TNR Proofs

The proofs offered here were originally published on Peirce-L in 1997. Some of them are presented in a more concise and more elegant form in the 2014 paper written for the conclusion of the series on Biosemiotic Entropy (2012-2014) which can be downloaded in its entirety for free by clicking here. The rest of what appears here is from unedited email exchanges on Peirce-L:

From joller@unm.edu Tue Mar 4 12:36:42 1997 Received: by ttacs6.ttu.edu; id AA09000; Tue, 4 Mar 1997 12:36:42 -0600

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Date: Tue, 4 Mar 1997 11:36:02 -0700 (MST) From: John Oller <joller@unm.edu> To: peirce-l@ttacs6.ttu.edu Cc: Multiple recipients of list <peirce-l@ttacs6.ttu.edu> Subject: TNR-Theory Intro Part 1 In-Reply-To: <BMSMTP8574312374tsander@popd.ix.netcom.com> Message-Id: <Pine.A32.3.91.970304092717.100852C-100000@callisto.unm.edu> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII

What follows here is specifically addressed to Tom’s question, which is quoted just below, but also to commentary by many persons of late including Hugo and Joe R., as well as anyone who may have an interest in these matters. I will try to be reasonably brief in each of the planned instalments on the subject of TNR-theory and will try to avoid excursions that would keep us from the goal of completing the exposition in a half dozen or so reasonably brief segments. I think the relation to Peirce will be obvious to those familiar with his writings and thought, but may need occasionally to be pointed out. Perhaps I can prevail upon colleagues on the list for help there. As for bibliography, I will be somewhat vague as in a conversation except possibly to refer occasionally to some published work where the specific references are given in detail.

On Mon, 3 Mar 1997, Tom Anderson wrote:

> John Oller wrote: > > >But that aside, TNR-theory develops fully the formal differences > >between representations that are based on the competent > >judgments of an intelligent observer and ones that are invented, > >imagined, owed to illusion, hallucination, etc. Hypotheticals also > >fall into the latter group of systems. > > What are those marks, John? > > Hypotheticals are also fictional, I take it? What about idealizations? For

> example, what about the idealization of a frictionless plane that’s proved so > fruitful in physics — I don’t think it would be an exaggeration to say that a > large part of the mechanics we all use and a good part of the theory about > mechanics is based directly on such idealizations. Furthermore, those > idealizations are false. What about idealizations in linguistics — an ideal > speaker-hearer? A uniform language? What about unobservables, as atoms were > for a long time, or electrons, photons, etc.? These were entities the positing > of which made sense in the context of explanatory theories and data, but could > not be independently observed? > > What are the marks that distinguish the true from the false, fictional, > hallucinatory, invented or imagined narrative? > > Does your theory — by distinguishing between the true and the invented — help > one to settle questions about whether mathematical work invents or discovers? > Perhaps, if there are marks of the true, one could just look around for them > among mathematical entities and settle the question that way? > Tom Anderson

Right. In a sense this is exactly what we do. But to address the deeper question rather than the rhetorical surface of it, we must look to the matter of how meanings are associated with signs in the most fundamental sense. This is a non-trivial problem, but it has often been either trivialized or side-stepped or swept under the carpet. Linguistics in America has tended to apply the Scarlet O’Hara approach–“I’ll think about that tomorrow.” But the question of how signs come to be vested with particular meaning requires to be addressed very explicitly. Also, just posing the question in this way shows that grammatical theory and the problem of language acquisition (by children) are logically interrelated so much as to address almost the same problem.

Peirce noted that the relation between mind and matter was an enigma and Einstein said there is an unbridgeable gulf between the realm of hard objects and events on the one hand and the realm of ideas and conceptualizations on the other. I call this separation “Einstein’s Gulf”. John Searle and others, of course, deny its existence, but I think Peirce always respects the fact that between the particular matter found in space and time and the realm of ideas there is a huge difference. For instance, he points out that we can easily do experimental manipulations with concepts, ideas, and signs, but with objects, events, and relations in the material world things go very differently. In the latter case our resources are limited by space, time, and by our own bodily constraints. We can, by comparison, do much less in the realm of hard objects than we can do in the realm of ideas. For example, falling from a height can kill you, but imagining such an event is not so costly. Still, the one bears some resemblance to the other and this is what needs to be worked out.

So, here is the question to be addressed: how comes it that signs acquire meaning and material content by their association with the world of hard objects and events?

Some might suppose that I have biased the outcome in advance by putting the question in this way, but I hope to show in what follows that this is not the case. Furthermore, I aim to apply Peirce’s logical method that introduces no distinction before it is shown to be necessary and adds no proposition to the developing theory before its necessity is demonstrated. I think this fairly sums up his “exact” logic as contrasted with less necessary (less deductive and less mathematical) varieties. At any rate, it is the method I aim to follow and if I deviate from it, any one of you would do a great service to show where. The beauty of the method is that it presupposes next to nothing and its results are both the proof and deduction of the method itself. That may sound like the proverbial traveler who pulled on his (or her) bootsraps to get over an obstacle, but the remarkable thing about signs is that the apparent circularity of Peirce’s method is only illusory. The method works, or so I believe and will endeavor to show.

(Peirce’s method, however, unlike Euclid’s and its derivatives is *not* axiomatic. It does not beg our indulgence to accept a list of possibly doubious and unproved assumptions before the arguments proper of the system can be launched. Hence, Peirce’s use of the term “exact” as opposed to, we may suppose, “inexact” methods. Incidentally, Tom, Tarski’s approach to truth in formal systems, I believe, involves a kind of question-begging in the distinction between formal and other systems, but an excursion along that road would lead us astray of our objective here.)

I begin a few steps prior to the introduction of the definition of a true narrative representation. I begin, in fact, about where I think Peirce did in his remarks in 1865 about things, forms, and representations. My vocabulary choices are a little different and I take a couple of steps that he did not bother to make explicit. I begin where he did with representations, or signs.

Now, suppose someone says they have no idea what we are talking about here and that they doubt even the existence of signs. After reflecting they may even get around to denying that any signs exist. It will be found in all arguments of that kind that the complainant (skeptic) is obliged to say something like, “I show by these signs, etc., etc., that the existence of signs can be reasonably doubted or denied.” But for all such arguments, purely examined on a formal basis, it will be found that the skeptic invariably employs (and to get the argument stated, must employ) a profusion of exemplars (signs) of the very kind of which they claim to doubt the existence. Therefore, it cannot reasonably be doubted that signs exist. Q.E.D. That is, a first step in our argument is taken and has been proved necessary before it was taken.

It is true that I left out the step that shows it necessary in any theory to aim at a bare minimum at consistency. This can also be proved in a very simple, but not quite trivial way, by showing that the only other alternative (inconsistency) is unreasonable and leads immediately to an unnecessary deadend. That is, denying the need for consistency leads immediately to an inconsistency. Hence, consistency is necessary. Q.E.D. (If one chooses to call this, i.e., the need for mere consistency, an a priori presupposition, I will not object much.)

Next, one might question the existence of objects such as the sort that might conceivably provide a material basis for signs. Let the skeptic, then, come forward with an argument against them. It will be found that on a purely formal basis the skeptic will be obliged to employ signs in the denial or doubt about the existence of objects and further that these signs will be formed as bounded objects. If, for example, the argument is written down (as was the former argument claiming to doubt the existence of signs) that it consists of various bounded objects including letters, spaces, words, phrases, sentences, and the like. Moreover, since it is utterly impossible to frame any argument against objects as existing entitites without employing such objects as are found in every sign, every such argument is absurd. Objects exist. Q.E.D.

But, the skeptic may come back with a further kind of doubt: suppose the skeptic chooses to doubt that signs have intelligible relations with objects. Yet, here again, as before, the skeptic will be obliged to show his or her reasons in the form of signs that are like this one: “I show by these signs and for these reasons that intelligible relations between signs and objects are subject to doubt.” Yet every such argument, it turns out, by its very form either establishes intelligible relations between its signs and the person or persons using them or is unintelligible. Further, every such argument consists of signs that are employed as objects in an intelligible relation to each other and to the person(s) using them. In all these ways, the claims of the skeptic’s argument must be judged absurd. Intelligible relations between signs and objects cannot reasonably be doubted. Intelligible relations between signs and objects exist. Q.E.D.

Now, it turns out that the foregoing arguments provide a solid ground for each of the three principal elements of a true narrative representation (TNR). Here I must insert a scolium which is no part of the proofs upon which the theory of TNRs rests.

Scolium: A narrative, of course, is merely a story that unfolds over time. A true one is merely one that asserts nothing false about its facts and in which whatever is asserted is actually delivered materially by the facts (including material objects, events, and relations between them, relations between relations, etc., in space and time as facts). Also, upon examination it will turn out that every TNR has a well-balanced form in three parts where certain sensory signs are set in correspondence by a competent observer of the reported facts such that they correspond also to the acts of that observer in noting the facts and to the conceptual signs (linguistic ones if they are published in a communal form) which are produced by the narrator. Nota bene, however, that this scolium has no place in TNR-theory. It is merely an aid to comprehension of the theory.

Next, let us suppose that the skeptic, our unwitting assistant in all of these endeavors, should wish to deny or doubt the existence of TNRs. It will be incumbent upon such a person to come forward with his or her argument: “I show by these signs, etc., that TNRs are doubtful and may not even exist.” Oddly, it turns out on account of the obligatory (the inevitable) form of such an argument that it constitutes a perfectly acceptable examplar of the very sort of thing of which it purports to doubt the existence. Since, every such argument is absurd, TNRs cannot reasonably be denied existence. Hence, TNRs exist. Q.E.D.

It remains to examine in the next several instalments (about 5 more, I think) how TNRs are structured in ways that make them utterly unique among all possible representations. The task sounds daunting, but it is not so difficult as it sounds. Further, in pursuing it much is to be gained. In the meantime, perhaps someone may wish to show how or where any of the foregoing demonstrations is flawed. Failing that, I will assume it safe to proceed, after a brief respite, to instalment 2.

Here ends Instalment 1 of TNR-theory.

**********************************************************************

Another thought. For whoever may be wondering just what steps I added to Peirce’s argument of 1865, they are these. He showed that no inconsistency arises in a view of things, forms, and representations if we assume that all three exist. This he proved by the argument that things and forms are known only through representations so that if the latter have consistency among themselves they have, as he put it, all the truth that the case admits of. My extra steps involve the assistance of the skeptic who would deny the consistency of the signs that give rise to the common belief that there is a world and that we are in it. I merely show that a necessary (absolutely inevitable and fatal to the mere consistency of any theory purporting to be about signs or their subject-matter) must arise if the mere consistency of signs should be denied or otherwise abandoned. I take this step with respect to signs first (which Peirce says we know “absolutely”, and I agree). Next, I take it with respect to objects. Then, with respect to intelligible relations between signs and objects. Then, with repect to the existence of what I am calling, loosely, at this stage of the argument, but to be greatly tightened up and refined at subsequent stages, the TNR.

Finally, I want to point out a couple of other sources for all of the foregoing and what is to follow in the exposition of TNR-theory. Since graduate school, for me, thirty-odd years ago, I have been aware that there must be some kind of very special relation between truth and meaning. I was encouraged to believe this through the influence of John Dewey on my father and later through reading Russell and Reichenbach in grad school. That was when I met Peirce for the first time, but finding him difficult and a bit too thick for the time I had on my hands, I set him to one side. Back then I tried to approach the subject from several directions on the basis of the working assumption that language acquisition cannot proceed at all without reference to appropriately applied signs, “true ones” in the most common sense of this phrase. I thought set-theory was somehow going to provide the apparatus I needed and was somewhat encouraged along that line by Reichenbach and others. In 1972 I had the good fortune of meeting Carolyn Eisele at a conference in Europe. I mentioned Peirce, James, and Dewey in a talk I gave on induction, mind, and the contextualization of materials to be learned. She took me aside to tell me a bit more about Peirce, and to correct my pronunciation of his name. I’d said something of Cantor’s set-theory and she urged me to look at Peirce’s views.

I continued to pursue the idea from various directions over the next three decades or so, but it was not until I read Dewey’s book on Logic: The Theory of Inquiry (in the late 70s, I think), that I was pointed back to Peirce by Dewey’s remark in the preface that he’d found himself obliged to disagree with all his mentors except for Peirce. With Peirce the going is difficult if all you have is his Collected Papers, not to fault the editors of those, but that collection was and is comprehensible only to the dedicated Peirce scholars, which I wasn’t yet. Not till I began to read the Chronological Edition in the 1980s did the lights begin to go on. Somewhere along the way I began to get a glimmering that a solution to my long-standing problem with traditional approaches to grammar (including the Chomskyan varieties and most of his critics) was possible.

Peirce’s method of reasoning was the key, but at first I understood it through his applications and only later came to study the method itself. When I did get around to the method, at first I studied his statement of the method as interpreted in How to Make Our Ideas Clear, and related writings on what James and Dewey popularized as pragmatism. But that was not really the soul of Peirce’s method–nor yet is it found in that popularized (because of editorial influence I think) version of his method in his own words known as the pragmatic maxim. The real heart of his method, I think, and I here I offer only an opinion, was his use of what he called exact logic. In the dynamism of that method, I believe, was found all the gold that he mined from that outcropping of a rich vein that led him to a mental variant of the proverbial “embarrassment of riches”. As he applied the method, in whatever field of endeavor, it seems, the riches came out faster than he load them up and haul them to a smelter. This, I believe, was what he meant by his “gift to the world”. Now on all this last, I only am giving hunches subject to correction,

dispute, etc. I even believe that it was his logical method that he saw as his goal in life to develop (I refer to the cryptic entry in his diary).

Dear Peirce-scholars, colleagues, and friends,

What follows here is actually a third try at e-mailing the second instalment of an introduction to TNR-theory. Let me pick up a couple of loose ends left in the previous instalment and then proceed with proofs of the pragmatic perfections of TNRs. The loose ends to be tied up involve the following quotes from Peirce and Einstein which I referred to but did not provide in the previous version:

Einstein wrote:

We have the habit of combining certain concepts and conceptual relations so definitely that we do not become conscious of the gulf—logically unbridgeable—that separates the world of sensory experiences from the world of concepts and propositions (1944, p. 287).

Peirce expressed much the same problem in a different way in 1902:

It is a fundamental position of logic, without which there can be no distinction of truth and falsity, certainly no falsity, that being and being represented are entirely different (Manuscript L75, p. 11 in Ransdell, 1994).

I cannot resist including here an illustration. Remember the Richard Prior routine

where he plays the man caught in the very act of unfaithfulness with another woman by his own wife? He says, “Well, what are you gonna believe, me, or your lyin’ eyes?” He could have made his case a little more extreme by saying, ” . . . me, or these lyin’ facts.” What the illustration shows is that facts in and of themselves just are what they are and can neither stand in error, nor can they lie. As a result, Einstein’s Gulf is quite genuine. John Searle argued sometime ago in a lecture here at the University of New Mexico (in the 1980s) that “there is no gap” between representations and material things. But Prior’s joke shows just how wrong Mr. Searle is.

Peirce described the problem of Einstein’s Gulf more briefly:

Mind [is] quite as little understood as matter, and the relations between the two an enigma (1896, in Hartshorne and Weiss, 1931,

p. 47).

It is the enigmatic middle ground that demands attention. Einstein (1941) wrote of science and everyday thinking that

everything depends on the degree to which words and word-combinations correspond to the world of impression (p. 112).

I refer to that enigmatic middle ground between abstract and general representations on the one hand and concrete particular facts on the other as “Einstein’s Gulf”. Our problem in understanding language acquisition and the growth of concepts in general is *to account for the vesting of abstract signs in general, i.e., all of them that get vested, with material content*.

How are relations between abstract signs on the one hand and concrete facts on the other initialized? This is the key question addressed in TNR-theory.

In TNRs as we saw from the last instalment, an essential triad of relations emerges: 1. There are the facts themselves which are independent of however we may represent them to be. 2. There are the acts we perform as sign-users to notice and represent the facts. 3. There are the signs we produce relative to the facts through our actions.

TNR-theory shows, explicitly, that the formal structure just described (every part of which, and the whole as well, was proved to exist in the previous instalment), leads to certain “pragmatic perfections” that are found in TNRs but not in any other sign systems whatsoever. Moreover, the perfections in question are strictly a consequence of the formal structure of TNRs as contrasted with the formal structure of other sign systems.

In describing these particular perfections, we use the term “pragmatic” because the formal completeness at issue in each instance concerns actual particular facts that have pragmatic (practical and active) status in the experience of some person or community or group of communities. The term “perfection” usually draws fire, but it is really quite innocuous. It only means “completeness” in a formal sense. That is, a pragmatic perfection is merely the opposite of a pragmatic “degeneracy” (or incompleteness) in the mathematical or logical use of these terms. In the same way that a triangle used to show a certain view of a cone as seen from the side, is degenerate with respect to one of the three dimensions of the cone, so every fiction can be shown to be a degenerate (imperfect, incomplete) in comparison to one of the three elements of a TNR. This is all that is meant by the term “perfection”. It is used as an opposite of “degeneracy” (both terms being applied in a strictly formal sense). In other words, I use Peirce’s sense of the term “degeneracy” and “perfection” only in contrast to that.

(The next three proofs lean on the definition of “general” which is not thoroughly justified until we limit and divide the *universe of signs*. However, once that is done, every step of these proofs is demonstrated and there is no lapse. I have presented them in this order merely for the sake of comprehensibility.)

*The first pragmatic perfection*, and the one on which all the other perfections (pragmatic, syntactic, and semantic) of TNRs can be shown to depend, is the *determinacy perfection*. By determinacy all that is understood is the difference between something completely undetermined, e.g., a point on a great circle or a point in a great sphere or anywhere in the space-time continuum that is unmarked with respect to its place, as contrasted with a point marked by coordinates relative to some situated observer. To proceed we need to show that this perfection actually inheres in the intrinsic structure of every TNR. Now, it is clear that every TNR has this sort of determinacy because every TNR involves particular facts (i.e., material content) determined by the viewpoint of a particular observer situated in space-time. Q.E.D.

*The second pragmatic perfection”, which derives directly from the first, is the *connectedness perfection*. By connectedness all that is meant is that the sign that has it must be connected not only to the particular facts that it singles out for attention but to all the other facts with which those particulars are joined. That is, if point M stands between L and N on a line, a sign that designates M and is connected to it must also be connected to L and N and to any points with which the latter are also connected, ad infinitum. But every TNR has this connectedness property because every TNR is determinately connected to some particular(s) situated in the matter-space-time continuum and the latter, owing to the nature of the continuum, connects the particulars singled out for attention to all the other particulars that the continuum contains. Q.E.D.

The *third pragmatic perfection* derives from either and both of the foregoing

and is the *generalizability perfection*. By generalizability, all that is meant is that any particular sign having it can be generalized with respect to whatever particular content it may have so as to apply to all possible particulars similar to the one (or ones) at hand (i.e., the ones singled out for attention by the TNR). Now, we are required next to prove that TNRs have this generalizability perfection. But, owing to the determinacy perfection, and the connectedness perfection of TNRs, their generalizability necessarily follows, for any determinate content of any TNR is connected to the rest of the continuum and, therefore, is generalizable thereto. Q.E.D.

(Here we may need to add the following scolium to make the immediately foregoing proof quite clear: The generalizability proof does not entail that dissimilars are generalizable to each other, but that similars are. Thus, any material content whatever that may be found in a TNR, which generalizes as the proof shows to the whole of the matter-space-time continuum, can be applied to any similar content wherever it may be found throughout the whole of the continuum. This, I believe, is the correct understanding of the proof.)

Next we must show that the foregoing perfections are absolutely unique to TNRs, i.e., that they are not found in any other system of signs among all those that are known to exist in the whole of the universe of signs. Therefore, with the foregoing proofs in hand, including the fooundational proofs from the previous instalment, we proceed to the next step by comparing TNRs against all other possible sign systems. This sounds next to impossible, but it is less difficult than it might appear to be at first. Based on the prior results, we can mark a limit, or boundary, to the universe of possible representations.

(Owing to the fact that this is the half-way point in the argument underway both from the view of length and also, more or less, its structure, I will conclude instalment 2 here and continue the argument immediately in instalment 3.)

Yours truly, John Oller

********************************************************

John Oller Phone office

Department of Linguistics home

University of New Mexico Fax 505-277-6355

Albuquerque, NM 87131-1196 e-mail joller@unm.edu

********************************************************

“The best maxim in writing, perhaps, is really to love

your reader for his own sake.”

C. S. Peirce, Mar 17, 1888 published in the opening of the first volume of the CE).

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Tom, the questions that you put about the special marks of truth will be addressed more fully, I think in the whole collection of about six instalments, but here I have given the most important mark which is consistency. As for hypotheticals, fictions, etc., these will be fully treated in subsequent instalments, provided only that there is any remaining interest.

Kindest regards to all, John Oller

******************************************************** John Oller Phone office

Department of Linguistics home
University of New Mexico Fax 505-277-6355
Albuquerque, NM 87131-1196 e-mail joller@unm.edu

******************************************************** “The best maxim in writing, perhaps, is really to love your reader for his own sake.”

C. S. Peirce, Mar 17, 1888

.

What follows from here forward is instalment 3 which continues immediately from instalment 2.

Based on the prior results we know that anything altogether unnoticeable to any sign-user cannot serve as a sign. This limit was noted by Peirce. A sign-candidate that was completely unnoticeable (by all possible means) would be the sort of thing that could relate to nothing but itself and so could not be a sign to any sign-user (nor community) nor to any object other than itself. Such a sign-candidate would implode, sucking up meaning like a black hole sucks up light (I owe this metaphor to my friend and former student Steve Krashen in a discussion over dinner about two years ago). For the same reason, a sign-candidate utterly void of meaning, i.e., of any intelligible relation with any object outside itself would also be outside the limit. Hence, a limit is both set and proved to exist. Q.E.D.

Next, we must divide the universe of sign systems (only meaningful ones being included and nonsense being excluded) exhaustively into signs of particulars and signs of generals. Now, the existence of particulars was demonstrated in the preceding proofs (instalment 1) about (1) signs, (2) objects, (3) intelligible sign-object relations, and (4) TNRs. Further, particulars are attested both in the parts (all three of them of every TNR) and in the whole of every TNR. Moreover, all actual pluralities that consist of particulars must also be particulars. Thus, a general (i.e., a representation applying to all possibles of a given kind) is entirely distinct from a particular or a plural. This is because, no particular and no plural can apply to all possibles of a given kind owing to the particularity of particulars and plurals. Hence, generals are a distinct and non-overlapping system of sign systems. But since particulars stand at a limit of particularity and generals stand at a limit of generality, and owing to the already established fact that particular plurals fall short of generality, these two grand systems, particulars and generals, are exhaustive and fill up the whole of the universe of possible sign systems.

Q.E.D. (Here I note that the Peircean distinction between existential and universal quantifiers, now an accepted part of modern logic, is accounted for in a simple and straightforward manner.)

Next, it is necessary to examine all possible particulars and all possible generals to see if any of them, in addition to TNRs, have the pragmatic perfections already proved to reside in TNRs.

First take particulars. These can be exhaustively subdivided into TNRs, and fictions. Fictions are produced by relaxing the requirement on TNRs that they must have some particular material content that is manifested in space and time. Fictions, therefore, are less determinate with respect to their material content than are TNRs and lack the determinacy perfection. But, they also lack the connectedness perfection for the same reason, and because generalizability of particular content cannot be attained without any determinate particular content, fictions also lack generalizability. We hardly need to examine errors and lies because each of these can be shown to be more degenerate in all three respects than fictions are. Errors are produced by degenerating for the factual element and the representational part of the TNR structure. Lies are produced by degenerating all three elements of the TNR. Next, consider generals. But we have already proved that among all particulars only TNRs have the pragmatic perfections, therefore, the only source of any particular meaning in any general whatever is a TNR. Thus, the pragmatic perfections are absolutely unique to TNRs. Q.E.D.

********************************************************** Scolium: The foregoing proof can be most easily understood by seeing that every TNR has three crucial parts: (1) it has particular facts involving objects in space and time; (2) it has coordinated actions of a competent observer who notes the facts by coordinating his or her actions in such a manner as to produce a sequence of signs suitable to mark those facts perceptually; and (3) every TNR has a publicizable representational form for the signs that corresponds as perfectly as it purports to correspond to both of the foregoing elements. Call these elements F-LR where F stands for facts known in the sensory way, L for links between those facts and certain conventional representations, and R stands for the representations.

In a TNR all three elements are present in a non-degenerate form, i.e., each TNR has the full form, F-L-R. But in a fiction, F is degenerate yielding a structure that resembles a TNR but does not attain to its perfections. Let *F*-L-R show the fiction (including hypotheses not yet tested against particular facts, fantasies, imaginings, etc.) where the asterisks mean merely that the element in question is missing a complete material instantiation while the other two parts, L, and R, remain in tact.

Then, for an error (including all illusions, hallucinations, and the like as special cases) we find a structure that looks like this: *F*-L-*R*, where the facts are not what they are thought to be and the representation of those facts is not what it needs to be to represent the facts correctly, though the linking between the presumed facts and the supposed true representation remains in tact.

In a lie (or any deliberate and intentional deception, setting aside distinctions that can be made concerning degrees of deviousness) all three elements are degenerated including the linking.

As an example of a fiction consider the difficulties associated with measuring or determining the properties of the fictional raft of Tom and Huck, in Twain’s novel. Note that the same difficulties do not accrue to a similar factual case represented in a TNR. As an example of an error, take the case where a certain person is mistaken for another. For the case of a lie, suppose that a certain person, let’s just say, O.J. committed a certain crime or crimes that were later denied by that same person.

Now, I know that many people will be wondering about the same sorts of mixed cases that most troubled me when thinking through the foregoing somewhere near the first glimmerings of the possibility of developing such a series of proofs. What happens when fictions, errors, and lies get mixed together with TNRs very thoroughly? How are we to tell the wheat from the tares, so-to-speak? The simple and correct answer, I believe, is that we cannot. (I mention this especially in response to Tom Anderson’s query about the “marks” of the truth.) However, the logical proofs do not break down in the least with respect to mixed cases. An error embedded in an otherwise true story remains an error. An exaggeration of an event does not necessarily make the event go away. And so on. If anyone will take the time to think through such cases, all the way to the bottom (as Peirce would want us to do), that investigator will discover, I believe, that there are no cases that do not yield at least in principle to the analysis suggested by TNR-theory. This, Tom, is also the best response, I believe, to the general line of your commentary on the previous instalment.

Some might regard the idea that we need to seek consistency above all else as too stringent a requirement (I hark back to conversations previously with Bill E. and Will O.) This is just an error of thinking. Consistency is the least stringent of the possible requirements to be put on signs, though for that same reason it is the most essential for that very reason of all conceivable requirements. That is why “mere consistency” as Euclid, Pythagoras, Archimedes, Nichomachus, Reimann, Lobachevsky, Peirce, Einstein, and others have often argued, and have demonstrated in their many mathematical proofs, is the foundational basis for mathematical reasoning.

Just as we would not trust a banker who said, “I really don’t believe in squaring accounts or balancing budgets,” or a statistician who invented data, or a witness who denied the need to report what actually happened rather than what he or she imagined might have happened in a world of his or her choice, a theoretician who denies the need for consistency (as I believe the deconstructionists end up doing, though I think no genuine Peirce-scholars can really want to do) might as well withdraw into his or her own private fictional world.

The trouble for the rest of us, however, remains as it was before the pretend withdrawal of the fictionalists. As TNR-theory shows in an unmistakably clear way, we are, evidently, stuck in the same space-time continuum as are those would-be theoreticians who think that “anything goes” is a perfectly good rule for making sense of signs. TNR-theory shows with all due respect to their persons and viewpoints (both of them situated as they are in the matter-space-time continuum per TNR-theory) that those folks are mistaken. This is what I think Joe R., Alan M., Leon S., Peder C., Robert M., and a good many others have been saying in response to the attempted defenses of poststructuralism and its would-be cognates known under so many different names.

As before, in the first instalment, I want to invite the most careful scrutiny of the proofs of TNR-theory. Pardon me for saying it, but I am not so much interested in a rambling commentary examining the literary style, allusions to other works, conformity to the thought of so-and-so, and the like, though I am not totally disinterested in the latter. I feel sometimes (selfishly) flattered by all that and at the same time frustrated (if I am as honest as I can be) that anyone would suppose that such commentary was relevant to arguments that purport to be based in rigorous proofs. What is needed in response to the latter is not literary criticism (though it might improve the style, etc.), but rigorous demonstration of some flaw in the reasoning, i.e., in the proofs. Perhaps I should say “alleged” proofs, except that I sincerely believe them to be valid or I would not present them under the label “proof”. If I did not think them to be genuine I would not have the temerity to present them in this forum at all. I cannot say how uplifted I was by the remarks of Peder Christiansen on February 5 in response to the two little proofs about time and the discussion showing their connection with TNR-theory. I really suppose that if there were a flaw in those, Peder would see it and be able to point it up as soon as he had considered the argument carefully. I think the same should hold for the rest of TNR-theory and its corollaries owing to the Peircean method of exact logic as described previously.

At the risk of sounding self-congratulatory, I sincerely believe, based on my very limited acquaintance, that some of the best minds in the world are represented right here on this list. I have no doubt that the examination of Peirce’s thought alone is sure at least to help separate the cream from the rest and, there can be little doubt that the other thinkers to whom he directed us in his writings are also a cut above the average. For this reason alone, I often lie awake thinking and rethinking the conversations of the day, testing and re-testing, going over and over the details of the proofs I claim to have presented or the thoughts voiced by some colleague here in the ether-waves or in some other venue. Still, having admitted all this, some of it quite selfish and hardly worthy of much consideration, I can say without guile, I believe, that I cherish this interaction precisely because it purports to be a discussion that is philosophical in the best sense of the term–i.e., in pursuit of whatever may be true and worthy of our efforts. ************************************************************

In the next instalment, number 4, unless some fatal flaw should be shown in this one or the previous, I propose to deal with the syntactic and semantic perfections of TNRs.

With many thanks for listening and for all the comments, literary criticisms included, I am for better or worse still the same guy on the line.

Yours truly, John Oller

********************************************************

John Oller Phone office

Department of Linguistics home

University of New Mexico Fax 505-277-6355

Albuquerque, NM 87131-1196 e-mail joller@unm.edu

********************************************************

“The best maxim in writing, perhaps, is really to love

your reader for his own sake.”

C. S. Peirce, Mar 17, 1888

P. S. At the great risk of overstaying my welcome, I want to add a post script concerning a comment made by my much respected colleague, the linguist Dr. Joan Bybee, at yesterday’s brown-bag meeting of our department here at UNM. At that meeting I gave an overview of TNR-theory. Dr. Bybee pointed out that the representational part of a TNR, or any statement whatever, can, as she understands the case, be applied to any number of situations. She asked how TNR-theory can account for this.

For instance, I noted in an illustrative comment yesterday that Melissa Axelrod was present at the meeting to illustrate in part what was intended by a TNR. Joan observed, correctly, that this same statement could as easily have been made in reference to a meeting that took place on Monday, March 3 (yesterday being Wednesday, March 5). She also noted that what differentiates the two cases is mere “deixis” (a cognate in its root form, I think, for “index”) which, of course, is true. It does not satisfy her question merely to point out that the two cases are distinct TNRs, though this was part of my response at the meeting.

At the time the question was put, our hour for the seminar room had expired and I did not do justice to Dr. Bybee’s question. It is really, I think, a question about how TNR-theory can deal with the evident generality of signs. This was not clear enough in my talk, and it was both insightful, I believe, of Dr. Bybee to point this out and to ask for clarification.

A complete answer is given along the lines of the generalizability perfection of TNRs. Further, the proofs pertaining to that element show that only TNRs can be applied in determinate ways to particulars (e.g., Melissa’s being at the two meetings, and others), and that the power to apply any signs whatever in the manner described is entirely dependent on TNRs. That is, the generalization to cases other than the one at hand relative to any given TNR is only possible for these and other signs (including all fictions, errors, lies, and generals) to the extent that the TNR at hand and others like it provide a meaningful and determinate basis for such generalizations. All fictions get their meanings only by resemblance to more perfect TNRs and all generals (e.g., all human beings are mortal) also get whatever particular meanings they may have from TNRs. In this way, the kind of ambiguity (or polysemy) brought up by Dr. Bybee is completely accounted for.

Dear Peirceans,

We proceed next to the *syntactic perfections* of TNRs. Here, as in much of the preceding, the discussion would benefit greatly from access to a diagram. However, not being able to present one, I resort to a description of an imaginary diagram.

The syntactic perfections of TNRs are three and involve three dyadic relations. It turns out upon examination that the *pragmatic perfections* examined in previous instalments are essentially monadic. That is, the pragmatic perfections pertain to every TNR taken as a whole, while the syntactic perfections require its analysis into three distinct dyadic relations obtaining between each of three distinct kinds of sign-objects.

The first sign object is the one that stands in the position of the facts of the TNR. It is the *logical object* of the TNR-sign. Call it 1 and place it to the left side of our imaginary diagram.

(Scolium: The existence of such sign-objects as we refer to here has been thoroughly justified by previous proofs. Further, note that we generalize across all possible cases in this imaginary diagram that we are constructing so that all logical objects that might stand in position 1 are accounted for by the generalization. Therefore, the *logical object* of our generalized TNR can be construed as any singular object or any plurality of objects that can be represented in a sensory sign.)

Based on proofs given earlier concerning the nature of the phenomenal present, it is easy to see that this logical object can be extended indefinitely so as to include anything whatever that might be represented in an icon, up to and including the whole universe of matter-space-time, or any portion thereof. Let this position be represented in an icon. To be sufficiently neutral with respect to what this icon might represent, call it “O” for any logical object.

The second sign-object is the *bodily sign-user*. Call this position 2. Proof that it exists directly follows from the fact that all TNRs are produced by competent observers or pluralities of them. Let this position be represented by an icon of a “talking-head” as commonly seen on television or in films. Let this icon stand to the right of the *logical object* in 1; thus, 1 2.

The third sign-object is the *manifest sign* that forms the representational part of the completely generalized TNR. Let it be represented by an oblong in a vertical arrangement divided into two squares at the middle. Let the top half of this oblong be labeled as *the material content* of the *manifest sign* and the bottom half be labeled as *the noticeable form* of the *manifest sign*. Let this dyadic oblong be called position 3 and let it be placed to the right of the other two positions; thus, 1 2 3.

(Note that the two parts of the *manifest sign* have been proved to exist and fully justified by the prior proofs in connection with the pragmatic perfections of TNRs. Also, we may as well also note in passing that the top half of 3 is very near what linguists call the “semantic value” of linguistic signs and the bottom half is the “surface-form” of such signs. In our conception, however, surface-form must allow for the full complexity of the manifestations of linguistic signs as uttered, heard, written, read, thought, or otherwise manifested–hence, the vague term, *manifest sign*.)

Now let an arrow labeled *alpha* be drawn over the top from 1 to the top half of 3 so as to mark what we will call the *alpha-perfection* of TNRs. Proof that this perfection is unique to TNRs is directly derived from the already established pragmatic perfections in view of the fact that only TNRs have particular *logical objects* in the sense defined here in this instalment and proved in the preceding three instalments. But something else needs to be said about the *alpha-perfection*: we can prove that it is the only possible source of the particular *material content* of any sign whatever.

Proof of the uniqueness of the *alpha-perfection*: For suppose that the particular content of the *manifest sign* of any TNR comes from its own *logical object*. In that case, the particular material content must be valid *of* the object for it is found manifested *in* the object. But suppose that the particular material content of the manifest sign, 3, should come from some other source than its own logical object, 1, e.g., from 2, or from any other object that we might choose. In the latter case, either the particular material content in question must be in 1 or not. If not, then that content would be invalid in 3. Otherwise it would be valid. However, since in all cases we must check 1 to see if the content of 3 is valid, and if it is valid, it must be in 1, the case is made. The only source of valid material content in 3 is 1. Q.E.D.

(Also, since only TNRs have *logical objects* it follows from the prior pragmatic proofs that the valid material content of any sign whatever must be found at first in TNRs, and strictly according to their alpha-perfection.)

Next, referring back to our imaginary diagram, let a second arrow be drawn from 2 to the bottom half of the oblong 3. That is, let it connect the *bodily sign-user* with the *noticeable form* of the *manifest sign* in position 3. Call this dyadic relation as drawn from 2 to 3 the *beta-perfection* of TNRs. Now, it is already clear from the pragmatic perfections of TNRs that if a sign-form is to have a particular sign-user as an object that it necessarily is by virtue of this fact already a TNR, for no fictions or generals are constrained to be connected to particular sign-users in this way. This can be perfectly clearly seen in the case of a general or fiction that someone attributes to a particular sign-user or plurality of them. In the latter case, the particularity is not in the fiction or general but in a TNR that is asserted of the fiction or general in such a manner as to include the latter also as an object, (e.g., that Mark Twain, alias Samuel Clemens, wrote the stories about Huckleberry Finn and Tom Sawyer, is presumably a suitable illustration of this general fact). Q.E.D.

Furthermore, it comes out that the *beta-perfection* has an interesting property. It shows the *bodily sign-user* to be the only valid source for the *noticeable form* of the *manifest sign* in position 3.

(It is important to bear in mind that this position is generalized to its limit in the same manner as the rest of the diagram. That is, the *bodily sign-user* in 2 is construed to refer to any single user or to any plurality of users or to any plurality of singles or plurals.)

The uniqueness of the beta-perfection can be proved in a way quite similar to the proof of the uniqueness of the alpha-perfection. Proof: For suppose that the *noticeable form* of the *manifest sign* at 3 comes from the *bodily sign-user* at 2. Then that form would be validly owed to that sign-user. But suppose it came from somewhere else, e.g., from 1 or 3, or some other user. In the latter case, the form would either conform or not to the form at position 3. If so, then it would be attributable validly to 2, otherwise not. In either case, the sole authority for the determination of whether 3 has a form produced by 2, is

2. Q.E.D.

(Sociolinguists should be slightly amazed at this result, I think, because, so far as I know, no one has ever before shown that the noticeable form of signs is solely attributable to communities of sign users, or to individuals in those communities. But this result is obtained. Also, phonologists would be interested, I suppose, in this result. It shows a perfectly satisfactory basis for differentiating the idiosyncrasies of an individual sign-producer from the communalities of phonological forms shared by individuals, families, communities, and other groupings. Similarly, it shows that the form of any given dialect can be owed to nothing other than the community of sign-users who employ that dialect. Experimental hypotheses are readily derivable, but I will not pause to pursue them here, or anywhere, in these several instalments of the introduction to TNR-theory, though I note that a number of rather surprising hypotheses have been tested experimentally and all of the ones tested so far have panned out.)

Finally, to conclude the discussion of the syntactic perfections, let a third arrow be drawn in the imaginary diagram from 3 back to 1. Let this arrow connect the whole of 3 to 1. Call it the *omega-perfection*. It is clear that it pertains only to TNRs on account of the fact that only they have logical objects in the required sense per the pragmatic perfections. However, the omega-perfection has a peculiar uniqueness worthy of examination. It shows that a *manifest sign* (of the TNR variety) is the only valid source for the determination of the material character of any *logical object* whatsoever.

(This result has some surprising implications for physics which I explored in the recent paper in Semiotica on free will, determinacy, and Einstein’s unified field theory.)

The proof of the uniqueness of the omega-perfection proceeds similarly to the prior proofs concerning the other syntactic perfections. Proof: By the proof of the first syntactic perfection, it is known that the *manifest sign* gets whatever valid material content it may have from its *logical object* and from no other source. Therefore, whatever content it gets from that object must enable the sign to validly determine that content in that object. But suppose some determinative content might be attributed to some other source than such a *manifest sign* (that is, something other than a TNR relative to the logical object in question). In the latter case, either the content to be attributed to the logical object would be found in the object and the sign would thus be rendered a TNR, or it would not be found in the object, and the sign would have to be a fiction relative to that object. Q.E.D.

(The main consequence that follows for physics from the foregoing result, which I suppose cannot be known except by a theory that makes explicit the pragmatic perfections of TNRs, is that determinacy per se does not reside in physical matter, but only in signs. I will not follow that thread further here, though we may take it up later if there is interest.)

At last, we come to the semantic perfections. As the pragmatic perfections are monadic and the syntactic ones dyadic, the *semantic perfections* are triadic. Let these be called *aleph*, *beth*, and *thav* after the first, second, and last letters of the Hebrew alphabet. Let them be conceived as follows: Each is seen in the triadic relation pictured in the just completed imaginary diagram (as described above in this instalment per the syntactic perfections) where aleph highlights the *logical object* of the TNR, beth highlights the *bodily sign-user*, and *thav* highlights the *manifest sign* itself. But since there is only one grand system of signs among them all that have these triadic completenesses (as seen in each of the pragmatic perfections, and in each of the syntactic ones), it follows immediately that only TNRs have the semantic perfections. Q.E.D.

I believe that the main uniquenesses of these semantic perfections consist in the fact that each of them is such that when any element of the triad appears, the other two are perfectly implicated.

In the next instalment, I propose to show that this peculiarity–which is the mathematical perfection of a genuine trinity–not only obtains for the semantic perfections, but also for every TNR (contrary to what I said in an earlier post, Feb 4, 1997, responded to by Peder Christiansen on Feb 5–where I speculated that the TNR might be a mere triad rather than a *genuine trinity*) and for each of the systems of the pragmatic, syntactic, and semantic perfections of TNRs. Each of the latter forms a genuine trinity and the three of them form in fact a trinity of trinities.

As I mentioned in the earlier post of Feb 4, I think these last properties might be a bit esoteric for most subscribers to Peirce-l and mainly of interest to logicians and mathematicians. The proofs concerning these mathematical properties, however, can be developed in a surprisingly elegant and straightforward way, so I’ll try to put them up in the next instalment. In brief, it appears that genuine trinities have the peculiarity that each of their parts perfectly contain and are contained in the whole trinity and in each of their other parts and pairs of parts. Furthermore, it is relatively easy to show that among all possible relational systems only genuine trinities have this peculiarity.

************************************************************* Afterward, I hope to do a better job of responding to comments that have come up in the interim. Thanks to all of the respondents for their patience, and especially to dear Tom Anderson. As I noted in a post to Victor T. earlier today, I tried to post this fourth instalment of the introduction to TNR-theory along with responses to Charles ? (last name not given in the previous post) and to a generous comment from BugDaddy (Bill Overcamp, I think, but I tend to confuse him with Bill Everdell). Also, I should acknowledge recent postings by a number of other individuals that at least obliquely indicates that they too are participating in this thread–especially, Leon S., Cathy L., Bill E., Victor T., Hugo A., Peder C., and our gracious host and colleague, Joe R. *************************************************************

Kindest regards to all, John Oller

******************************************************** John Oller Phone office Department of Linguistics home

University of New Mexico Fax 505-277-6355

Albuquerque, NM 87131-1196 e-mail joller@unm.edu

********************************************************

“The best maxim in writing, perhaps, is really to love

your reader for his own sake.”

C. S. Peirce, Mar 17, 1888

Next it is possible to prove that every TNR is a genuine trinity and that its perfections form a genuine trinity of trinities. In doing so, we also prove the existence of genuine trinities and provide a basis for examining the further proposition that only genuine trinities and no other relational systems meet the requirements on genuine trinities. Take first the TNR to see if it meets the requirements of a *genuine trinity*–*that each of its parts contains the others and the whole.*

To develop the series of proofs it is necessary to examine more closely what the relation of a TNR is to the matter-space-time continuum. First, we must consider the nature of that continuum. We can show immediately that *whatever may be construed as the physically present time of any given observer located in the continuum must be, with respect to its purely physical aspect, unextended and thus infinitesimally brief*.

This is proved as follows: if the physical hypostatic present were extended even in the slightest degree it would form a segment of time analogous to a line segment and could be divided at the middle. However, a line segment divisible at the middle would have, in addition to its hypostatic present moment, an infinitude of moments prior to that one and an infinitude of other moments subsequent to that one. The ones after the hypostatic present moment would have to be designated as future, and the ones before the present as past. Therefore, all that can be left between these extended segments of past and future time relative to the present time must be an infinitesimal unextended present moment, or else, if extended, it would be divisible into past and future. Thus, *the hypostatic present moment cannot be extended and must be infinitesimally brief*. Q.E.D.

Next, we prove that *the phenomenal (experienced) present of any observer must be extended and absolutely cannot be infinitesimally brief*. First, we may observe that this proposition is obviously true relative to perception. This holds, because, as C. S. Peirce argued (in volume 5 especially of CP, early in that volume), perception requires some passage of time. A more rigorous proof, however, is possible.

Suppose we take the time of any given bodily observer as manifested in the real motion of that body relative to the heavenly clocks (e.g., the earth relative to the moon, sun, and so forth) within the matter-space-time continuum. During any such reference motion-conceived as a vector of movement defined by some pair or more of moving bodies (e.g., the earth and sun) relative to a third body (e.g., our observer), let us arbitrarily designate some particular moment on the observer’s time vector as “the present”. It immediately comes out that whatever moment we choose as “the hypostatic present moment” must have moments extended behind it that we call “the past” and moments extended ahead of it (just after the hypostatization has occurred) that we call “the future”. These other moments are defined by the motion vector defined by the bodily objects in space-time. Thus, we cannot pick a point along the vector of spatio-temporal movement of the bodily experiencer (moving as he or she must be relative to the heavenly bodies), where the argument just stated will not hold. It holds for every possible point along any such vector. Thus, *the phenomenal present of any observer must be extended.* Q.E.D.

Now, the proof just given shows why it must be the case that any observer’s present appears as a moving vector-call it a *matter-space-time smear*-such that the phenomenal present necessarily is smoothly joined to the past and future of any conceivable observer. It remains to show that this same phenomenal present *can be extended indefinitely in either direction (past or future) relative to any given moment in the phenomenal present of absolutely any observer we choose*. The fact that no definite boundary to the phenomenal present can be found either in the past or future of any observer is proved in a manner similar to the preceding proof. Let us take any moment on the time vector of the observer’s phenomenal present that we like. It will be found that no matter what moment we select, it will have the smear-property of the phenomenal present and is therefore part of it. Since this property can be found in both directions proceeding away from absolutely any arbitrarily selected moment, it follows of necessity that *the phenomenal present of any conceivable observer can be extended indefinitely in either direction without limit*. Q.E.D.

As a result, since the phenomenal present cannot be definitely bounded by any particular moment in its own past, nor by any particular moment in its own future, it can always be extended indefinitely to cover as large a portion of the matter-space-time continuum as can be represented. In fact, since the smear can be extended without limit, any portion of the continuum, right up to any bounded part, or the whole (whether it be bounded or unbounded), can be represented. Thus, the observer’s phemonemal present logically entails the whole of the matter-space-time continuum. The foregoing proofs provide a connection, then, between the theory of time just developed and TNR-theory.

From the foregoing proofs it follows that every TNR, which must refer to some portion of the matter-space-time continuum (in order to have the pragmatic perfections as proved above), logically also entails the rest of the continuum.

Further, because of its pragmatic perfections, every TNR necessarily connects the phenomenal present of one or more observers to the whole of the matterspace-time continuum. And, moreover, since every TNR is necessarily part of the phenomenal present of some observer (or plurality of them), it follows that every TNR is naturally extended in the manner of every observer’s phenomenal present (as proved above). As a result, it follows that every TNR by virtue of its actual attachment to the extended phenomenal present of one or more observers and thus to the matter-space-time continuum partakes of the peculiar mathematical properties that inhere in any genuine continuum. The foregoing results are critical to our being able to see that TNRs must also be genuine trinities.

Now, *a genuine trinity is such that each part must contain the other parts as parts of itself, and that each part thus contains the whole*. Let us next see if this is so of TNRs. According to Figure 2 above, every TNR consists at a minimum of three interrelated parts: (1) sensory signs (icons) corresponding to facts in the matter-space-time continuum; (2) motoric signs (indices) corresponding to the sensory signs and thus also to the facts; and (3) conceptual signs (usually linguistic ones; or symbols) corresponding to the motoric signs, the sensory ones, and the facts. It is already clear that the conceptual part of every TNR contains the motoric part and that the motoric part contains the sensory part. In other words, we may write: 3 >- 2 >- 1, where the symbol ” >- ” is used to signify the “containment relation” such that the element to the left of this sign “>-” contains the element to its right.

It is obvious that 2 >- 1 because a motoric sign with no sensory signs connected to it is impossible even to conceive, and every motoric sign (always an indexical action) must be connected to the actor that produces it through an icon of that actor, and every such motoric sign must also be connected through one or more additional icons (sensory signs) to whatever else it signifies, e.g., to whatever object is indexed or otherwise acted upon. Similarly, it is obvious that 3 >- 2 because there are no conceptual signs without surface-forms, but the surface-form of every such sign must be produced in a motoric way by some actor (sign-user) which follows directly from the syntactic perfections of TNRs. Moreover, it is obvious that 3 >- 1 because 2 >- 1 and 3 >- 2, therefore, 3 >- 1.

It remains only to show, therefore, that 1 >- 2 >- 3. This last proposition is not at all obvious on account of the fact that we normally think of sensory signs as being something less than motoric signs, and both of them as being less than linguistic (or other conceptual) signs. The idea that the lower ranking signs ought to contain less than the higher ranking ones goes back to the old Euclidean (and Leibnizian) notion that the part must be less than its whole. However, as Peirce and other mathematicians showed more recently, that idea was always incorrect. The part, in many cases, may be equal to its whole, and thus we should say that the part must always be less than *or equal to* its whole.

With that in mind, let us examine the proposition that 1 >- 2 in every TNR. This follows immediately from the fact that the motoric signs of the TNR, the 2 element, cannot be known apart from the sensory signs, the 1 element, and that furthermore, the motoric signs always themselves consist of dynamic sensory signs involving moving icons over space-time. Hence, upon examination it turns out that the iconic signs that move around, 1, constitute the very motoric signs, 2, and that they thus must and do contain them in the requisite sense. Next consider the proposition that 2 >- 3. Upon examination of the motoric sign part of every TNR, we discover that it absolutely must be connected not only on the one side to icons of the sensory kind, but on the other to the conceptual symbol(s) of the language or language-like system of conventional signs found every TNR. If this were not so, then the conventional sign would lack its semantic component and its surface-form, but this is impossible owing to the syntactic perfections of TNRs. Finally, consider the proposition that 1 >

3. This follows from the fact that every conventional sign must have a form that is produced by the community of sign-users who apply that conventional sign (per the BETA perfection of TNRs). However, that surface-form in every TNR, which consists of one or more motoric signs, owing to the proof that every motoric sign must have an iconic manifestation, absolutely must be found in the sensory signs of the TNR. Thus, *every TNR is a genuine trinity*.

Q.E.D.

Next we consider in its turn each of the trichotomous sets of perfections of TNRs to see if they also form genuine trinities. The first proposition to be examined along that line is that *the pragmatic perfections of every TNR form a genuine trinity*. If this proposition is true, it must be the case that determinacy (D) contains connectedness (C) which also contains generalizability (G). The proof proceeds in the manner of the preceding proof that TNRs form a genuine trinity. It is clear in advance that (1) G >- C >- D, but what must be proved is that it is also true that (2) D >- C >- G, which is not at all obvious. In the first instance, it must be the case that C >- D because a TNR cannot be determinate with respect to any particular material object except that object be connected to the matter-space-time continuum. Similarly, G >- C and G >- D because a TNR cannot be generalizable except it be connected to the matter-space-time continuum, and we have already proved that C >- D. Therefore, (1) is proved. Next we examine proposition (2). Taking first the instance of D >- C, we find that it is impossible for a TNR to be determinate and not also to be connected to the matter-space-time continuum. Similarly, no TNR can have any generalizable meaning except that meaning be determined, so that D >- G. Also, if a TNR is connected to the matter-spacetime continuum by the preceding cases examined, it follows that C >- G, and that proposition (2) obtains perfectly. Thus, *the pragmatic perfections of TNRs form a genuine trinity. Q. E. D.*

Next consider the proposition that *the syntactic perfections of TNRs form a genuine trinity*. Take the ALPHA relation. Since the semantic content of the sign is owed exclusively to the sign-object as known perceptually to the sign-user, we find all three syntactic elements of the TNR right here: thus, ALPHA >- BETA, and ALPHA >- OMEGA. That is, the valid content of any percept comes only from its material object as seen in the ALPHA relation. However, the ALPHA relation contains the BETA relation because the semantic content of the percept is produced exclusively by the act of the perceiver that produces that percept. Similarly, the ALPHA relation contains the OMEGA relation because the percept is invariably applied to its own material object by the perceiver in such a manner as to determine the character of that object. Thus, it is proved that ALPHA >- BETA >- OMEGA. Next consider the propositions that BETA >- ALPHA and that BETA >- OMEGA. The second of these proposition is a necessary result of the immediately preceding proof, for if ALPHA >- BETA >- OMEGA, it follows that BETA >- OMEGA. The BETA relation is also necessarily contained in the ALPHA relation because the percept (sign-form) is actually produced exclusively by the sign-user. Next, we take the propositions that OMEGA >- ALPHA and that OMEGA >- BETA. We find that the OMEGA relation contains ALPHA because the content of the sign that is applied to the object, for that particular TNR, is always found exclusively in that object. Moreover, the OMEGA relation also contains the BETA relation because the act of applying the sign-form of any TNR to its object can only be attributed to the actor who makes that application and to no-one and nothing else. Therefore, *the syntactic perfections of TNRs form a genuine trinity*. Q.E.D.

Next consider the proposition that *the semantic perfections of TNRs form a genuine trinity*. In view of the fact that the ALEPH relation involves all three of the elements of the BETH relation and also of the THAV relation, and the reverse is true; thus, THAV >- BETH >- ALEPH and ALEPH >- BETH >THAV.. *Q. E. D. *

Next we consider the proposition that *each of the foregoing trinities just described must also contain each of the others*. Let the semantic trinity be designated as M, the syntactic as S, and the pragmatic as P.

It is clear that it must be that case that M >- S >- P. This follows directly from the fact that the monadic pragmatic relations are necessarily contained in the dyadic syntactic ones and that both of the foregoing are necessarily contained in the triadic semantic relations.

However, the proposition that P >- S >- M is not obvious. Do the pragmatic perfections contain the syntactic and semantic ones? Do the syntactic perfections contain the semantic ones?

Take determinacy, D. D >- ALPHA because the determinacy relation logically stands between exactly the sort of dyad seen in ALPHA. That is, every TNR determines some material particular(s). Also, D >- BETA because every TNR stands in a relation of connectedness to the matter-space-time continuum such that the sign-user puts the TNR into a BETA relation to its particular(s) and the space-time continuum by creating the TNR so as to determine the material particular(s). D >- OMEGA because in determining its particular(s), the TNR shows precisely the OMEGA relation. Further, since D >- C >- G, it follows that both C and G stand in the requisite relations to all of the relations in S.

It remains then, to demonstrate that S >- M. Consider first the proposition that ALPHA >- ALEPH. This follows from the fact that contained within the dyadic ALPHA is the perfected triad seen in ALEPH. That is, as the object conveys semantic content to its abstracted sign, through the intelligent act of the sign-user, ALEPH is quite perfectly revealed. Consider next the proposition that ALPHA >- BETH, but this follows from the fact that ALEPH >- BETH. Similarly, that ALPHA >- THAV is assured by the fact that ALEPH >- THAV. Next we can infer that BETA >- M because ALPHA >- M and we have already proved that ALPHA >- BETA. Similar reasoning reasoning enables the inference that OMEGA >- M, because ALPHA >- OMEGA.

Thus, we have proved that *all three trichotomies of perfections-the pragmatic, syntactic, and semantic perfections of TNRs–are genuine trinities and form a genuine trinity, i.e., they contain each other mutually and are contained in the TNR.* Q.E.D.

Next we come to the proposition that *only a genuine trinity (or combination of them) can show the remarkable properties proved to accrue to TNRs and their perfections*. This proposition can be proved by showing that one or more of the properties requisite to genuine trinities are missing from all other relational entities.

Clearly the complete absence of a relation (e.g., a mathematical zero or nothing) cannot be a genuine trinity because it lacks any definite parts and manifests nothing. A mere monad does not have the properties of a genuine trinity because it appears in only one form, and not three distinct forms. It resembles a trinity in an imperfect way by virtue of the fact that its one part is the whole, but since every monad shows the whole only in its one part, and this part can in no way be distinguished from any other part, every such monad fails to achieve the mathematical properties of a genuine trinity. Next, consider any dyad. Every such relational structure must contain two distinct parts, but neither of these parts contains the other, nor is either part the whole. And what of triads and higher polyads? They encounter the same problem as any dyad,

but accentuated more and more as the polyads achieve higher and higher *adinity* (i.e., an ever increasing number of parts), the parts resemble the whole less and less. *Therefore, only genuine trinities possess the properties shown to accrue to TNRs.* Q.E.D.

In the sixth and final instalment of the introduction to TNR-theory, I propose to merely note some of its consequences. I will also, per the recommendations of Charles ? , make an effort to say more specifically where TNR-theory contributes something new to Peircean thought and where it reiterates what he had previously demonstrated.

Here follows instalment 6. It has three purposes. First, to prove that the traditionally recognized requirements on an adequate theory (namely, consistency, exhaustiveness, and simplicity) are a genuine trinity. Second, to make a few remarks on what TNR-theory may add to Peircean thought. Third, to refer to a couple of hypotheses derived from TNR-theory that have panned out as predicted and to point to some of the relevant empirical research.

*Requirements on any Adequate Theory* Let O be a definable object or at least one that can be represented through a finite number of TNR signs and sign-operations which we shall call theory, T. Let it be stipulated that O need only be represented and thus defined by T such that T sets the logical limits of O.

Let the following definitions be introduced:

  1. T is S (simple) to the extent that it uses signs necessary to represent O and avoids signs not necessary to O.
  2. T is E (exhaustive, i.e., complete relative to O) to the extent that it includes all that is in O but nothing extraneous to O.
  3. T is C (consistent) to the extent that it contradicts nothing true of O and asserts nothing false of O.
  4. If X >- Y, X contains and represents within itself all that Y contains and represents. This is the containment-relation.
  5. If for any triplet of properties, X, Y, Z, the containment-relation holds for all possible pairs, XY, XZ, and ZY and between each individual element, X, Y, and Z, and the whole, that triplet is a genuine trinity.

Given that T is S, E, and C, required to prove that these are a genuine trinity:

i.e. that 1. S >- E; 2. E>- S; 3. S >- C; 4. C >- S;

5. E >- C; 6. C >- E.

  1. S >- E because anything necessary omitted from T, or superfluous included in T, results in an unnecessary complexity in T. Q.E.D.
  2. E >- S because any superfluity in T relative to O (excepting empty repetition or meaningless signs not really superfluous) must either make T too narrow for O, or too broad for O. Q.E.D.
  3. S >- C because any inconsistency in T is invariably unnecessary and therefore a superfluous complexity in T. Q.E.D.
  4. C >- S because any superfluity in T (that is not merely repetitious or empty) must be inconsistent with O or else not superfluous. Either way such a superfluity, if non-empty, must involve some inconsistency between O and T.

Q.E.D.

  1. E >-C because any inconsistency in T must involve O or not be a genuine inconsistency in T and thus it must either include or exclude too much of O from T. Q.E.D.
  2. C >- E because any T that is not-E includes or excludes too much and is thus inconsistent with O. Q.E.D.

Thus, C, E, S, form a genuine trinity. Q.E.D.

*Differences or Additions to Peircean Thought* Whereas Peirce described himself as a *scholastic realist*, TNR-theory insists on the genuine reality of things, forms, and representations, owing to the fact that if the latter be denied, an arbitrary and unneeded inconsistency is certain to arise immediately. Thus, the first departure from Peirce, if it is one, involves the deliberate demonstration that the matter-space-time continuum must be granted reality up to the limit of the consistency of our signs. (Peirce did not deny this, but he did not seem to insist on it either. TNR-theory insists on it rather than merely permitting it. The reason? Otherwise, a virtual infinitude of inconsistencies is certain to arise.)

Next, the marking of the limit of the universe of signs, I believe, is unique to TNR-theory, though developed along Peircean lines, and enables a division of signs that is also Peircean, but that has not been spelled out in such detail previously. Certainly, it has not been applied in the way that it is applied in TNR-theory.

From there forward, the proofs pertaining to the formal structure of TNRs, fictions (including errors and lies as increasingly degenerate cases), and generals (true, false, or indeterminate ones) are very Peircean but were not envisioned by him. Similarly, the theory of abstraction and the theory of systems grammar which follow from TNR-theory are also very Peircean but will not be found explicitly in his writings. Many parallels exist, but many differences also arise. Among them are differences of purpose. While Peirce wanted to develop a “moving picture” of thought in his “existential graphs” (and in the “entitative graphs” that preceded the latter), my aim in TNR-theory is to explain the sine qua non of the vesting of signs with meaning (the material content of signs). In the theory of abstraction I aim to explain how TNRs are formed. The other kinds of signs are proved in TNR-theory to be parasitic on TNRs (I refer here to all fictions and generals) and dependent on them to obtain any meaning whatever, so it is only necessary to explain in the theory of abstraction how TNRs are formed. The latter theory, therefore, aims to account for the underlying logic of sign development. It generates a 30-layered hierarchy of considerable richness that appears to be complete in profoundly logical ways as well as universal. Finally, TNR-theory together with the theory of abstraction leads to the theory of systems grammar which aims to account for the dynamic application of signs in experience which makes possible the ground for language acquisition, etc.

*A Few Applications*

  1. In the theory of photographic meaning, J. Roland Giardetti and I performed a number of manipulations of experimental data based on a test he developed from photos taken from National Geographic. In that study, a great many hypotheses derived from one or another aspect of TNR-theory were tested with various statistical approaches ranging from specific analyses of variance to a comprehensive exploratory factoring procedure. I’ll give one example: we predicted that photos dependent mainly on iconic information, as contrasted with those that contained symbols, e.g., words or other conventional signs, or those that contained revealing indices, would benefit from coloring while the others would not. No one had ever dreamed of this hypothesis prior to applying a Peircean conception. It was sharply confirmed (rejecting the null hypothesis of no contrast between the three kinds of photos).
  2. Another application is in the processing of textual or other syntactic arrangements of signs. Here I will cite two applications: one in cloze test research that Jon Jonz and I published in 1994 (Cloze and Coherence, Bucknell University Press), and the other on IQ testing, both verbal and nonverbal, in my paper shortly to appear in *Applied Linguistics* titled “Monoglotossis: What’s Wrong with the Theory of the IQ Meritocracy and Its Racy Cousins?” Some subtler hypotheses derived from TNR-theory in connection with the latter are being tested in IQ research underway in Korea presently, but in a very general way TNR-theory shows that it is impossible that non-verbal IQ items should be answered without access to abstract ideas and relations attainable only through a conventional language system. Many other consequences follow including a satisfactory explanation for nine major puzzles in the statistical findings of IQ testers–e.g., why language minority children are over-represented in classes for the mentally retarded, and the like.
  3. Another line of research involves examination of the genetic code with reference to the sign hierarchy revealed by the theory of abstraction. So far, isomorphism between the first tier of signs and the signs employed in the process of translation (genetic), has been established. The remaining two tiers are under examination. Dr. J. Omdahl, Professor of Biochemistry at UNM School of Medicine, is collaborating in this research.
  4. Finally, I mention again the work on autism. Evidently is involves a breakdown in indexical as contrasted with other kinds of signs and these involve special neurological mechanisms (the latter have yet to be detailed, but some reasonable hunches have been confirmed). That work remains to be published but has been completed in its initial phases and is under consideration for publication. My collaborators in that work include Dana Rascon, a graduate student and former fellow at UNM School of Medicine and Jack Damico.

What is difficult about all of the foregoing is getting lay-persons relatively unacquainted with Peirce to understand what we are doing and the results already brought to light. I’m sure that the Peirce-scholars on the list have an idea of what we are going up against in view of the difficulty of even getting experts to understand the gist of the arguments, much less to really become fluent in applying them. No offense, but these are the facts. (Even Richard Robin, editor of the Transactions sees “nothing new” in any of this according to a review of three distinct pieces submitted to him for consideration–one on systems grammar which is entirely new!)

Many thanks for allowing me the opportunity to share some of our work with you folks and to solicit your reactions.

Sincerely, John Oller

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John Oller Phone office

Department of Linguistics home

University of New Mexico Fax 505-277-6355

Albuquerque, NM 87131-1196 e-mail joller@unm.edu

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“The best maxim in writing, perhaps, is really to love

your reader for his own sake.”

C. S. Peirce, Mar 17, 1888

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