Jon S, Ben, Jeff, List, I'm going to take one last shot at explaining why I think the 3 inference patterns should follow the paths (or vectors) I've argued for in recent posts (and for the past decade and a half). But first some general vectorial preliminaries which I hope will be helpful in following my argumentation.
In the chart below the six possible vectors are reproduced from a 2012 paper of mine, arranged simply by the category from which they commence. So, working from top to bottom, 2 begin at 1ns, 2 at 3ns, and 2 at 2ns. In addition, I will refer to those vectors which reverse a categorial path as in "inverse vectorial order." So, for example, at the very top of the chart, 'process' (1ns/3ns/2ns) is in 'inverse order' to 'aspiration' (2ns/3ns/1ns) at the bottom. (Additionally, a vector may 'mirror; another--such as 'representation' and 'analysis' do at the middle right of the chart--while mirroring has little bearing on the present discussion, 'inverse order' most certainly does). [image: Inline image 1] The first two vectors at the top of the chart, process and (dialectical) order, commence at 1ns. 1. 1ns/2ns/3ns, *vector or order* (more precisely, dialectic or, as Peirce terms it, "dilemmatic" order, see CP 1.491). In one sense, which Peirce outlines in "The Logic of Mathematics," Hegel's dialectical order (1ns/2ns/3ns) can be seen as a kind of *ur*-vector. Peirce even suggests in some places that his own categories were inspired by Hegel's order, but appear "in new costume." The Hegelian order (thesis/antithesis/synthesis) is, in Peirce, something/other/medium. As I see it, this is not an inferential path, but logically and categorially much more primitive. 2. 1ns/3ns/2ns, *vector of process*: While in "The Logic of Mathematics" Peirce refers to the vector just discussed as Hegel's *evolutionary order *(CP 1.490), Peirce's own order for biological evolution has 3ns (not 2ns) as the second step or stage. I have also discussed how this is as well the order of at "complete inquiry" (N.A.) with hypothesis generation (1ns) followed by the deduction of the implications of the hypothesis for the purpose of testing it (3ns) leading to the divising of an inductive test and the actual experimental testing of the hypothesis (2ns). As important as both of these vectors clearly are, as I see it, none of the 3 inference patterns follow either of the above two paths. The next two vectors at the middle right of the chart,* representation* and *analysis*, commence at 3ns. 3. 3ns/1ns/2ns, *vector of representation*: As I pointed out in my first paper on Peirce's trichotomic (category theory) http://www.iupui.edu/~arisbe/menu/library/aboutcsp/richmond/trikonic.htm , along with Peirce's discussion of Hegel's order and it's reverse (the vector of analysis, or, involution, to be discussed below), the secondary source material which got me most keenly interested in categorial vectors was Robert Parmentier's "Signs' Place in Medias Res: Peirce's Concept of Semiotic Mediation" (1985). In it Parmentier argues that, in his opinion, Peirce has placed too much emphasis on the *vector of determination *(2ns/1ns/3ns) as opposed to its inverse, the *vector of representation *(3ns/1ns/2ns)*.* While I did not then, and do not now agree, with Parmentier on this point of vectorial emphais, I did lift from that paper theterminology he used for these two paths (i.e., vectors). Parmentier saw the *vector of representation* as that path taken by the scientist in hypothesis formation (or the artist in creating a new work), and on this point I have always agreed with him. So, as I see it, this is the path of one of the patterns of inference, namely, abduction. In this variation of the bean example, I will *presume* that all the beans from this bag are white *because* I see a handfull of white beans lying near that bag, and so I suppose that a sampling of the bag will show them *possibly* to be white. (But, in fact, the present bag of beans may turn out to be black, and the bag of white beans from which they came had, in face, been earlier removed, and, although it has no actual bearing on the inference, perhaps, later found to be the missing bag from which the handful came). 4. 3ns/2ns/1ns, *vector of analysis* or, better, *vector of involution*: Although Peirce uses both analysis and involution to describe this path, which he offers as the inverse of Hegel's dialectical 'evolution', I have more or less come to prefer the expression *vector of involution *as it suggests how each of the categories *necessarily *involves the others when analyzed: 3ns involves 2ns and 1ns and, in turn, 2ns involves 1ns, but 1ns 'involves' only itself. This is offered in "The Logic of Mathematics" as a mode of derivation of the categories themselves, peculiar in that 'first' one has 3ns (Peirce remakrs that it may be difficult for certain minds, indeed, some of the very best, to grasp this involutional order). I argue that it is also the path of deductive inference since, as I see it, the rule (a law) involves a case (the sampling, as of the beans) which in turn involves the result (a character, in our example, *whiteness*). That character will necessarily follow from the rule. [Rule: All the beans in this bag are white; Case: this *sample *of beans is from this bag; this sample will *necessarily* be white.] 5. 2ns/3ns/1ns, *vector of aspiration*: As I see it, this vector is not involved in any of the inference patterns, but represents, for example, a community of, say, scientific interest, pooling its intellectual resources towards the accomplishment of some shared goal. 6. 2ns/1ns/3ns, *vector of determination*: Perhaps the most familiar of the vectors (although Hegel's dialectical order may be a close second). This, of course, is the path of semiosis itself: the object (2ns) determines the representamen, or sign (1ns) which, in turn, determines the interpretant (3ns) to stand in the same relation to the object which it (the representamen) stands. This appears to me to be the path which inductive inference follows. A large sample (2ns) of beans from a bag reveals a character (all white) which suggests that *probably* the entire bag is white (the rule). Comment: As I see it, 3., 4. and 6. above are the only vectors involved in inference. Most importantly, t o me, 2ns is , in particular, associated with *case* principally because of the idea of *sampling*. Now a sample, in at least physically based experiments (but one can extrapolate to other types of experiments as well ) , is an *existential* something : Sampling is, as I read Peirce, associated with 2ns , *not* with 1ns. So this is still how I see things for the three inference patterns: 1ns character (for, as I see it, 'result' only applies to deduction) |> 3ns rule 2ns case (i.e., existential case, a sampling, for prime example) So, i n the bean example, in the case of *deduction *the actual sample (2ns) will *necessarily* be of the color (character, 1ns) of the bag of beans which represent s the rule (3ns). In *induction*, one begins with the existential sample (again, 2ns as I analyze it), then finds the sample to have a certain character, viz., whiteness (1ns), and posits that the rule (3ns) will *probably *be that all the beans in the bag will prove to be white. In *abduction*, my abductive variation o f the bean example , one needs in a n important way to see all three phases all-at-once-together (as Matthias Alexander might have put it ; or as Ben Udell recently wrote, "you have to look at the inference as a whole" ), so that I *presume* a rule (3ns) is in effect, that is, that all the beans in this bag are white, *because* I see a handful of white (1ns) beans nearby which I imagine to *possibly* be from that bag *were* a sample (2ns) to be taken. [ As a further step in my inquiry, I might take that sample and find that all the beans are , in fact, not white but black . I now look for another explanation and discover that some of the bags of beans were earlier removed including the one with all white beans ; in this case my hypothesis turned out to be incorrect . ] If one reverses 1ns and 2ns as Jon has suggested, making 1ns the case and 2ns the resul t , then one arrives at very different, and as I see it, virtually impossible vectorial paths. This is how Jon may be seeing it: 1ns case |> 3ns rule 2ns result (perhaps, 'fact') Following Jon's suggestion, deduction would be 3ns/1ns/2ns, following the *vector of representation* (rather, than what I'm arguing, the *vector of involution, or analysis*). But while a clear case can be made. I think, for the rule *involving* a sample (2ns) *revealing* a character (1ns), I don't see how representation , at least as seen by Parmentier and me, makes any sense for deduction. Indeed I have argued rather pretty much the 'opposite': that abduction--and* not *deduction--follows the *vector of representation*. For in abduction one imagines a rule (forms a hypothesis) which may or may not represent the facts of the matter *when* a sampling is *actually* taken. So, from my standpoint Jon incorrectly reverses the categorial order for deduction and abduction. Similarly, if one reverses case and result for induction, one is following the Hegelian *vector of order*, and, rather than starting at a sample and arriving at a *probable* rule, one starts at the result (1ns, a character in my analysis). I must say that this just doesn't make any sense to me. As I see it, you have to *start* with an actual, existent sample in induction. And, to repeat, Hegelian dialectic seems far too primitive (basic) a vector than to be any given inference pattern. I agree with Jon that we may not reach agreement on this matter. But this is how I continue to see the 3 inferences patterns vectorially. I have reviewed his (and other comments) and find that I'm content to stick with my original vectorial associations for the 3 inference patterns. On the one hand, this might not amount pragmatically to a hill of beans. But on the other, consistency ought to count for something in the vectorial part of Peirce's trichotomic. Best, Gary R [image: Gary Richmond] *Gary Richmond* *Philosophy and Critical Thinking* *Communication Studies* *LaGuardia College of the City University of New York* *C 745* *718 482-5690 <718%20482-5690>*
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