index.html  Translation as Contradictory Complementary Convergence:
Language, Science, and Culture


Floyd Merrell

Purdue University






All signs becoming other signs are signs in the process of translation (1) within the same language (intralinguistic), (2) across languages (interlinguistic), and (3) between sign types (intersemiotic).[1]

Intralinguistic translation consists of a sign’s repetitive use within different contexts that, as contexts vary, so also the sign’s meaning, and thus the sign becomes a variation of itself.  An extreme case:  ‘Atom’ from Greek times as a minuscule solid, changeless, impenetrable sphere became transformed, after a series of differentiations over the centuries, into a wave amplitude of myriad possibilities one of which might at some instant become a particle.  Interlinguistic translation is the customary use of the term ‘translation’.  In such cases ‘atom’ in English becomes ‘átomo’ in Portuguese.  Intersemiotic translation, between sign types and sign contexts, is a matter of a relatively simple sign’s becoming a more complex sign, or vice versa.  A typical case:  The image of a cross, icon, by hyperbole, is taken as part of a scene including other religious images, and in this context where signs are indicating other signs it is thus translated into an index, and subsequently the thought or invocation of the icon-image becomes translated into a linguistic symbol, ‘cross’.  The sign becomes translated from one sign type to another, and then another in a sequence that, as far as the sign’s interpreter can tell, occurs in the mere blink of an eye.  Yet the sequence occurs in time, however brief the increment.  Time occurring within experience, a primarily one-dimensional process, becomes of utmost importance, as we shall note below.



Sign types


Charles S. Peirce developed a rich tripartite concept of the sign (CP 2.227-307).  The most basic of Peirce’s three sign types consisting of icon, index, and symbol is the icon, which, in its most fundamental form, is self-contained, self-reflexive and self-sufficient, because it has not (yet) entered into inter-dependent, inter-related inter-action with anything other than itself.  In other words, with respect to the three sign types, the icon is a First, of C. S. Peirce three categories, Firstness, Secondness and Thirdness.[2]


Briefly:  (1) Firstness is what it is, irrespective of anything else (like the icon, it is self-contained, self-reflexive and self-sufficient), (2) Secondness is what it is insofar as it is inter-active with something else, and (3) Thirdness is what it is insofar as it mediates between Firstness and Secondness and brings them together in the same way that it mediates itself and them, bringing it into inter-relationship with them.  Thus the categories are inextricably linked; they are inter-dependent.[3]


An icon is a sign bearing possible similitude with something else.  The similitude is no more than possible, for the icon qua icon has no other; that is, it enjoys no explicit element of Secondness—inter-action with something else.  Icons in the purest sense come in the form of what Peirce labels hypoicons.  They are so called because they have not (yet) been endowed by their sign makers and takers with any form or fashion of linkage with anything else, nor have they been given any sort of meaning (CP 2.274-83).  Peirce subdivides hypoicons into three categories:  images, diagrams and metaphors (in terms of pure sensations, whether visual, auditory, tactile, olfactory or gustatory).  Figures 1, 2, and 3 as visual examples of hypoicons.


Notice that:  (1) the first figure is a matter of Firstness, insofar as it remains uncontextualized, though it enjoys possible contextualization in inter-dependency with other possible contexts, (2) the second figure is that of Secondness as a possibility that can enjoy realization only if it becomes inter-actively linked with something else, and (3) the third figure includes two images that enjoy possible linkage, and if this linkage is realized, given the two images’ inter-relation of similarity, they can take on the character of Thirdness, though the inter-relationship remains no more than tacit at this level.  Notice also that the notion of hypoiconicity entails no more than possibilities, since at this stage of a sign’s development there is no more than self-containment, self-reflexivity and self-sufficiency.

An index is a sign that has inter-actively entered into inter-dependent inter-relationship with something other than what it is (a Second).  Examples of his type of sign are given in Figures 4 and 5.  An alarm clock obviously having been triggered by the time is an index inter-relating and inter-acting with the function of the alarm, obvious to jerk someone out of their slumber.  A flash of lightning is an index inter-relating and inter-acting with that which constitutes the effect caused by the lightning:  thunder.  Peirce terms indices ‘natural signs’, since their inter-related inter-action is the result of contiguity, part-to-whole, container-to-contained, or cause-effect, all of which consist of physical world or mental inter-connectivity.


A symbol is the consequence of sign mediation between icons and indices in the same manner in which it, the symbol, mediates between itself and them:  a democratic process of inter-dependent, inter-related inter-action (as such the symbol is a Third).  Figure 6 is, in and of itself, a visual icon.  If one is so inclined, within one’s particular context at the moment, one might be motivated to think of the icon as a produce of some McDonald’s franchise, the indexical extension of the icon.  And if one happens to feel hunger pangs and is in the company of a friend, one might blurt out:  ‘Big Mac!’ (a simple symbol or a term insofar as it inter-relates, by way of indexicality, with the visual image).  Linguistic signs are the prototypical example of symbolic signs.  The symbol mediates between the icon and its indexical extension and at the same time it, as a linguistic sign or a symbol, mediates between itself and them to create the background for it as a symbolic sign in order to endow it with some form of meaning.  Now, if one happens to have taken a liking to fast food, and especially to that of McDonald’s establishments, one might justify oneself to the other person with the utterance in Figure 7.  This is a sentence string, or proposition, consisting of various linearly linked symbolic signs the combination of which makes up a complex symbol.  If the other person present questions one’s culinary preference, one might then create a rudimentary argument such as that found in Figure 8, a simplified syllogism of sorts, which makes up a more complex symbol, or an argument.[4]


The set of sign inter-dependent, inter-related and mediated inter-action gives us the wherewithal for a Peircean conception of the sign as illustrated in Figure 9, the basics of which consists of the diagram in Figure 2.  Take notice of its ‘democratic’ nature according to which in spite of the difference of complexity between sign, semiotic object and interpretant (the later term, roughly speaking, consists of interpretation coupled with meaning), like the difference between icon, index and symbol, does not alter the fact that the three sign components are in their own way of equal value.  The model is hierarchical, for sure, but each element of the hierarchy is of equal importance.  Without either iconicity or indexicality, or without sign function or semiotic object, there can be no symbol or meaning.  The combination all three components stands tall; a lack of any one or two of them, and the entire structure falls.  Thus, the ‘triad’ of elements should be more properly conceived as a ‘tripod’ of three dimension rather than a two dimensional drawing on a flat plane.  And the swiveling, swerving bidirectional arrows should be conceived as giving the sign incessant movement, ongoing change, for it is process rather than fixed product.




Sign ‘degeneracy’


Just as signs become other signs of increasing complexity, from icons to indices to symbols, and from words to sentences to texts, so also signs can become ‘degenerate’.  By ‘degenerate’, I must hasten to point out, Peirce meant the mathematical use of the term, and it has nothing to do with the ordinary implications of the word as ‘deterioration’, ‘decay’, ‘corruption’, ‘perversion’, or whatever.[5]

‘Degenerate’ signs involve the process of signs going from greater complexity to lesser complexity or relative simplicity.  Signs having become ‘degenerate’ are signs made and taken by semiotic agents, whether human or some other form of sign making and taking organism, as a consequence of ‘habituation’.  That is, ‘degenerate’ signs entail conscious and conscientious sign activity that has become ‘sedimented’ and ‘entrenched’, through repeated use, such that they are made and taken in a relatively simple matter-of-fact manner, but as if they were signs of greater complexity.  The relatively simple signs are made and taken in ‘habituated’, rather ‘automaton’ fashion; yet their meaning, their interpretation, is rich in implications of complexity.  In a manner of speaking, (1) ‘degenerate’ and relatively simple signs embody the possibility of what they once were in all their complexity, (2) they can once again become (be translated into) those complex signs they bear implicitly, and (3) if and when they have become translated into the signs they once were, their ‘habituated’ character will have been at least partly revealed.

I should offer a mathematical-geometric example of ‘degeneracy’ in good keeping with the spirit of Peirce’s notion of the term.  In schematic form, just as a point, by multiplying itself an infinite number of times in linear succession, can become (can be generated into) a line, so also a line can become a plane, a plane a solid, and a solid a hypersolid.  In other words, it is a matter of zero-dimensionality becoming one-dimensionality, one becoming two, two becoming three, and three becoming four-dimensionality, as in Figure 10.  De-engenderment, or de-generacy, then, would reverse the process.  In this respect, just as a point has the potentiality to become a line, a line a plane, a plane a solid and a solid a hypersolid, so also dimensionalities can, topologically speaking, be compacted or compressed (by ‘degeneracy’) to a point (in technical terms, a ‘singularity’).

In terms of the general semiosic process, ‘degenerate’ implies that symbolic signs can be made and taken as if they were signs of indexicality or iconicity.  In reference to Figure 10, a symbolic sign having become sedimented to the first degree of degeneracy is tantamount to an indexical sign, and a symbolic sign having become sedimented to the second degree of degeneracy is tantamount to an iconic sign (Peirce CP: 2.246-72).  In another manner of putting it, with respect to Figure 11, first degree mediation takes signs from Firstness to Secondness, iconicity to indexicality, and from feeling-sensitivity to sensation-perception, while second degree mediation takes signs from Secondness to Thirdness, indexicality to symbolicity, and sensation-perception to conception-articulation.  Degeneracy, then, would reverse the arrows.




The cubolator:  a possible model of generacy-degeneracy


Playing around with dimensionalities inevitably takes one’s mind to the realm of cubist art.  What is it that the cubist is doing?  In essence, presenting a three-dimensional object on a two-dimensional plane from varying perspectives of the three-dimensional object as if the artist were look at the object now from the front, now from the back, now from one side or the other, now from an upper angle, now from a lower angle, and so on (for further, Shlain 1991, merrell 2007).

Of course during the process of physically or mentally walking around the object in order to attain a diversity of perspectival grasps, the artist is required a series of time lapses.  But where is time in the finished painting?  It’s there, in an instant, crystallized, as in the hypersolid of figure 10.  This is metaphorically like the point becoming a line and the line a plane and so on during successive increments of time, and then it’s all there all at once, in the four-dimensional hypersolid—a ‘timespace continuum’.  The hypersolid, then, is metaphorically what we see when we gaze at a work by Pablo Picasso.  We gaze at the cubist object as if it were of three dimensions of space and one dimension of time (as if it were a genuine sign of generacy) but actually, it is no more than a set of two-dimensional images on a flat plane (a sign of degeneracy).[6]

A simplified illustration of the cubist method is available by considering a Necker cube from its two possible incarnations, then putting them together, as in Figure 12, where the letters, R, O and I, or ‘Representamen’ (sign), ‘Object’, and ‘Interpretant’ merge and blend into one in order to give the impression of their being so sensed, perceived, conceived and interpreted in all their variations in an instant.[7]  It is as if one were viewing the sign from one angle within one context and interpreting it, then viewing it from another angle and context and interpreting it as differentiated with respect to what it was.  Within the juxtaposed cubes, what we have is a blended, hybrid combination of both possible interpretations.  Figure 12, in this respect, suggests that:  (1) within the lowermost image, we have Firstness as no more than possible interpretations (emerging from 0, the realm of all possible possibilities) of the ambiguous image, (2) within the middle images we have an actualization, or realization, as Secondness, of the alternative ambiguous interpretations, and (3) in the uppermost image we have the two interpretations converging and blending and thus potentially revealing, by way of Thirdness, the cubes’ ambiguity.[8]



Cubotalogy-Degeneratology and the Peircean sign


What, more specifically, does this ‘cubotological’ talk have to do with degeneracy?  Much, I would suggest.  The standard Saussurean concept of the sign, derived from Ferdinand de Saussure’s theory of language, and especially the practice of Saussure-inspired ‘structuralism’, is indelibly binary in nature; that is, it holds true to bivalent logic.  It allows a choice between one alternative and another:  truth or falsity, good or bad, male or female, black or white, and so on.

What I attempt coming to grips with in this essay is an alternative ‘logic’ of more than two values, namely:  (1) imaginary (giving rise to viable new possibilities that can emerge and take their respective place in our perceived and conceived world in terms of how we distinguish it into our familiar bivalently ordered eithers and ors—chiefly of the nature of Firstness), (2) true or false (of the nature of bivalent logic, classically conceived—chiefly a matter of Secondness), and (3) emergent (within the range of likely, probable, or necessary timespace frames or contexts—within the parameters of Thirdness).  In other words, I wish to account for a context-dependent contradictory complementary ‘logic’ of inter-dependent, inter-related, inter-active sign processes.  These processes have been depicted in Figure 12.  We have:  (1) at the lower level, viable possibilities (Firstness), emanating from the virtually infinite range of all possible possibilities (0, ‘nothingness’, ‘emptiness’), (2) at the middle level a pair of these possibilities is actualized (Secondness), each within a separate temporal moment, and (3) at the upper level these possibilities are given interpretative implication as contingently emergent blended, hybrid viabilities (Thirdness) within their respective timespace context.  There is generacy, from Firstness to Secondness to Thirdness as we move up the levels, and the possibility of degeneracy, such that one might be able to capture the duality of interpretations in an instant, at the lower level in terms of Firstness.  There are imaginary possibilities, bivalent actualities, and emergent combinations of blended signs.

But when everything is taken into consideration, is all this not reducible to just another form of bivalency?  Well, in a manner of speaking, yes.  The Necker cube is doubly ambiguous, for sure.  But this example is a simple one, for illustration.  For an exceedingly complex example, in comparison, all one need do is contemplate a Picasso painting and one will become aware of the multiply ambiguous nature of the work.

So, what does this have to do with an ‘alternative logical’ formulation to which I have alluded?  Consider the following.



A nondual, contradictory complementary lattice


Intralinguistic translative  practices

Figure 13 entices us with the notion of the range of all possible possibilities, 0.  Then we have:  (1) a first level representamen or sign, R1, its respective semiotic object, O1, and mediation, M1 giving rise to the emergence of an interpretant, or interpretation, I1, (2) a second level, consisting of the doubly mediated offspring of the first level in R2, O2 and M2-I2, and (3) a third level, culminating in S (counterpart to the uppermost portion of Figure 12), a contradictory complementary convergent fusion, a hybrid or alloy, of R, O and I, which includes everything in the lattice that has been engendered from the lowermost point, 0.

But what is going on here?  There is no smooth transition from the bottom to the top of the figure.  There is no direct path from M1-I1 to M2-I2, from O1 to O2, or from first level mediation to second level mediation.  So how can it be that everything is inter-dependent, inter-related and inter-active, according to the premises I’ve put forth in this essay?  The paucity of direct, linear pathways is precisely of the nature of the nonlinear, context-dependent lattice.  We cannot go directly from any one of the first level sign components to its counterpart at the second level because there is no classical linearity.  Everything is inter-connected, but through mediation by way of contradictory complementary convergence.

Now what do I mean by that?  I mean that the lattice is not exclusively bivalently ordered but ordered by an ‘alternative logic’.  This alternative is defined by the following processes—and I write ‘processes’, because what is becoming by virtue of the lattice is always becoming something other than what it was previously becoming.  That is to say, everything is incessantly changing; nothing is fixed.  The lattice operates in terms of a departure from Boolean principles according to the following:

1.         R1 x O1 = 0

            O2 x R2 = M1-I1

            M1-I1 x O1 = 0

            M2-I2 x R2 = 0

(In other words, the ‘disjunction’, or product of any two signs within the lattice calls for a move downward until a common point is reached.  The ultimate product, or the range of all possible possibilities, 0, entails inclusion of everything that might emerge from within an unlimited range of timespace contexts.  It includes both ‘this’ as possibility … and ‘that’ … and ‘that’ … n.)

2.                   R1 + O1 = M2-I2

            O2 + R2 = S    

            M1-I1 + O1 = S  

                  M2-I2 + R2 = S

(In other words, the convergence, ‘conjunction’, or sum of any two signs within the lattice calls for a move upward, and movement upward creates more inclusive contexts.[9]  Thus, converging signs entail alternate contexts capable of including the range of all signs involved.  Such contexts embody neither ‘this’ previously existing context … nor ‘that’ one … nor ‘that’ one … n, but something else.)

If you wish for further details regarding the nonlinear nature of the lattice, consult the Appendix.  If not, let us proceed with the observation that 0 is the range of all possible possibilities, and S is the contradictory complementary convergence of all disjunctive or common elements between the terms of the lattice.  Notice that:  (1) the uppermost level of the lattice is nothing without the lowermost level, which allows for the emergence of actual signs from within the range of all possible possibilities, (2) inter-dependency regarding the uppermost level and the lowermost level is impossible without inter-action at the middle levels, and (3) the middle levels give rise to the possibility of emerging inter-relations regarding the annealed, hybrid elements making up the uppermost level.

Notice also, in this light, that:  (1) R, O and I are no longer separate elements, but rather, they have merged or converged into one another to yield a hybrid or alloyed signifying process, (2) R, O and I at all levels of the lattice can be of the same language, which is to say that each sign within a given language, during successive instantiations of its use within different contexts, is always becoming something other than what it was becoming in the previous instant, which is to say that intralinguistic translation is germane to everyday language use, and (3) R, O and I can be interlinguistic or intersemiotic, or a mixture thereof, but in whichever case there can be no translation without inter-dependent, inter-related inter-action of iconicity and indexicality as well as symbolicity; in other words, no translation can be exclusively linguistic (symbolic).

The process of signs becoming something other than what they were becoming during everyday language use is apparently contradictory in nature, since R (the sign), is what O (the semiotic object) is not, and both R and O are what I (the interpretant) is not.  However, the contradiction is the appearance that hides the underlying nature of the semiotic process, whose processual character is actually that of complementarity (regarding the concept of complementarity, Bohr 1961, Folse 1985, Plotnitsky 1994, merrell 2004).  There is complementarity in the semiosic process, since, as intimated repeatedly in the above paragraphs, R, O, and I are thoroughly inter-dependent, inter-related and inter-active, such that there is a tinge of O and I in R, of R and I in O, and of O and R in I.  This falls in line with the familiar Yin-Yang image: (Merrell 2002).

With the lattice in mind, let us turn to interlinguistic and intersemiotic transformations during sign translation.



Contradictory complementary convergence viewed through cultural processes


Interlinguistic translation always includes some form of intersemiotic translation

As an illustration of what was asserted in the preceding section, let us consider the metaphorization process.  We begin with an image (iconicity, Firstness), which is then inter-related inter-actively with a particular object it motivates in the physical world and/or the mind (indexicality, Secondness), and then a word is evoked (symbolicity, Thirdness).  This makes up the first tripartite tier of the Figure 14 lattice consisting of the schematic image (representamen, sign—on the left) of a dog, an actual dog (semiotic object—on the right), and the word ‘dog’ and its meaning (interpretant—in the middle).

The second tier incorporates the process of metaphorization wherein the original image is endowed with a particular subspecies of the sign in question (the schematic form of a bulldog), which is then figuratively inter-related and inter-acted with something which the first tier sign, object and interpretant are not (the photograph of Winston Churchill), both of which are mediated by the word ‘bulldog’.  This combination yields the mediated metaphorical equation at the lattice’s uppermost level to imply:  ‘Winston Churchill (is a bulldog)’.  The man is not the animal in question, but the man evinces some aspect or other of what the animal is; hence the metaphorical homology.  Tier one entails literal interpretation and tier two figurative interpretation.  Mediation one gives the customary symbol or word while mediation two entails the creation of a variation of the image, the inter-connection, and the metaphor from 0—which entails the range of all possible possibilities within an indefinite number of possible contexts.  The final symbolic evocation, ‘Winston Churchill (is a bulldog)’, contains neither Winston Churchill as ordinarily perceived and conceived nor a bulldog as ordinarily perceived and conceived; rather, it contains a fused hybrid mix, a contradictory complementary convergence of signs, objects, and interpretations in a slightly to radically distinct manner (all of which is depicted by the symbol, S).

Let us try another figurative device:  a portmanteau word.  At tier one of Figure 15 we have the sign-word, ‘chocolate’, the object, what it ordinarily what ‘chocolate’ is not and is contradictory to what it is, ‘alcohol’, and we have the first mediation, Mediation1.  The format for this lattice is at variance with that of the Figure 14 lattice since we begin with sign-words, or symbols, rather than images.  At tier two we have one of the consequences of alcohol when consumed by humans, such consumption acting as the mediator between alcohol and chocolate at Mediaton2.  Mediation3, at the lattice’s apex, gives us the contradictory complementary converging hybrid, ‘chocoholic’, a portmanteau word.  I would respectfully submit that by this procedure it is possible to give a diagrammatic illustration of any and all rhetorical devices.

At this juncture I should reiterate the assumptions I put forth at the outset of this essay, namely, all signs, in the Peircean sense, involve translation:  (1) of signs of the same language within different contexts, (2) of signs from one language to another language, and (3) of signs of distinct semiotic types.  It has become more forcefully apparent that Peircean translation involves, in one way or another, iconicity and indexicality in addition to symbolicity in case the translation is intralinguistic or interlinguistic.  In other words, it involves signs as pre-linguistic images, diagrams and metaphors, whether visual, auditory, tactile, olfactory or gustatory (second degree degeneracy), as well as signs in inter-related inter-activity with respect to their objects (first degree degeneracy).

The very notion of degeneracy is in this manner germane to the process of signs incessantly becoming something other than what they were becoming, that is, to the process of sign translation.  As a further example of degeneracy, if one alludes to one of the four appendages holding up the nearest table as ‘leg’, one is using a doubly generate sign.  One uses the image (Firstness, icon, image) of a ‘leg’, and the function of the appendage (Secondness, index, connection) in one’s allusion to the ‘leg’ (Thirdness, symbol, word).  And one comes forth with the verbal evocation, ‘table leg’, as if the ‘dead metaphor’, along with its iconic and indexical nature, were a word used in literal fashion.  But it is not.  Use of the word ‘leg’ involves implicit, tacit, sedimented imagery or iconicity, and indexicality or allusion, in addition to symbolic articulation:  a ‘habit of mind’.



Interlinguistic-intersemiotic translation:  Brazil, the football country


Now comes the moment to begin pondering over the intricacies of interlinguistic translation through illustration of the actual complexity involved in what might at the outset appear as a simple one-to-one correspondence between a word from one language and its sibling from another language:  ‘futebol’, of the specifically Brazilian variety, versus ‘soccer’, in the general sense of the term in English.

What should be apparently simpler than translating ‘soccer’ (English) into ‘futebol’ (Portuguese), or vice versa?  The game is obviously the same in both languages:  rules, number of players, techniques and methods, strategies on the playing field, screaming fans in the stands.  As the saying goes by those apparent onlookers who are not true-blood fans, if you’ve seen one game you’ve seen them all.  Yes, quite obviously, translation of ‘soccer’ within an English language context is ‘futebol’ within a Portuguese language context.  Case closed.

But not so fast.  The apparently obvious conceals fluctuating, scintillating, effervescent sign processes incessantly becoming something other than what they were becoming.  Cultural analyst António Risério (2007) presents the convincing case that Brazilian ‘futebol’ is unique within the Brazilian cultural milieu.  It is not just ‘soccer’, nor is it the name of the game in any other of the world’s languages.  It is thoroughly Brazilian.  Originally from England and transported to Brazilian soil by upper middle and upper class Brazilians obsessed with  ‘macaqueando’ (‘parroting’, ‘imitating’) things foreign, ‘futebol’ was initially embraced as the product of a more ‘advanced’ and ‘civilized’ society that would surely render Brazilian society more ‘respectable’.  And so ‘futebol’ became one of the chief sources of Brazil’s emerging national identity.

But is ‘futebol’ really ‘soccer’ as it is ordinarily conceived outside Brazil?  Risério’s argument suggests a negative response.  Candomblé and other Brazilian hybrid contradictory complementary convergent religions, along with Capoeira and Samba, make up a set of repositories for identity making within the country’s cultural flux and flow.  Futebol-Candomblé-Capoeira-Samba:  like the sign components discussed above, together they merge into one; any one of them isolated from the entire set becomes of little consequence, and furthermore, the entire set to disintegrate.  United they stand, divided they fall.  United, they reveal Brazil’s rich mestiço (‘variegatedly mixed’, ‘fused’, ‘hybrid’) culture, a mind-boggling fusion and confusion of races and ethnicities and linguistic variations and cultural practices.  Separate any one of the cultural tendencies from the whole, and Brazil is no longer what it was.  Futebol-Candomblé-Capoeira-Samba is inter-dependent, inter-related, and inter-active.  Everything is convergent with everything else.[10]

What has this to do with the lattice as developed above?  It provides further demonstration of the radically nonlinear, multivalent, flowing and fluctuating nature of cultural processes in general.  Figure 16 affords an expanded version of the lattice to illustrate how ‘futebol’ cannot be considered in isolation, and much less translated as a context-free word.  ‘Futebol’ is part of Brazil’s rich cultural heritage, and as such it is certainly not the same as what goes as ‘futebol’ in England, Germany, Italy, France, Spain, or wherever.  It is Brazilian and only Brazilian.  Consequently, translation of ‘futebol’ into the English language as ‘soccer’ simply doesn’t do the trick.  If one strives to offer a precise translation, one would be compelled to incorporate the convergent phrase, ‘Futebol-Candomblé-Capoeira-Samba’, into some sort of English counterpart to ‘SamCapCanFut’ (that is, a counterpart to the uppermost point in Figures 12, 13, 14 and 15).  But how?  There is no Capoeira, Candomblé or Samba Brazilian style in any English speaking country.[11]

Of course, Brazil has influenced cultures the world over.  Capoeira schools abound in the Americas, Europe, Asia and Africa.  Candomblé and other Afro-Brazilian hybrids are practiced in many areas of the globe.  And who isn’t familiar with Samba and Brazilian Carnaval?  But the ordinary perception and conception of these cultural practices from people outside Brazil is not the same as the manner in which Brazilians perceive and conceive their own cultural practices.  There is no collusion of all these cultural tendencies into one, Brazilian style.  So to remain true to the cherished idea of translation in order to attain maximum precision (clear and distinct translation as the ideal), one must resort to vagueness (ambiguity as the actual consequence).

Strange as it might seem, this might remind us of Niels Bohr’s ‘Complementarity Principle’ and Werner Heisenberg’s ‘Uncertainty Principle’ in their interpretation of quantum theory according to which a particular manifestation of a quantum event (as ‘wave amplitude’) can be precisely described, but when that manifestation is combined with the event’s other manifestation (as ‘particle’), uncertainty pervades.  And we are now at the crux of the issue.  The lattice presented in previous sections is none other than a variation of what is termed ‘quantum logic’, which was invented by John von Neumann in an attempt to come to grips with the nonclassical (hence nonbivalent) nature quantum phenomena (Suppes 1971).

Time and space limitations do not allow for expatiation on his topic.[12]  However, I would ask you to notice that if we connect Fut2 to Sam1 and Sam2 to Fut1, and Can2 to Cap1 and Cap2 to Can1 in Figure 16, we will have a strange blended image comparable to that of the two Necker cubes in Figure 12.  This is not mere coincidence, I would submit.  The juxtaposed image presents an alteration of what is termed a ‘Tessaract’, as depicted in Figure 17.[13]  The ‘Tessaract’ involves an alternate method for patterning four dimensionality on a two-dimensional plane—along the lines of cubist methodology as illustrated in Figure 12.  Notice also that if you rotate the ‘Tessaract’ ninety degrees, and stretch it out horizontally, you will end up with an image comparable to that of Figure 16, as portrayed in Figure 18.

Whereas the generation of one-, two-, and three-dimensionality from an original point requires an increment of time, the ‘Tessaract’, like the point, exists atemporally.  In other words, metaphorically speaking, and commensurate with contemporary physics, the four-dimensional timespace continuum contains three dimensions of space and one-dimensional time compacted into an instant:  a timespace singularity.  The importance this act of mental gymnastics is to suggest that the lattice doesn’t model ‘reality’ or ‘objective semiotic reality’, but rather, it is the result of a visual attempt to explain semiotic flux and flow within broad cultural practices.[14]

In this light, another illustration of the contextual, nonlinear, contradictory complementary convergent nature of translation would behoove us.



Brazilian ‘jeito’


Speaking of the quandaries and quagmires of translation, Brazilian ‘jeito’ practices are enough to give any respectable translator nightmares.  How in the world can the word be converted into English?  We are hard pressed to find a remotely comparable English word, let alone a precise equivalent.  How can put ‘jeito’ practices into a solitary English word, in view of the fact that ‘jeito’ the Brazilian ‘jeito’ has been the subject of an entire book, numerous book chapters and a spate of articles in attempts to explain the term? (see the excellent studies by Barbosa 1992, 1995).

Once again we find ourselves thinking in terms of contradictory complementary convergence.  Brazilian ‘jeito’ requires great skill in manipulating cultural values and standards, legal laws and regulations, and generally influencing and sometimes taking advantage of people.  In order effectively to get something done by means of ‘jeito’, one is required a massive dose of skillful and tactful behind the counter wheeling and dealing involving charisma, cunning, and conning.  Lívia Barbosa tells us that ‘jeito’ involves a method for making some people more equal than others.  The point is well taken.  In a hierarchical society laying claim to the label ‘democracy’, the havenots must enjoy some means for coping with the haves, and the haves must have some means for asserting themselves as haves vis-à-vis the have-nots (see Ituassu and Almeida 2003).

Admittedly, an adequate account of ‘jeito’ in this essay is out of the question, for obvious reasons.  Yet there is the problem of translating the word, from Portuguese to English in this case, which I must address in order adequately to illustrate my semiotic model of contradictory complementary convergence.  Thus the question:  how is it possible to offer an English equivalent?  In a word, it’s impossible, strictly speaking.  There is no translational precision to be had here.  So how about an English word that at least gives a vague idea of the Brazilian vocable?  A ‘certain charismatic way’ about one’s manners and behavior?  A particular ‘knack’ for getting things done by bending the rules slightly?  A ‘special aptitude’, ‘ability’, or ‘talent’?  Too vague, all of them.  We are left with the possibility of a convergent, blended term, perhaps ‘SkilTacManConCun’, as portrayed in Figure 19.

There is some semblance of precision in the hybrid concoction, perhaps.  But whatever precision there may be, it is offset by pervading vagueness.  This puts uncomfortable and perplexing demands on the English language user with little to no knowledge of the Portuguese when striving to come to grips with the hybrid term.  She surely remains hardly any more knowledgeable than previously regarding her notion of ‘jeito’ within a Brazilian context.  And context is the key.  Indeed, the nonlinear contradictory complementary converging ‘logic’ revealed through the Figure 19 lattice is just that:  context dependent.  Adequate translation of ‘jeito’ would ideally require the entire range of Brazilian context of the word’s use, both possible and actual, and in the past, the present and the future.  The whole of this range of uses cannot be encompassed within any particular finite conception of word meanings.  Attempting to do so exhausts one’s patience.  So why should one even make the effort?  Perplexity grows, it would appear.

In order to further elucidate the issue of contradictory complementary convergence, perhaps we should turn to a scientific example:  quantum events, as mentioned above in passing.



Interlinguistic-intersemiotic translation:  the quantum quandary


In Figure 20 we have the bottom tier depicting the classical Newtonian-Cartesian corpuscular-kinetic view, and the top tier depicting the quantum theoretical view.  According to the classical view, particles are particles and waves are waves, and that’s that.  The quantum view plays havoc with this notion of the world.  There is no absolute distinction between wave amplitude and particle:  they are complementary.  Putting the two terms together in contradictory complementary convergent fashion, and we have the vague notion of a ‘wavicle’, which, by the way, was actually suggested by a respectable physicist during the quantum theory turmoil of the 1930s (Peterson 1985).

Now, how, more specifically, does the lattice diverge from classical linear, bivalent logic?  By the very nature of the lattice.  At the lowermost level of possible possibilities, a quantum event as a whole is made up of both a wave and a particle, as if A and Not-A were inconsistently combined, which is to say that within the sphere of all possible possiblities the classical Principle of Non-Contradiction is tossed in the trash can.  At the lattice’s middle portion, the focus can remain either on one or the other of the possible options, so here the bivalent either/or Principle of Non-Contradiction holds.

At the uppermost level, there is neither particle nor wave, strictly speaking, but something else, a ‘wavicle’.  In this manner the classical Principle of Excluded-Middle teeters and collapses, for actually, there is a certain nonlinear ‘Principle of Included-Middles’ allowing for the emergence of at least one of a range of possible or probable alternatives within some timespace context or other.  According to the ‘many worlds interpretation of quantum mechanics’ the entire universe is a wave function, so the very idea of ‘particles’ takes on a new countenance (DeWitt 1973); according to the ‘hidden variables’ concept notion of wave-particle complementarity-uncertainty is no more than an incomplete account (Bohm 1980); according to John A. Wheeler’s variation of Bohr’s Copenhagen Interpretation, there is no physical consequence of a quantum event as humans perceive it and conceive it, which elevates their role as participators rather than merely observers of the universe’s becoming (Wheeler 1980a, 1980b, 1990).  And so on.  This is to say that a host of alternate versions of the quantum world are in competition with one another, and each of them emerged from the ‘Included Middles’ between what was taken within some context as the one and only correct version in contrast to its erroneous opposite.[15]

All this is to say that the lowermost level of Figure 20 represents the range of all possible possibilities, however inconsistent, and the uppermost point represents possibilities that have been selected, but other possibilities could have been selected, are at the moment in the process of being selected, or they will have been within some future context selected, as alternatives.  So incompleteness is inevitably the watchword, since no matter what possibility has been selected in the past, there is always some other possibility awaiting its opportunity to seep up from the ‘Included Middle’ to make its presence known.

Inconsistency and/or incompleteness:  it reminds one of logician Kurt Gödel’s proof demonstrating that any and all formal systems of sufficient richness are either the one and the other or both the one and the other such that total account of the whole of mathematics, logic, or by extension the physical world, is beyond grasp (Goldstein 2005, Nagel and Newman 1964).  That, by the way, is another facet of nonlinear contradictory complementary convergence as portrayed in the lattice.

With this in mind, let us combine a broad cultural notion with an equally broad scientific concept, historically considered, in order hopefully to get a better picture of the lattice’s importance.



Language, science, and culture:  why bivalentism is insufficient


Figure 21 puts us in the sphere of apparently mutually exclusive interpretations of:  the ‘center of the universe’ on the one hand, and Brazilian ‘racial and ethnic relations’ on the other.  However, the apparent irrelevance of the two phrases with respect to one another is blended when they are placed together in the light of our lattice—if the lattice indeed bears universal application, that is.

As possible possibility, the ‘center of the universe’ is the ‘Earth’ (pre-Copernican theory), the ‘Sun’ (Copernican theory), ‘relative to the frame of reference’ (in Einsteinian theory), and only the future will tell whatever other possible alternatives.  Possible possibilities regarding race and skin color, and by extension ethnicity, are too numerous to quantify here.  However, I refer to the work of two investigators, who have pegged the number at 136 regarding race and 492 regarding skin color, and who knows what the future might hold.  At this particular time and place, the ‘center of the universe’ is by and large deemed to the ‘relative to the frame of reference’, and, at least from the perspective and conception of militant Afro-Brazilians, their community consists of ‘Afro-descendants’ and ‘blacks’, without question—though other ethnic groups might have a bone of contention with this idea.

Notice that at the uppermost level of the lattice, the ‘center of the universe’ is neither the ‘Earth’ nor the ‘Sun’ nor ‘relative to the frame of reference’, but, construed within the broad historical panorama, past, present and future, it ‘will have been’ something else at some time and place, and then something else at another time and place, and so on.  That is to say, science is itself in flux, always becoming something other than what it was (Agassi 1975).  Think about it.  Every scientific theory that has been accepted in the past has later been rejected except the theories that are generally accepted in our times, and there is by no means any iron-clad guarantee that our theories will stand the test of time; hence in some future time and place, some alternative theory will surely have been found more adequate than those we have embraced during our times.  At the uppermost level also, the range of all Brazilians conceived in racial and color terms are neither ‘black’ nor ‘white’ (Degler 1971); rather, each and every Brazilian will fall within at least one of all the alternatives.[16]

Thus, covering the lattice from bottom to top with respect to everything and anything, we have:

1.      Both-And … And … And … n, as possible possibilities (implying failure of the Principle of Non-Contradiction).

2.      Either/or, according to the context and in line with classical bivalent logic.

3.      Neither-Nor … Nor … Nor … n, as likely alternatives (implying failure of the Principle of Excluded-Middle).[17]

The semiotic flux and flow will go on, whether we know it or not and whether we like it or not.


On the Lattice:  a more formal description


A nonlinear, non-Boolean, orthogonally complemented, context-dependent lattice has been depicted in Figures 13, 14, 15, 16, 19, 20, and 21, which are in this appendix qualified in greater detail, in light of Figure 22.  The lattice, of which I present an alteration here for descriptive purposes, has the following characteristics:

1.         The connectives, ‘and’, and ‘or’ (or ‘+’ and ‘x’ as previously illustrated in this inquiry), are ‘quantum logical’ not ‘Boolean logical’.  This is to say that that ordering is no more than partial.

2.         Partial ordering entails broken linearity.  Regarding the connectives, ‘and’ and ‘or’, nonlinearity is the case, though linearity is the case in the following sequences, beginning with ‘0’:

            0 à R1 à I2 à I3 à If-1

            0 à I1 à O2 à I3 à I f-1

            0 à O1 à I2 à I3 à I f-1

            and so on, where ‘à’ denotes ‘implication’, 0, as described in this inquiry, is the range of all possible possibilities, and I f-1 depicts the non-linear, partially ordered process toward the sign’s final destination—the ‘final’ or ‘ultimate interpretant’ in Peirce’s terms—though  in this finite fallible world we can realize no more than an incomplete account of it.

The symbol ‘I f-1’, indicated by the subscripted ‘f’, which is accompanied by ‘-1’, specifies the sign’s incompleteness.  Incompleteness there will always be in one form or another; the probability will always exist that somewhere and somewhen along the semiosic stream, what a sign is and what it is not according to how we take it will be discarded in favor of some alternative sign that is neither what the previous sign was nor what it was not, but something else (recall the case of the ‘center of the universe’ or of ‘race’ and ‘skin color’ in Figure 21).

3.         However, the following paths are and are not possible:

            0 à O1 à I2, but not O1 à O2

            0 à R1 à I2, but not R1 à O2

            0 à I1 à O2, but not I1à I2

            and so on.

In other words, only the paths of ‘implification’ specified within the lattice can be followed; a shortcut or direct ‘leap’ is not possible from O1 à O2.

4.         The lattice entails context-dependency.  For example, context I1 is within context O2, but not the other way round, context O1 is within context I2 but not the other way round, and context R2 is within context I f-1, and so on.  Thus, moving up the lattice yields greater, all-inclusive contexts, because it entails the whole context of both one entity and another entity.  For example, I2 contains everything contained within both R1 and O1.  Combination of all entities in the lattice proceeds toward I f-1.  At the other extreme, ‘0’, ‘emptiness’, that is, the range of all possible possibilities, is the source of any and all contexts.

5.         Moving down the lattice limits a particular context, since it entails inclusion of no more than what one entity and another entity have in common.  For example, the product of R2 and O2 is I1, which is to say that the context of I1 includes only what is common to both R2 and O2.[18]  However, the lowermost point of a lattice, ‘0’, represents the fountainhead or source of any and all possibly possible contexts:  it knows no limitations.

6.         Moving down the lattice is roughly tantamount to Boolean ‘disjunction’, written ‘or’, or ‘Ç’.  It still involves particular context dependency, but of increasingly limited breadth within a particular context.  Yet, when all possible possibilities within all possible contexts are taken into account, at ‘0’, breadth is unlimited.  Thus we have the following:

            R2 Ç O2 = I1

            I1 Ç R2 = I1

            I1 Ç O1 = 0

            and so on.

7.         Moving up the lattice is roughly tantamount to Boolean ‘conjunction’, written ‘and’, or ‘È’, and, like moving down the lattice, it involves context dependency, but of increasingly greater breadth, to I f-1, and beyond.  This ‘beyond’, if extended to the maximum, entails the realization of all possible possibilities:  it is tantamount to the sum total of everything within ‘0’ having emerged into the light of day, which, as suggested in above, lies outside the capacity of finite semiotic agents in a finite world.  Thus:

            R1 È O1 = I2

            O2 È I1 = O2

            O2 È R2 = I f-1

            and so on.

8.         ‘Disjunction’ regarding the tangentially complemented context dependent lattice is far removed from the Boolean notion of either/or bivalently ordered sets of signs:  true/false, black/white, self/other, and so on.  Given the complementarity inherent in any and all signs in the lattice, the one sign implies the other sign, and as such there is something of the one in the other and vice versa, much like the Yin-Yang concept, [, which was instrumental in Bohr’s development of ‘quantum complementarity’.[19]  Thus, at the most inclusive level, ‘0’, or ‘emptiness’, the ‘disjunction’ of any and all possible possibilities includes the entire range of signs that might emerge into the light of day from within a virtually unlimited range of timespace contexts, past, present, and future.

9.         ‘Conjunction’, in contrast to ‘disjunction’, includes the totality of all the possible implications of two or more processes, which is to say that the door is always left open, and the possibility exists, for some alternate process that was implicit (tacit, sedimented, entrenched ‘habits of mind’) to become explicit.  In this sense, ‘conjunction’ allows for inter-dependent, inter-related inter-action between two incomplete processes such that some new process may emerge that is neither the one nor the other but something else.  Hence novel combinations can always arise, given the context, and then they can move up the lattice to make way for new possible combinations.  This is evident in the development of the lattice accounting for rhetorical tropes (Figures 14 and 15) and the contradictory complementary convergence of complex signs when in the process of becoming other signs (Figures 16, 19, 20, 21).

10.       According to classical Boolean distributive logic:

A + (B x C) = (A + B) x (A + C).

In contrast, the lattice’s non-Boolean nature yields:

R1 È (I1 Ç O1) = R1 È 0 = R1 (tantamount to A + [B x C])


(R1 È I1) Ç (R1 È O1) = I f-1 Ç I2 = I2 (tantamount to [A + B] x [A + C])


R1 ¹ I2!

In other words, unlike Boolean logic:

R1 È (I1 Ç O1) ¹ (R1 È I1) Ç (R1 È O1) (tantamount to A + [B x C] ¹ [A + B] x [A + C])

Distributivity does not hold!  Thus the lattice’s nonlinearity.  Thus the ‘logical’ connectives bringing about no more than a partial ordering of tangentially complemented, context-dependent fields of signification, require vague rather than ‘classical logical’ connectives.


Is this to say that the lattice gives free rein to relativism?  No.  That is, not exactly.  The Boolean based middle section of the lattice is antipathetic toward any and all forms of relativism.  The lattice, however, does imply a certain form of relativism at the lower and the upper ranges.  But this is not relativism of the Protagorean sort according to which ‘Each individual is the measure of all things’.  Nor does it imply incommensurabilism (incomparability and untranslatability between conceptual schemes, paradigms, and languages).[20]  Thus translation is always at least to an extent possible, however improbable it may appear at the outset, given the virtually unlimited range of possible possibilities (at ‘0’) and actual possibilities Firstness offers.  Moreover, the lattice is by and large foreign to the philosophical notions of nihilism, and irrationalism, for, of nonlinear, tangentially complemented, context-dependent, and above all processual nature, it allows for self-organization, self-correction, and melioration around every bend in the stream.






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[1] This tripartite division of translation comes originally from Roman Jakobson (1956), as further developed with the aid of Peirce’s concept of the sign by David Savan (1987-88).

[2] I offer a highly condensed description of Peirce’s rich and complex formulation.  If one wishes for more detail, one can consult Peirce’s Collected Papers (CP 2.219-444, also Hookway 1985, Stearns 1952).

[3] I use the italicized terms ‘inter-dependent’, ‘inter-active’, and ‘inter-related’, though they are not Peircean in origin.  Nevertheless, as I have argued in detail elsewhere, citing derivation of these terms in Buddhist philosophy and quantum theory, I believe they effectively portray the spirit of Peirce (see merrell 2000, 2002, 2003).

[4] Peirce elaborates at length on the engenderment of nine signs from the three basic signs, on ten signs from nine signs, and from the ten sign categories, on 27, 66, and, he speculates, developing the semiosic process to its maximum, one could ultimately reach the grand total of 310 signs, an adequate discussion of which for obvious reasons lies outside the focus of the present essay.

[5] Kauffman (2001), in particular, is keen on the implications of Peirce’s philosophy of  mathematics and its importance to the notion of ‘generacy’ and ‘degeneracy’ of signs in everyday life.

[6] In this vein, Edwin Abbott (1952) and Dionys Burger (1968) create imaginary fictional accounts regarding the problems involved in viewing worlds from within perspectives that are spatially limited.  Yet these fictions are based on topological principles, regarding perspectivism within worlds contained within distinct dimensionalities, which complement the notion of dimensionalities developed throughout this essay (see also Stewart 2001).

[7] For further on this model of the sign in light of Peirce’s general process philosophy, see merrell (1995, 1996, 1997).

[8] The term ‘blending’ bears on the notion of ‘conceptual blending’ as developed by Gilles Fauconnier and Mark Turner (2003), as I develop the term in merrell (2007).

[9] Movement upward along the lines of the lattice involves the Boolean notion of ‘implication’, though unlike Boolean principles, nonlinearity is the norm.  Thus O1 implies M2-I2, and M2-I2 implies S, such that each successive stage increases the context.  Signs at the lower tiers of the lattice have their own unique context, but when these signs move up the lattice their context merges and blends into another context.

[10] For diverse views of the four cultural practices in question, Futebol-Candomblé-Capoeira-Samba, see Almeida (1986), Browning (1995), Bruhns (2000), Filho (2003), Guimarães (1978), Lewis (1992), Santos (1995), Serra (1995), Soares (1994), Sodré (1979, 1988), Vianna (1999), Vieira (1995).

[11] For a series of anthropological studies that bear on a comparable process of ethnic and culture mixing (mestiçagem), see the work of Roberto DaMatta (1991a, 1991b, 1994, 1995).

[12] I have developed this topic in greater detail in merrell (1995, 1996, 2000, 2007).

[13] The ‘hypersolid’ in Figure 10 is another alternate manifestation of four-dimensionality.

[14] I by no means wish to imply that the topological model of semiosis I am developing here is tantamount to the notion of the four-dimensional continuum.  Rather, that the continuum, and the absolute absence of everything, that is, ‘nothingness’, or ‘emptiness’ in the Buddhist sense—to which Peirce occasionally alluded—are precisely that:  absolute, and timeless, in contrast to we finite fallible humans, who cannot but exist in time.  Moreover, the continuum is dependent some particular frame of reference (as illustrated in Figures 16 and 19).  In this sense, and metaphorically speaking, just as a point becomes a line, a line a plane, a plane a solid and a solid a hypersolid, the latter a four-dimensional mental, hypostatical construct including three dimensions of space and one dimension of time, so also a three-dimensional object portrayed on a two-dimensional plane is timelessly there, as two-dimensions of space and one temporal dimension crystallized—which, once again, is essentially the cubist technique as presented in Figure 12.  I repeat, we are in time and time is in us; hence the fourth dimension, as well as ‘emptiness’, or 0, are beyond the concrete world of our experience.

[15] See also, in general, Bohr (1961), Cole (1984, 2001), Heisenberg (1958, 1974), Peterson (1985), Smith (1995).

[16] For further on this model as it involves cultural processes, see merrell (2005).

[17] For a ‘logic of inconsistency’, Carnielli, Coniglio and D’Ottaviano (2002), Costa (1974), Melhuish (1967), Priest (1979, 1984, 1987, 2002, 2006), Priest, Beall and Armour-Garb (2004), Priest, Routley, and Norman (1989), Rescher and Brandom (1972); for a ‘logic of vagueness’ Nadin (1982, 1983), Putnam (1983), Rosenthal (1994); for an outline of ‘quantum logic’ Heelan (1970a, 1970b, 1971, 1983), Putnam (1971, 1976).

[18] Metaphorically speaking, what two terms have in common is like the complementary interdependent, interactive interrelations between Yin and Yang, where the black area contains a little whiteness and the white area contains a little blackness.  In other words, the complementarity between Yin and Yang is not possible without some degree of commonality.

[19] In fact, there is a close affinity between Buddhist philosophy and quantum theory which is considerably deeper that the outpouring of books and articles on the ‘Tao of Physics’ have thus far indicated (Balasubramaniam 1992, Mansfield 1989).

[20] The incommensurability thesis, notoriously developed by Thomas Kuhn (1970) and raised to a screaming pitch in the social sciences—though most practicing scientists commonly ignored it and went on with their work—was recently placed under a critical light, especially by Donald Davidson (1986).


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