strip them all away and see the general character of the argument.
So suppose we want to investigate variations on this argument. We could vary the middle term (exhibits signs of design); or we could vary the major term (has a designer) or we could vary the minor term (the world as a whole). It's natural to think of terms along the lines of classifications, however; for instance, classifications allow you to identify appropriate middle terms for arguments that are explanatory. And we can think of each term along Aristotelian lines again. Every term is a (logical) species in some kind of classification relating it to other terms; it consists in a (logical) genus with a specific difference distinguishing it from other terms that share the same genus. So one way we can vary a term in a regular way (i.e., in a way that does not involve arbitrarily replacing it with another term) is by making it more general. For instance, we can move from the term has a designer to has a cause, since having a designer is one way to have a cause. Likewise, terms can be divided into (logical) species. (It is important to keep in mind that we are talking about logical divisions of a term here. For instance, in a metaphysical classification, there is no subspecies of Brandon, since I am just me; but we could logically treat me as a species by dividing me up into Brandon yesterday, Brandon today, Brandon tomorrow, and so forth, or maybe Brandon happy, Brandon sad, Brandon irritated, and so forth. That these are not real subspecies of Brandon would make no difference in the context of logical terms.) So we can go downward by adding specifications.
Thus we have three terms in our argument, and each can be either generalized or specified. In practice, however, it makes sense to leave the minor term out when looking at this aspect of arguments, because the minor term is always the application, and in practice when looking at the general structure of argument, what kind of argument it is, we ignore the particular application. So, for instance, if I replaced the world as a whole with biological organisms, it's still the same kind of argument, it's just applied to a different context. So unlike the major and middle, we will let the minor term vary freely. (It can still, of course, vary by generalization or specification; we just won't require it.) Changing major terms or middle terms, however, can alter the argument drastically, so we will require that they be either generalized or specified.
There is another way an argument can be varied, however, and that is by playing with modalities. Modalities in the sense used here modifications of the way in which a predicate applies to its subject. For instance, we could turn "What exhibits signs of design has a designer" into "What exhibits signs of design may have a designer". Modality typically has a clear direction (those that don't are usually indistinguishable from specifications of terms), so that we can move to a stronger modality or to a weaker modality. In the example just given, for instance, we moved to a weaker modality. The weaker modality takes the same terms, but it applies the predicate term more weakly to the subject than the original premise. Let's call a shift to a stronger modality boxizing and the shift to a weaker modality diamondizing, after the usual way we represent these. Modalities can be any kind of modification -- time, space, possibility, permissibility, or what have you, as long as it affects the connection of the predicate term to the subject term. (The reason we treat these variations as different from variations of terms is that treating them as the latter causes complications when dealing with middle terms.)
So let's coin a phrase and say that any argument A is in the local n-neighborhood of any argument B if you can start with argument A and change it to argument B with n and no more than n of these allowed modifications: generalization of major term, specification of major term, generalization of middle term, specification of middle term, substitution of minor term, boxizing of major premise, diamondizing of major premise, boxizing of minor premise, diamondizing of minor premise. Since generalization/specification and boxizing/diamondizing are simply pairs of opposing logical directions, n is always less than or equal to 5.
So if we look at the following argument (just giving the premises):
Whatever exhibits signs of design must have a cause
Computers always exhibits signs of design.
This is in the local 4-neighborhood of our original argument: you can get it from the first by generalizing the major term, substituting the minor term, boxizing the major premise, and boxizing the minor premise.
Here's another argument:
Whatever is an effect has a cause.
Biological organisms are effects.
This is in the local 3-neighborhood of the original argument: you can get it by generalizing the major, generalizing the middle, and substituting the minor. It is in the local 4-neighborhood of the argument we just gave, since you'd have to generalize the major term, diamondize the major premise, diamondize the minor premise, and substitute the minor term. In both these last two we are seeing the connection between design arguments and causal arguments more generally.
Whatever outside of me that exhibits signs of intelligence has a mind other than my own as one of its causes.
Other people's faces are things outside of me that exhibit signs of intelligence.
This is in the local 3-neighborhood of the original argument, in the local 5-neighborhood of the second, and in the local 3-neighborhood of the third. And we could go on, of course.
There's obviously a sense in which this is a crude measure of the 'neighborhood' of an argument; we often treat arguments differingly only by modality as much more similar than arguments with different terms, and arguments with specified terms to be in some way 'closer' to the original argument than arguments with generalized terms, and so forth. But part of the reason for this is the reason why we would be interested in talking about the neighborhood of an argument at all. One context in which neighborhoods of an argument play an important role in our reasoning is when we are dealing with refutations or possible refutations. For instance, if I refute argument A, this may have implications for other arguments in the neighborhood -- refuting one argument is a fortiori refuting another. But this is not an uncontrolled chain reaction: refuting an argument only refutes other arguments a fortiori in a few neighborhoods at most. The reason, of course, is that a fortiori reasoning is often a form of reasoning using genus and species or a form of reasoning between modalities, both of which are captured by the concept of the local n-neighborhood. On the other side, if an argument is refuted, the obvious question is: is there an argument in the neighborhood that is not refuted? And one way we could answer this question is by determining arguments that are unrefuted by the refutation that are in the local n-neighborhood with the lowest n. This is actually a form of analysis all reasonable people already engage in to some extent: local n-neighborhood is just one way of thinking about the way in which arguments are similar to each other in non-superficial and non-arbitrary ways.
There is another way in which this is relevant. If we look at the last example that we showed above, for instance, it is not a design argument but what would be called an argument about other minds, which is often treated as a distinct philosophical problem. We can obviously distinguish the two kinds of problems, design and other minds, but they are distinguishable in ways that nonetheless don't make them easy to separate. The two are connected historically; ever since Berkeley, other mind arguments and design arguments are mutually spawning -- if a particular kind of other mind argument develops, it is often followed by a design argument working in a similar way, and if a particular kind of design argument develops it is often followed historically by someone developing an other mind argument along similar lines. Even taking the history out of the question, we can easily see that there are lots of similarities, and when we're dealing with different kinds of arguments, we can often use the same classification system to classify either. And it's an interesting exercise to pick some random design argument and turn it into an argument for other minds and vice versa: the arguments won't always be equally plausible, and the design arguments, like the computer one above, won't always be theologically suggestive (they usually won't, in fact) but you can always do it very easily. They are in each other's neighborhoods. Indeed, I think that every possible design argument has some other minds argument in its local 1-neighborhood, and vice versa. When we're dealing with summary-syllogisms of the sort we're considering, the existence of a designing cause would have to be in the major term of the design argument, so all you actually need is a generalization of the major term, going from minds causing as designers to someone else's mind causing somehow; starting with other minds it's just the reverse.
It's worth noting that both design arguments and other mind arguments are related in a similar way to causal arguments for the existence of one's own mind, which tend not to get the same philosophical press. But one thing this means is that a causal argument for the simpler case of how one knows one's own mind exists could potentially shed light on possible arguments for other minds. This is another potential use of thinking along these lines.
There are no doubt other things one could mean by one argument being in the neighborhood of another; they too are worth developing, but this one seems a fairly easy concept to explore, and whose exploration seems to have some possibility of leading to new discoveries about arguments.