**Sketcher of various interrelated fourfolds.**

# Nontrivia

**Recentest significant edit: December 19, 2013.**

In my previous post "Unsettlings" I discussed a double opposition, or "double chiasm" as I called it, among the (cognitive) lights in which a given phenomenon would seem (1) simpler or (2) more usual or normal or (3) clearer, more clarificatory, more significant or informative, or (4) deeper, less trivial:

1. Simplicity, optimality, etc. 2. Likeliness, probability, etc. | 3. Informativeness, significance, etc. 4. Nontriviality, depth, complexity, etc. |

The first three correlate pretty obviously to mathematics of optimization, mathematics of probability, and mathematics of information. The fourth (nontriviality, depth, etc.),in at least one sense, seems to me to correlate to mathematical logic.

## Inverseness between probability and information.

A message's quantity of information, its amount of informativeness or "newsiness," reflects the improbability of that message before it was sent. The information quantity is not usually quantified simply as 1 minus the message's erstwhile probability (e.g., 1 minus 1/8 probability equals 7/8 improbability), but still it's pretty simple, the logarithm of the*reciprocal*of the erstwhile probability (for example 1/8 probability, ergo 3 bits). The smaller the erstwhile probability, the larger the logarithm (to a given base), and we can think of information as a kind of inverse of probability.

## So, inverseness between optimality and nontriviality?

I mean "optima" as in mathematical programming a.k.a optimization. If one puts, as I do, optimality and nontriviality/depth likewise into an opposition, one might expect a similar kind of inverseness. Lloyd's and Pagels's idea of*thermodynamic depth*is "the entropy of the ensemble of possible trajectories leading to the current state" (from Cosma Shalizi's notebook on complexity measures) and gets us the idea of some sort of opposite or inverse of the shortest path (simple, optimal, etc.). Then there is the idea of algorithmic complexity, the shortest program capable of obtaining a given result, which complexity is uncomputable because of the halting problem, and anyway the general idea that you can't get a ten-pound theorem out of five pounds of axioms (as discussed by Chaitin). So by merely looking for "big-picture" patterns (and roving through things like the Mathematical Subject Classification), I seem, despite my amateurish ignorance, to have found myself in the right neighborhood.

Shalizi, with a bluntness that is helpful to the general reader, starts out his above-linked notebook "Complexity Measures" with this striking paragraph:

Now, one may note that there also seems no general quantification of "optimality," either — instead, one seeks specific optima. As for a quantity of feasibility, it might just be a roundabout way of locating an optimum (a located feasible solution getting characterized, say, by a distance and direction from the optimum). If feasibility is considered just lowness of cost (compared to a highest feasible cost) or size of net benefit (minus some lowest feasible net benefit, I guess) or some unifying generalization of those ideas (I don't know what), it still isn't like a ratio, comparable across disparate cases. If we set the optimum to unity in order to get that comparability, then can feasibility come out like probability? There's a duality between optimization and probability where cost corresponds to probability. (One would think it more intuitive that cost would correspond to improbability formulated somehow, but I don't know whether that leads to problems or is merely less convenient for expositing the duality. Here's a paper (PDF) of which I understood maybe three sentences and one formula.C'est magnifique, mais ce n'est pas de la science.(Lots of 'em ain't that splendid, either.) This is, in the word of the estimable Dave Feldman (who taught me most of what I know about it, but has rather less jaundiced views), a "micro-field" within the soi-disant study of complexity. Every few months seems to produce another paper proposing yet another measure of complexity, generally a quantity which can't be computed for anything you'd actually care to know about, if at all. These quantities are almost never related to any other variable, so they form no part of any theory telling us when or how things get complex, and are usually just quantification for quantification's own sweet sake.

**Update 5/17/2013:**: I think I get it now. An extremal solution is generally less likely to be chosen at random. A straight line is the opposite of a random walk. Something like that. The first paragraph of the linked paper "Duality between Probability and Optimization" ends: "To the probability of an event corresponds the cost of a set of decisions. To random variables correspond decision variables." End of update.)

Anyway, so maybe it's the same for the nontrivial as for the optimal. One doesn't typically seek an amount of nontriviality, instead one typically seeks nontrivia, complexuses, etc. Now, it's not so hard to understand what constitutes an optimal case, a probable case, and an informative or "newsy" case. But, if nontrivia are to be considered as being on some sort of par with optima, probabilities, and information, then what constitutes a nontrivium, a nontrivial case? Insofar as the words "nontrivial" and "deep" are typically used in a sense that does not invite precise determination, I think it better to speak of contingent truths pertinent to a question at hand, i.e., which help support non-vacuous conclusions. These could be *data* in a typical sense, or some sort of *standing givens* such as mutually independent (and consistent) postulates - one way or another, *givens* (for which the word "data" is the Latin).

## Now, there are some other big-picture considerations here.

I'm thinking philosophically, analogically, so please bear with me. There are temporal issues involved with the conceptions of optima, probabilities, and information.### 1. Optima and feasibles

are, for lack of a better word,*potentialities*(with the optima as "debentialities," lowest or most efficient potential expenditures, what would really be owed) for what

*could*happen or be done given things as they stand; the impact of directly revealing or acting if one were to reveal or act now (the moment of decision); correlated more or less to the surface of the future light cone.

### 2. Probabilities

pertain to what is*going*to happen in the course of a

*future*in virtue of repetitions; that which does happen thereby reaches 100% probability.

### 3. Information

is newsiness and pertains to what*is coming to light*or being actualized (correlated more or less to the surface of the past light cone) but not already settled; if the message's information is already known, then the information is zero.

So we have this pattern (of characterizations, not definitions):

**1. Optima, most feasible, simplest, most efficient, etc. — things**

2. Probabilities — things

3. Information — things

*worth supposing, imagining*, etc.2. Probabilities — things

*worth expecting*.3. Information — things

*worth noticing*.Ergo (by completing the analogy):

### 4. Independent givens / data — things *worth remembering*.

This associates truth or fact in some sense with the nontrivial or deep, as well as with the complex, the complicated, etc.; an extra and independent given is a complication. Some mazy and labyrinthine complications have a kind of triviality when they don't teach real lessons, still they can be worth remembering — ask any lab rat. The idea of that which offers lessons worth learning, remembering, etc., that which is "educational" in some sense, that from which lessons or more or less secure conclusions can be drawn, is another thing which distinguishes the nontrivial from the distinctive, informative, etc. We learn from the past; experience is the great teacher. But can it be that logic's concern is simply that datum, fact, or basis (e.g., some postulates) from which one can draw conclusions? What is the complexity in it - simply that it is non-tautologously true? This seems to be missing something in that which mathematicians mean by "nontrivial" and "deep."
### There's still another big-picture issue — what you might call that of *subjective nontriviality versus objective nontriviality*

but which would better be called **aspectual nontriviality versus transpectual nontriviality**. I alluded to it above in distinguishing contingent relevant givens from the "nontrivial" in mathematicians' sense. This could be a newsy distinction, since I haven't found any notice of it as a possible source of confusion.

Take a nontrivial equivalence between mathematical propositions — its nontriviality is a nontriviality in outward *aspect* for the same reason as that behind mathematicians' joke that anything proven is trivial. I don't want to call it "subjective" since that would imply incorporating a subjective judgment into the reasoning itself about a mathematical structure, just as "subjective probability" suggests trying to quantify one's subjective expectation in a specific case. As for "transpectual," I just mean that as the opposite to "aspectual": if two statements are different in form but logically (or as it is sometimes said, "formally") equivalent, then they are different *aspectually* but the same *transpectually* (i.e., when you look through them enough). Equipollence (equivalence between propositions) is a transpectual simplicity; mutual non-implication is a transpectual complexity. Independence (or as some express it, independence and consistency) among axioms or postulates is a transpectual complexity. Nontriviality as a criterion of value of equipollential inferences is *ironic*, and is ironic and aspectual in the same way as analogous criteria for other modes of inference. (The ironic aspectual criteria may be used merely intuitively in devising methods of reasoning; whether one employs a method of incorporating specific subjective judgments of amount of likelihood or whatever into reasoning is another question, one which I'm not addressing.) Examination of the following pattern in the next table lend some subjective probability to my claim of a valued aspect that is ironic in relation to the character of the inference.

### Were it not for these various aspects in which various conclusions put their premisses — aspects natural, verisimilar (in Peirce's sense), new, and nontrivial, — we would see no point in reasoning at all.

I think I've said that somewhere before, but I don't remember off-hand.(Note: mathematical conclusions are often through equipollential deduction. For a common example, the induction step in mathematical induction is equipollential: the conjunction of the ancestral case and the heredity is equipollent to the conclusion. The conclusion is a universal hypothetical (in form) while the ancestral case is an existential particular, but the equipollence is intact because the existence of the well-ordered set to whose elements the hypothetical conclusion refers is already assumed and usually actually already proven. In a simpler case than mathematical induction, in a nonempty universe "whatever there is, is blue" (hypothetical in form) validly implies the existential "there is something blue." Proofs of the ancestral case and the heredity are often through equipollential deductions, though sometimes not so, especially when greater-than or less-than statements get involved.)

Modern science has been builded after the model of Galileo, who founded it on il lume naturale. That truly inspired prophet had said that, of two hypotheses, the simpler is to be preferred; but I was formerly one of those who, in our dull self-conceit fancying ourselves more sly than he, twisted the maxim to mean the logically simpler, the one that adds the least to what has been observed, in spite of three obvious objections: first, that so there was no support for any hypothesis; secondly, that by the same token we ought to content ourselves with simply formulating the special observations actually made; and thirdly, that every advance of science that further opens the truth to our view discloses a world of unexpected complications. It was not until long experience forced me to realise that subsequent discoveries were every time showing I had been wrong, while those who understood the maxim as Galileo had done, early unlocked the secret, that the scales fell from my eyes and my mind awoke to the broad and flaming daylight that it is the simpler Hypothesis in the sense of the more facile and natural, the one that instinct suggests, that must be preferred; for the reason that unless man have a natural bent in accordance with nature's, he has no chance of understanding nature at all. Many tests of this principal and positive fact, relating as well to my own studies as to the researches of others, have confirmed me in this opinion; and when I shall come to set them forth in a book, their array will convince everybody. Oh no! I am forgetting that armour, impenetrable by accurate thought, in which the rank and file of minds are clad! They may, for example, get the notion that my proposition involves a denial of the rigidity of the laws of association: it would be quite on a par with much that is current. I do not mean that logical simplicity is a consideration of no value at all, but only that its value is badly secondary to that of simplicity in the other sense.
— Charles Sanders Peirce," A Neglected Argument for the Reality of God." |

**point**. The hypothesis points a way that seems ready to lead to further indications. The aspect of naturalness and facility is not only for hypotheses in the usual sense. Insofar as any theory's bad match to experimental results (in physical, material, and biological sciences and in human and social studies) can be explained away by additional hypotheses, there's always a role for the simplest expanation — the simplest "hypothesis" to account for a theory's persistent bad results is that the theory is wrong.

Now, in order to distinguish the naturalness and facility desirable in a surmise, I will call it

**(Latin**

*viatility.**viare*means to travel, take a path, from the Latin word

*via*which means "way" or "path.")

2. An induction seeks a kind of aspectual probability or likeliness. At early or crude stages when there is insufficient data to support its conclusion with high confidence, one seeks at least an aspect or appearance of likeliness. Yet an induction, in adding unpremissed information into the conclusion (and omitting no premissed information from the conclusion), increases not probability but information. That's ironic. It's as if induction were seeking to compensate for its informativeness by seeking likeliness, by sticking to the assumption that the larger or total population will resemble the currently available data. The word *verisimilitude* has taken on meanings which differ from that which Peirce meant by the versimilitude of an inductive conclusion, a verisimilitude consisting in that, if pertinent further data were to continue endlessly to have the same character as the data supporting the conclusion, the conclusion would be proven true. That is not what is often currently meant by verisimilitude, the closeness of a theory to an issue's full truth that would be found by sufficient investigation; instead it means likeness to the less-than-whole truth or data garnered up until now. *Ceteris paribus*, the conclusion most faithful, most correlated, to the character of the currently available data seems the best candidate to hold up over time, increase in actual inveterateness as time goes by; it has a correlation that seems likely to lead to further correlations. The inductive conclusion should, aspectually, not seem to add anything new, but instead seem inveterate, typical, "conservative" in a sense, maybe even seem to remove or smooth information.

In order to distinguish the verisimilitude desirable in an induction, I will call it ** veteratility**, which is Peircean verisimilitude.

3. A categorical syllogism or other "forward-only" deduction seeks to bring information to light — but it doesn't really increase information, it reduces it. That's ironic. There's little that I can find about efforts to quantify the "psychological novelty" (as various folks have called it) or "new aspect" (as Peirce called it) or seeming informativeness of a forward-only deduction's conclusion. It's aspectual. Another way to look at it is that a forward-only deduction increases probability (if the premisses are assigned probabilities beween 0% and 100%); in order to be true, it doesn't absolutely need all (or sometimes any) of the premisses to also be true. Anyway, it's as if forward-only deduction were seeking to compensate for its decrease of information (or increase of probability) by seeking newsiness, especially in the sense of a meaningfulness, a difference that makes a difference. This kind of deduction elucidatively symbolizes formal implications drawn from premisses.

In order to distinguish the news-like aspect desirable in a "forward-only" deduction, I will call it ** novatility**.

4. So, continuing the pattern, the nontriviality of a deduction through equivalences or equipollences will be aspectual and ironic. Equipollential deduction neither adds nor removes information, and it's as if it were seeking to compensate for that simplicity with a kind of complexity in the sense here called aspectual. The transpectual complexity or complexus will involve independences, mutual non-redundancies, etc. It is *surmise* (by which I mean inference that both adds and removes information) that is transpectually nontrivial, even though surmise ironically seeks a kind of aspectual simplicity, naturalness, etc. Now, an aspectual informativeness (psychological novelty, new aspect, whatever one wishes to call it) is sought through a categorical syllogism or other forward-only defduction — an extrication of information by removing some of the clutter, so to speak, of the premisses. That (aspectual) informativeness is not to be confused with its kin, the (aspectual) nontriviality that is sought through equipollential deduction; that nontriviality consists (as far as I can tell) in the outward disparities of things bridged by a proven equipollence, a bridge which one may wish to cross and recross in either direction. The nontrivial or deep promises to stand as a basis for further such conclusions. The equipollential deduction stands as a further-transformable proxy or model of the premissed objects; if it is a true proxy (I don't mean a substitute index) or model, it is definitively determined by the same laws or postulates as the objects are amid transformations, but may be better for teaching their lessons through a given line of experimentation; indeed so, if it is nontrivial.

In order to distinguish the nontriviality or depth desirable in an equipollential deduction, I should call it *statility* or *tardatility* in keeping with an analogy with special-relativistic kinematics that I've been using here (as well as the analogy to the light cone's zones that I used earlier above), but the coinages "statility" and "tardatility" inherit some extra connotations that don't seem to work well enough, so I will call it ** basatility**.

Aspectual Merit that inferences in a given mode vary in having, but which seems hard to quantify or render exact, at least hard to do so fruitfully. | Transpectual Content, quantity, or status deducible from given parameters of a total population, universe of discourse, etc. | |
---|---|---|

Worth supposing if pertinent. | Viatile, natural, facile (surmise). | Optimum / feasible. |

Worth expecting if pertinent. | Veteratile, verisimilar in Peirce's sense (induction). | Probability. |

Worth noticing if pertinent. | Novatile, new in aspect ('forward-only' deduction). | Information. |

Worth remembering if pertinent. | Basatile, nontrivial (equipollential deduction). | Independent given, fact, datum. |

Now is a time at the Speculation Lounge when we **speculate**. One might ask, for example, does the "veteratility" of an induction consist in the probability that the premissed sample would have if the inductive conclusion were true? Likewise does the 'viatility' of an abduction consist in the optimality or high feasibility that a phenomenon would have if the explanation were true? Does the 'novatility' of a 'forward-only' deduction consist in the information (truthful news) that the conclusion would re-state if the (non-axiomatic / non-postulational) premisses were truthful news? I suspect that the answer is no.

The case of surmise is instructive. Now, what I mean by 'surmise' is much like what Peirce meant by "abductive inference". Peirce in his later years allows as a form of abduction the inference to a new rule combined with a special circusmstance, to explain a surpriing phenomenon. Although one can formulate an inference to a rule in such a way that it is neither automatically truth-preservative nor automatically falsity-preservative (my definition of surmise) such as the toylike example "EGH ergo A(G-->H)", I think that a surmise involving a new rule usually really involves a hypothetical induction, and that such an induction is involved whether it originates/modifies a rule or merely extends a rule to cover the surprising phenomenon. The surmise itself is the basis for a hypothetical induction, as well a hypothetical deduction of testable consequences. The *rule* in question may be an extremal principle or a combination of constraints, or a distribution or frequency, etc., or an informative rule of dependency, or something else. Note that I am not discussing at this point the inductive evaluation of tests of a prediction deduced from the explanatory hypothesis/surmise. The question at this point is whether the induced rule would continue to assign all the cases under it appropriate optima/feasibility, probability, informativeness, or whatever. So one can deduce the resultant optima/feasibilities, or probabitilities, or whatver, covering all known cases, and, in light of that, at least try to decide whether the induction is veteratile enough (I'm unsure about what sort of inference such a decision itself involves). As for viatility, one may consider that without regard to the veteratility of the induced or extended rule - would an elliptical orbit account for the surprising observations of a given planet, apart from whether there's a more general rule of elliptical orbits? That is like asking whether a person walked in an elliptical path, apart from whether many people do that as a rule. This can be a chiefly mathematical question, whose terms may be embodied approximately by embodying a model that works according to one's hypothesis; think of a professional magician like James Randi producing a supposedly paranormal effect just to show that it *can* be done by non-paraormal means; a kind of proof of concept; the surmise may involve not a pattern of motion but a composition of materials or attributes, and so on.

Anyway, at some point one deduces testable predictions (and here I think that novatility has a role), and inductively evaluates the repeated tests; and thoee reproductions of tests simiilar enough to be counted as repetitions; as for reproductions in different, though in some sense equivalent, forms, their collective evaluation becomes abductive, 'surmisual'. In the end, it's all surmise-based, soever cogent the surmises, insofar as the premisses in a special science come down to perceptual judgments; the special-scientific application, for example, of an induction implies the defeasible surmise that all its premisses are in fact true and depends for its validity on the idea that sufficient further research would correct its errors.

### Plausibility (viatility) and Peircean verisimilitude (veteratility) suggest truth; a novel aspect (novatility) and nontriviality (basatility) suggest falsity.

That answers the following symmetry-based objection:It might be objected that while 'viatility'(naturalness, plausbility) and 'veteratility' (Peircean verisimilitude) incline one at least a little toward holding a conclusion to be true, on the other hand 'novatility' and 'basatility' do not seem to do so (except in the sense perhaps of fostering the *hope*). Besides, if the deduction is valid, then its conclusion is true if its premisses are true; how would novelty or nontriviality increase that kind of assurance? There seems something inconsistent or non-symmetrical about it.

However, there's a deeper symmetry. To the extent that a deductive conclusion seems 'novatile' (novel) or 'basatile' (like an independent basis), that appearance may naturally incline one to *doubt* the conclusion, incline one to the conclusion's possible *falsity*, and towards checking one's premisses and inferences. It is not unheard of that one's premisses and/or inferences sometimes involve errors. Doubt is not always a bad thing. Why shouldn't occasions for doubt occur naturally in good necessary inference? I'd say that occasions for doubt *will* occur if the deduction is worth doing. A kind of dubitability is built, so to speak, into the nature of reason, reasoning that is deductively necessary as well as reasoning that is non-deductive and contingent.

Of course, in a certain sense, a solution or conclusion can seem all too 'viatile' - seem glib or *facile* in that word's present-day usual sense, - or all too 'veteratile' - seem too conservative, standard, nothing-to-see-here, etc. Still, I think that, to the extent that a conclusion is not skewed by hope, fear, etc., a surmise's viatility and an induction's veteratility properly favor their respective conclusions (variably but at least a little), while deductions' novatility and basatility properly disfavor their conclusions (variably but at least a little).

### Meanwhile, as to the nontrivial, the complication, the datum, can't I do better than I've done?

Here I'm describing as*aspectual*the typical sense of "nontrivial" in mathematical talk but, but what makes something "transpectually" nontrivial?

**Update December 28, 2013: I think that I have now done better. Arity, adicity (as in monadic, dyadic, triadic, etc., seems to be the "transpectual complication" that I was looking for. Scroll down to the update under "Correlated operations." End of update.**

Is that kind of nontrivial simply a set of independent facts or truths or givens, i.e. they couldn't have proven or disproven one another? Do they have to be "facts or givens worth remembering" or is it enough that their interrelations are facts or givens worth remembering? Are such complexuses really the core of logical ideas such that logic should have been named for them ("givens theory" or "complexus theory" or whatever), just as probability theory is named for probability, and so on? They may be optimal or otherwise (or more precisely, perhaps, they may be such that they would have been optimal or otherwise); but they are the paths which *have been traveled*, the structures which *have been built*. Is that it? Is a "transpectually nontrivial" statement simply one that is consistent and materially true, and perhaps, pertienent to a question at hand, and not just not tautologously true? But isn't logic about formal truth, not material truth?

Actually that's not what bothers me. Basic deductive logic is about deducing material truths from other material truths - more or less, facts from facts, be the basal facts postulated or established observationally or merely supposed as premisses for the sake of argument. In that sense deductive logic is about material or nontautologous truths in the same sense that probability theory is about probabilities (and optimization theory is about optima, and information theory is about information). I like that idea of transpectual nontriviality: it avoids suggesting that lengthy convolution of an argument is the essence of nontriviality or depth and somehow logically "better," more "logicful," when in real life such an argument is riskier, less likely to escape a weakest-link problem. Such convolution increases aspectual nontriviality (sometimes only in a superficial way, to boot), not transpectual nontriviality, much less security or factuality.

A deduction does not automatically turn its concluding proposition into a truth of logic. The fact that Socrates is mortal is a material truth even if deduced from other material truths, even if deduced from a postulate or axiom that Socrates is mortal. If it is postulated that Socrates is mortal in advance of premisses, a premissual proposition "Socrates is mortal" is part of the tautology "Socrates is mortal by the postulate that Socrates is mortal", but the fact, the datum, that Socrates is mortal is not tautologously true. The nontrivium is that basis on which conclusions - further bases - can be drawn. A set of such postulates, or, say Euclid's five postulates, independent (and consistent), have more transpectual nontriviality or depth than any single such postulate. So, if nontriviality can't be usefully quantified (except as number of independent givens or postulates or the like, whatever that tells you, given their varying internal complexities, even assuming that each is "indivisible" or "atomic" in some sense), maybe it can at least be ordered. Add a postulate, enrich or deepen the system - transpectually if not aspectually. (Should one say that Gödel statements are transpectually nontrivial but aspectually trivial in the mathematical system in which they are true but unprovable?) Even an axiom of propositional logic is not completely trivial, when it is introduced as an axiom, though from it by itself there follows little if anything. Those considerations may seem a bit slippery but they're not what bother me.

What bothers me is that in a sense I'm saying that "transpectual" nontrivia are basically data, givens, facts, i.e., such that one can draw conclusions from them (well, that's the good part), but data are often quantified just like information, in bits, bytes, etc.; so, are data really something different from information or are they merely information such that one doesn't demand that they be new, previously unknown, etc.?

Maybe I shouldn't make a big deal about it, and **I already fear that this is one of the most ignorance-parading posts that I've ever written**. After all, as I mentioned, there's a duality between optimization and probability where cost (a kind of lowness of feasibility) corresponds to probability. (To repeat myself: one would think it more intuitive that cost would correspond to improbability formulated as 1 minus probability, but I don't know whether that leads to problems or is merely less convenient for expositing the duality.) An amount of information depends in a sense on what question was asked. Did a given horse win a race? Yes or no? That's one bit of information, as if the probability of the horse's winning had been 50% when it almost certainly was not. So maybe I shouldn't worry about data's seeming like information any more than about feasibility's seeming like probability. Now, as to a datum *qua* datum, we're concerned not with how newsy it is, how improbable it was before it happened, given that which was already known, etc., but with the complication or complexification that it brings (what would have been its "suboptimal" character before it happened) and what conclusions can be drawn from it. Some say that information is a difference that makes a difference. Perhaps one could say that a nontrivium is a basis for a further basis. That's much like the online Merriam-Webster's first definition of *data*: "factual information (as measurements or statistics) used as a basis for reasoning, discussion, or calculation."

It also bothers me that this reduces complexity/complication to a kind of randomness. It's as if, in going conceptually from optima to probabilities to information to facts, one settles into a kind of heat death of material or non-logical truths. All I can think of at the moment is that the randomness is real in a sense, but that it's why it matters that the data be data, facts, givens in some sense, not just newsy announcements, or probables, or optima or feasibles.

### Achilles and the Tortoise and the Hare.

Here's another way to see optima (simplicities, efficiencies, etc.) and independent givens / nontrivia as opposites in some sense.A problem with the quantity called the *logical depth* of a theorem is that it involves the number of steps that it takes to prove the theorem. If somebody finds a shorter proof, then the quantity changes; it's not "hard-core." Shalizi asserts the shortfall or lack of general usefulness of quantities of depth/complexity, and again I note likewise the lack of a generally useful quanitity of optimality or feasibility; instead we wish to know what are the optima, what are the complications/data/nontrivia, etc. Now, we can think of a proof's steps as "middles," and more generally, any but the shortest proof as feasible but not optimal or shortest-path. But in the mathematical case, the shortest possible proof is the postulate set followed directly by the theorem as conclusion, even if we slowpokes don't see how the postulates lead to the theorem. *We don't desire the truly shortest proof in the case of a theorem.* We want something less short than shortest - but not the longest, most circuitous, either, which would just be the opposite extreme.

Consider Lewis Carroll's example of Achilles and the Tortoise. (I thought of Carroll's example because Cathy Legg has been writing about it). The Tortoise is slow and can't see how to get from the two premisses *all A is B* and *all B is C*, to the conclusion *all A is C*. So Achilles adds a third premiss, *if all A is B and if all B is C, then all A is C*. But the Tortoise doesn't see how to get from the now-three premisses to the conclusion that indeed all A is C. Now suppose that the Hare joins them, and they start discussing Euclidean geometry. Achilles can't get the Tortoise to infer from the five postulates to the Pythagorean Theorem. On the other hand, Achilles can't get the Hare to pause for the intermediate steps between the five postulates and the Pythagorean Theorem, or even for the Pythagorean Theorem itself; the Hare would always go straight from postulates to conclusion, except the Hare is even quicker than that. Insofar as the Pythagorean Theorem is itself an intermediate step to further theorems, the Hare sees no reason to pause with it. In fact the Hare rests satisfied with the five postulates as making sufficiently evident all the implied theorems without need for elaboration. For the Hare, all that we call theorems are but direct corollaries of the postulates and are not even worth mentioning. Some might say that it's as if the Hare naps and loses the race to Achilles and maybe to the Tortoise too. In any case, logic is not done from the Hare's viewpoint, nor from the other extreme, the Tortoise's viewpoint. Logic is properly concerned with givens and middles of various kinds, just as ordered sets are concerned with heredity, convergence, etc.

### Correlated operations.

Looking back at optima for a hint — maybe there's no standard way to quantify optimality, but one can often think of an optimum as a distance with a direction or directions — a shortest path for instance, or the location of a minimum of a curve, etc. Even if it's only a rough idea, still one discerns a pattern, one that I've noticed before:optimum — difference (with direction or directions)

probability — ratio

information — logarithm

Note that this blog's title does include the phrase "Speculation Lounge"! Now for a look into the Speculation Lounge's **Rank-Speculation Sub-Lounge**.

So one might expect, simply on the superficial appearance of the pattern, that for the nontrivial one might be able to think of it as the next in the series "difference, ratio, logarithm." As to the ordering "optimum, probability, information," I didn't reach that from considering the pattern "difference, ratio, logarithm." Instead I got it as part of a broad pattern (see table on right).

Some sort of proportion or analogy here. | |

Motion, forces. | |

Matter. | |

Life. | |

bases (for further conclusions). | Mind. |

Well, it's hard to decide the next term after "logarithm" with confidence, when one expects only a four-term series (I expect it for various reasons including the fourfold correlations outlined earlier in this post). Now, subtraction (finding a difference) is the inverse of addition, and division (finding a quotient or ratio) is the inverse of multiplication. Yet, finding a logarithm is one of *two* inverses of exponentiation (raising to a power); the other being to find a *root* or *base*. A root with a direction? (Now I'm thinking of complex roots). A base?

**Update December 18, 2013:** I now think that it's a root or base, corresponding to arity, adicity, valency, etc., as in *monadic, dyadic, triadic,* etc. A succession of relations of constant arity is a root or base raised to successively higher powers. In first-order logic, the first-power arity is the usual concern, and arity with more than one degree is what deepens its interest, that is, polyadicity is where such logic becomes nontrivial. Quine, in Methods of Logic, 4th Edition, p. 137:

We have our test of validity by existential conditionals, and little is left to be desired—until we move from absolute ormonadicterms like 'book' to relative ordyadicterms like 'uncle'. This is the move that complicates logic and makes for its stature as a serious subject.

Polyadicity is itself a kind of complication, and is a source of complication in logic. It seems to fill the bill as the "transpectual" kind of complexity or complication that I had been seeking earlier in this post. It is a property of special, perhaps even partly definitive, interest in regard to data, givens. Now, in a relational database, the order of the so-called attributes related by a relation does not matter, but in logic, questions of order in a polyad certainly can matter, so size is not the only thing that can matter for arity. Hence, direction can matter in the movement from a polyadic term to its objects, i.e., in a dyad, the direction of movement from the dyadic predicate to the left subject or the right subject.

So, if this is the case, then deductive logic is the deductive study of data, givens, bases for drawing conclusions, and processes of inference to conclusions as further bases, the data themselves understood as adically related or composed, i.e., as "datads" or "datumplexes" or "dedomenads" (words that I've just coined, the last one is from Greek). **End of update.**

Multi-valued logic? MVL has not been a big, thriving field, so far as I can tell, but on the other hand fuzzy logic is a kind of MVL, so maybe I shouldn't speak so fast. Anyway, if you have a higher numeric base, a larger alphabet, a larger lexicon, etc., you can express things with more concision, in a sense you have increased memory capacity too, do you have an increase in some sense in that which is worth remembering (learn the ABCs, expand your vocabulary, etc.)? (I resist this in part because of the terminological coincidence between a numeric *base* and a *basis* for a conclusion. Is it just a pun of ideas?) The other alternative seems to be the hyperlogarithm, or maybe an endless series (hyperlog, hyper-hyperlog, etc.), some sort of orders of nontriviality; one starts thinking of powersets and so on. Now, all that I'm seeking here is an idea in terms of which we can think merely *roughly* of the nontrivial, but this sort of thing leaves me shaking my head as usual.

So I have to leave it here for the time being as it stands. It's a difficult question that has me taking shots in the dark.

(Note on the double-chiasm image near post's top: I've given Hyatt Carter total permission to use the image freely as he pleases, for example here.)

# Unsettlings

**Recentest change: March 26, 2010.**

I can see four general inquiry-stimulating “unsettlings” at least:

1. Bafflement, perplexity, at the complex or complicated. E.g.: What

*has*happened?

2. Surprise at the anomalous, the seemingly unlikely. E.g: What

*is*happening?

3. Suspense, impatience, over the vague. E.g.: What’s

*going to*happen?

4. Hesitancy, inagency, about the unfamiliar, the uncolligated, that whose lessons have not been learnt. E.g.: What

*would*happen (if…)?

And searches for at least four kinds of answer: In what light would the unsettling phenomenon seem (1) simpler? (2) more usual or normal? (3) clearer, more clarificatory, more significant or informative? and (4) deeper, less trivial? – each of which can be helpful in any of the above questions.

Note the conceptual opposition or tension between (1) simpler and (4) deeper, less trivial. And also that between (2) more usual or normal and (3) more significant or informative. A kind of double chiasm.

1. Simplicity, optimality, etc. 2. Likeliness, probability, etc. | 3. Informativeness, significance, etc. 4. Nontriviality, depth, etc. |

C. S. Peirce says that all inquiry begins with irritation by doubt as a result of surprising observations and that it struggles toward belief - a fixation of belief, which is to say, a settlement of belief (a settlement at least for the time being). (See his 1877 "The Fixation of Belief"). So it seems to me appropriate to think of inquiry as beginning with an unsettling. But Peirce is more specific. Yet that word "irritation" irritates me there, it sounds unnecessarily negative. One could say that at least some inquiry begins with

*temptation*by doubt.

Is curiosity merely a kind of irritation? Some people regard all desires as irritations. Desire is the positive version of averseness, while pleasure is the positive version of pain. Yet desire - as we say, the

*pangs*of desire - correlates also with pain, desire as a pain of lacking something that one

*would*like. There's something negative or privative about the

*would-be*in contrast to the

*is*, whether one responds with desire or averseness. Averseness, as opposite of desire, could suggest a pleasure of lacking something that one

*would*dislike. The idea of averseness doesn't particularly suggest that to me, except when I consider the idea of contentment and satiety, but those aren't the same thing as positive pleasure. Of course, the idea of desire doesn't always bring the idea of pain strongly to mind, but still, something seems not quite symmetrical here. Looking it up, I find

*pang*defined as an ache or twinge, and its antonym given as

*tingle*. A tingle of aversion or disgust? A

*pleasurable*tingle of aversion? A delighted scorn? There are threads of sense in all this but somehow they don't come together as truistically as I'd like. Maybe one of these days I'll figure it out. Anyway, since curiosity is a desire to know, one can see it as involving desire's pain and irritation, yet desire is not merely pain and irritation, the feeling of actual ill. Instead it's to feel drawn toward the potential positive presence of something that would be pleasing. Desire and pain logically involve each other but are not flatly equivalent.

Now, Peirce frames inquiry generally in terms which do not portray the inquirer as searching for truth

*per se*, just for truth's sake, as if in some sort of idleness. Peirce holds that genuine inquiry is struggle and that inquiry based on merely verbal doubts is normally fruitless. He wants to show inquiry as a struggle driven by strong motivations yet capable of eventually attaining considerable objectivity. Quoting myself from a few articles at Wikipedia: "Starting from the idea that people seek not truth

*per se*but instead to subdue doubt's irritation, Peirce shows how this can lead some to submit to truth." One does often notice that people are concerned to maintain beliefs in which they are invested by practices built upon those beliefs - so, doubts irritate them. But suspicions also excite people, so that many a person seeks to dampen such distracting suspicions, excitements, sneaking hopes in himself or herself, or in others, i.e., tries to keep eyes on the ball. So, I think that "irritation" is too negative a word there. Unfortunately I can't think of a more general word that covers both irritation and excitement. It's like trying to think of a word that means specifically both desire and averseness, that is to say, a word that has the meaning of the phrase

*affectivity oriented toward the would-be*.

I disagree with Peirce's claim that surprise is the universal occasion of inquiry. Some sort of unsettling, yes, but not always surprise. Inquiry does not always begin with the improbable, the unlikely, the anomalous, as if a person had a

*plenum*of beliefs or expectations for all occasions. Instead one is sometimes aware of things about which one has no particular expectations, things which are standing mysteries. How did some island get where it is? Etc. One is puzzled or baffled, the puzzling thing

*complicates*one's understanding of things. One seeks a simplifying explanation. Peirce tries to account for such things as "passive" surprises, things that happen in the

*absence*of expectations specifically of their happening. (For Peirce, "active" surprises are things that happen in the presence of expectations specifically of their

*not*happening.) I don't think that it's a simplification to regard bafflement as passive surprise. That's like regarding fancy or supposition as passive expectation. The anomalous goes against particular expectations. Not only does the baffling complexity or complication elude particular expectations, it more actively goes against fancies or suppositions about what most simply would be.

From the start the claim of surprise as occasion for all inquiry involves idea of expectations about the future, the probable, the likely. So, from the start, in order to check that claim, one should consider whether other time ideas and modality ideas can also be found as occasions of inquiry. One such is bafflement, defiance of simplifying supposition about what most simply would be. Here the time is the would-be and the modality is simpleness, facility (to the point where it's hard

*not*to do it), a kind of optimality. Note that surprise occurs when that which comes to light defies expectations, and bafflement, perplexity occurs when that which is (more or less) familiar defies suppositions and simplifying fancies.

So there are two other times right there to consider: the present in its coming to light, and the past. The two correlated modalities are informativeness and nontriviality, depth. Sometimes currently gained information is insufficiently clarifying as to what is going to happen, what is "coming down the pike." One becomes impatient. And sometimes one's established facts, one's remembered past, aren't deep enough to offer lessons, conclusions to be drawn, about what in a current case would happen if one were to do certain things - so that one hesitates, for instance to walk across a log bridge, or to build a bridge out of some curious material.

So, once more from the top, unsettlements that stimulate inquiry:

1. Bafflement, perplexity, at the seemingly complex or complicated, defying one's best simplifying suppositions. E.g.: What

*has*happened?

2. Surprise at the anomalous, the seemingly unlikely, defying one's expectations. E.g: What

*is*happening?

3. Suspense, impatience, over the vague, the seemingly uninformative, defying one's discernment. E.g.: What’s

*going to*happen?

4. Hesitancy, inagency, about the unfamiliar, the uncolligated, the seemingly trivially-connected, that whose lessons have not been learnt, defying one's remindedness. E.g.: What

*would*happen (if…)?

Note: I understand the would-be, in those contexts, as corresponding roughly to the surface of the future light cone, in a sense the present, but the present to which one appears and into which one acts directly. That present, one's presence to others, is the surface of one's future. This is as opposed to the present as it comes to light to one - that coming-to-light present is the surface of one's past (corresponding roughly to the surface of the past light cone). I say "roughly" because, for example regarding the analogy to the future light cone's surface, one's outgoing most-feasible, optimal or extremal, or best "shots" don't always travel at lightspeed, yet the difference between such and the later future, the future as probabilities, parallels the difference between the future light cone's surface and the future light cone's inside. Thus a difference between desire and trying for what is almost now, on the one hand, and hope and pursuit toward a later goal on the other hand. In that sense, where one is distinguishing between that which has happened, that which is happening, that which is going to happen, and that which would happen, I'd like a conditional participle, so that it's clear that I'm talking about a would-be based in the concrete situation. (Informal Esperanto would allow it -

*tio, kio estas pas*.)

**unt**a# Symbols

Underlying my ideas of index, semblance, symbol, and proxy, are four categories:

1. 'Substance' or object. Primary substance is

*this man*,

*this horse*, etc. But it could be abstract, a mathematical structure for example.

2. 'Accident' or attribute or quality.

3. Modality, logical quality ('indeed', 'not'), probability, novelty (information), feasibility, optimality, etc. By those I mean

*mode*in its original definition though not usage (see Peirce, CP 2.382 and DPP p. 89) as referring to any qualification of a proposition or its copula, and I expand the idea to include straightforward logical quality - affirmative and negative.

**Update**(3/17/2010). I just found that Peirce leaned that way about modality and logical quality. See "Prolegomena to an Apology for Pragmaticism" (1906)

*The Monist*, v. XVI, n. 4, pp. 492-546, footnote on page 525 (last footnote of CP 4.552).

**End of update.**Obviously I need a distinct technical term for it but I'll make do with "modality" for the time being, though, again, I do not mean it only in the usual sense (necessity, possibility, impossibility, unnecessariness). It is something like a conduit of comprehension (intension) which may qualify the comprehension in terms of fulfillment in some portion or totality of cases, the distribution of the associated denotation, etc.

4. Mathematical relationship ('double of', 'inverted order of', 'indefinite integral of', etc.), particularly as conceived of as object(s)-to-object(s) relationships. In a way, it's like a router or re-router of denotation (or maybe I should say, denotational relationships).

The index represents an object by being connected to it (in the same larger object).

The semblance represents an object by sharing characteristics or qualities with it.

The symbol represents its object by having the same modal properties/relationships.

The proxy represents its object by having the same mathematical relationships in some sense, anyway by making the same "decisions," following the same rules under ongoing observation and experimentation.

So, consider a property which has a 50-50 chance of belonging to any given object in a total population. One side (it could be either side) of a fair coin symbolizes that property in its frequency. Surprising winners symbolize each other, in a sense, irrespectively of resemblance or connection or proxyhood to each other. There is something in common about their situations - not their locations or times, but about the alternatives in which they are involved. Again, not proxyhood: you can't use a fair coin to determine the outcome of an evenly matched two-horse race, but you can consider the fair coin in order to deduce horses' odds on the assumption that the race is evenly matched. And so on. Sometimes the odds overwhelmingly favor a given outcome. Two masses of particles can be proxies for each other if they make the same "decisions" thanks to the law of large numbers when their respective particles are not proxies for each other. These relationships of attribution, distribution, etc., are matters of interpretation, but are not always subjective or matters of mere convention. Implication itself depends on structures or processes of alternatives among cases; meaning is a phenomenon, if you will, of modality. The modalities lead to an effect on interpretation. Sometimes, habitual implications are translated into conventional symbols which have the same value as (are equivalent to) their objects despite disconnection and dissimilarity.

(At one time, Peirce distinguished comprehension or intension from implication; at later times, he seemed to hold that they're more or less the same thing. I need to look further into that. The problem is that comprehension is of characteristics, but Peirce does not seem to take resemblance as a kind of comprehension. And if one either denotes objects ("Seconds") or comprehends characters ("Firsts"), then, in Peirce's system, what mode of sign relation is there to "Thirds" - representation, sign relation, attribution, etc., themselves? It appears that, early on, Peirce would have said

*implication*, but not later on.)

As regards Jakobson's paradigm, it is an alternative among words that could be used in a given place and time. It is not the expression of such an alternative, e.g., "The horse

*cantered or galloped*." Such disjunctive compounds, as well as conjunctive compounds, are alternates forming an alternative or paradigm (if there's some flexibility about the length of the phrase to be chosen). Moreover, logical conjunctions and alternatives all express basically "paradigmatic" rather than syntagmatic relationships, insofar as paradigmatic relations reflect relations of logical quality and modality. (Note: the polyadized variables expression "

*xyz*" is not a logical conjunction like the conjunct propositions expression "

*pqr*", instead it is something else.) When we speak in a second-order way about the first order, then the "and" versus "or" distinction is like that of syntagmatic versus paradigmatic - you use one word (or phrase) AND another AND another, etc., in the syntagmatic relation, and one word (or phrase) OR another OR another in the paradigmatic relation, but the resultant sentence does not thereby

*express*relations of conjunction and alternation. Expressions of logical relations - "She wore a blue

*or*green dress" - "She wore a blue

*and*green dress" - etc., are expressions, explicitations, of options about things, options that also also underlie and help determine paradigmatic relations among words. That said, I'm unsure how best to analogize paradigm (to process? to function?) as I've analogized syntagma to structure. I guess it's not such a shortcut after all.

Do I feel that my idea of symbol is comfortably simple and consistent with my ideas of the other sign kinds yet? Not quite yet.

# Rosen, Saussure, Peirce

**(Recentest significant change: December 30, 2010).**

In Peirce's semiotic, an irreducible triadic relationship defines object (subject matter), sign, and interpretant. I've argued the need for a fourth element, which I've called a

*recognizant*. Trying to get away from such a psychological-sounding term and to follow the word pattern of "interpretant", I've also called the recognizant the

*verificant*but that's misleading, because I mean not the verificatory evidence but instead that of which the content is the verified or established or corroborated, etc., the

**lesson learned**. As the interpretant is, in a sense, an aspect or moment of an interpreter, so the recognizant is an aspect or moment of the recognizer (who in a sense is a verifier or corroborator, etc. - recognizing or acknowledging a legitimation). In the analogy of semiotic to Shannon's information-theoretic scenario: As interpretant stands to decoding, so recognizant stands to recipient.

Now, among kinds of sign, Peirce's most famous trichotomy is that of icon, index, symbol. The three are defined by how they stand for their objects: icon by its own characters, as resembling its object; the index by factual connection to its object; and the symbol by interpretive norm or habit of reference to its object. The point to notice is that the index represents in virtue its object-connection which it has had; the icon in virtue of its own representative characters as now presented; and the symbol in virtue of how it will be interpreted, i.e., in virtue of its interpretant. You see the pattern:

So I thought that to complete the pattern for my version of semiotic I would need a fourth kind of sign, one defined by some sort of relationship to the recognizant. The pattern seemed to generate the idea of a sign which represents in virtue of the case that it would be recognized to stand for its object by one observing the object if the object were available. Such would be a sign which stands for its object for observational and experimentational purposes. This I called aindex:object::icon:sign::symbol:interpretant.

*proxy*. I distinguished proxy from mere surrogate by this consideration: a proxy can make decisions on somebody's or something's behalf, by following some sort of rules for the decisions that that person or thing would make. Of course, in a corporate election, if you have somebody's proxy, you can vote howsoever you want. But I was thinking of things like power-of-attorney, a lawyer representing somebody by acting on that person's behalf according to the rules of that person's best interest, making the decisions that that person would make if they were conscious and present, grasped the law, etc. Eventually it occurred to me that that which Peirce calls

*diagrams*are also proxies. Now, Peirce classes the diagram as a kind of icon. The diagram is subject to the same transformabilities as its object. Peirce holds that the study of mathematics proceeds through experimentation with diagrams, observation of them, etc. Such diagrams may consist in geometric forms or in arrays of algebraic expressions, etc. They need not outwardly resemble their objects at all. So, I speak of

*semblances*, not icons, and class diagrams as proxies, not as semblances. It occurred to me that some physical objects can be proxies for others - electrons can be proxies in experiments for any electrons, they're all the same and follow the same rules. A statistical correlation between two things, on the other hand, may be a mere resemblance independent of underlying sameness of structure, rules, etc. It's the kind of relationship which conduces to inductive generalizations, subject to testing. Anyway, as signs, proxies are no more infallible than semblances and still need to be checked.

I've noticed that the word "proxy" is currently used to refer to surrogate indices, such as tree rings as so-called "proxies" for thermometers. I'm doubtful that that is a good use of the word "proxy." They are alternate indices for the same phenomena. Any, the scientific currency of the word "proxy" in such a sense is certainly inconvenient for me, but there's little that I can do about it.

### Rosen

Somebody who has read my Websites contacted me in late 2008 and, in our subsequent correspondence, mentioned Robert Rosen's modeling relation and gave me this link: http://www.panmere.com/?page_id=18. I read that and another page at the linked Website http://www.panmere.com/?p=56. The distinction which Rosen made between simulacrum and model is much like mine between semblance and proxy, except that I was unsure that Rosen models could be concrete and not only abstract. Later I learned from a comment by Rosen's daughter Judith that for Rosen a model could be an individual concrete object (http://www.panmere.com/rosen/mhout/msg02147.html), so, as far as I can tell, my*proxy*is Rosen's

*model*; and my

*semblance*is Rosen's

*simulacrum*. Well, I'm glad that Rosen beat me to it! It suggests that the idea is not just my whimsical notion. He worked it out in terms of considerations of scientific thinking, especially in biology. I arrived at it through consideration of Peircean semiotic, an area in philosophical logic. I've mentioned this fit of my ideas to Rosen's a couple of times on peirce-l, in a May 17, 2009 post and in a July 14, 2009 post. (My earliest peirce-l discussion of proxies that I can find is this December 2, 2004 post.) So which is the better word,

*proxy*or

*model*? Well, a lawyer can be a proxy for, or of, his/her client. But you wouldn't say that the lawyer is or acts as a model

*for*the client, because that suggests that the lawyer's function is to set an example for the client to emulate. On the other hand, as I've noted, the word "proxy" has taken on a specialized and weakened meaning in science. Well, I'll go on speaking of proxies rather than of models, so that it's clear that I'm speaking about the idea of proxy as I've been working on it, not Rosen's idea of model as he worked on it, howsoever they may coincide.

Anyway, for Rosen, a model has that which he calls a synonymy of structure of entailment with the thing or process that is modeled. I'm not sure what is the point of the word "synonymy" in that context. It suggests that the structure itself is some sort of word or symbol. Instead it's enough that the structures of entailment be the same, or isomorphic, or homologous, or whatever, between model and modeled thing, without calling those structures "synonymous." To have the same meaning is to reflect the same norms or parameters, to make the same difference; not necessarily to teach the same lessons through experimentational structural transformations. The reader may think that I'm splitting hairs but I'm dealing with four kinds of sign (index, semblance, symbol, proxy) and they involve meaning in various ways. I need to maintain some sort of careful distinctions, at least sometimes.

A symbol is a kind of sign that has the same value or import as its object even when they have neither a connection nor a resemblance nor a structural sameness (the kind of sameness that allows for parallel transformations). Now, since I can think of concrete objects which are natural indices, natural semblances, and natural proxies, why not symbols? So I have to broaden the idea of the symbol, to that of a sort of functional surrogate or functional equivalent for something, such that it serves as a sign about that thing, irrespectively of connection, resemblance, or homology. Wish I could think of an already existent name for this broadened idea of symbol.

### Saussure

In trying to regularize this tetrachotomy of index, semblance, broadened symbol, and proxy, thinking (as always) of Aristotle's four causes, and so on, I recently started considering the syntagmatic-versus-associative dichotomy of Saussure and, more specifically, Jakobson's refined version, the syntagmatic versus the paradigmatic. The syntagmatic has to do with the way things are compounded or connected in an orderly way; it's a broadening of the idea of syntax. The paradigmatic has to do with sets of options which exclude each other in some sense, anyway involving at least an alternative among words if not among things, as in "She wears a..." scarf or jacket or blouse, etc. They don't have to exclude each other literally, though they can. Paradigmatic relations involve differentiation.1. Index - thing which represents its object by being in the same syntagm, so to speak, with its object - it is connected to its object. (This involves broadening the idea of syntagm.)2. Semblance - thing which represents its object by having the same differentiae, the same qualities, differentiating respective parts or stages. (Here a sequence is seen not as a temporal syntagm but as a paradigm going through its alternatives, its various phases, in time). A very simple semblance is when two things simply have the same single quality across respective parts or stages. | 3. Broadened symbol - thing which repesents its object by being an alternative to it, but still having the same value, still making (as opposed to having) the same difference.4. Proxy - thing which represents its object by having the same syntagm, in some sense same structure, following the same rules, and teaching the same lessons. |

. . . . |