Vagueness: The Structure of Non-Existence.
December 29, 2011 § Leave a comment
For many centuries now, clarity has been the major goal of philosophy.
It drove the first instantiation of logics by Aristotle, who devised it as a cure for mysticism, which was considered as a kind of primary chaos in human thinking. Clarity has been the intended goal in the second enlightenment as a cure for scholastic worries, and among many other places we find it in Wittgenstein’s first work, now directed to philosophy itself. In any of those instances, logics served as a main pillar to follow the goal of clarity.
Vagueness seems to act as an opponent to this intention, lurking behind the scenes in any comparison, which is why one may regard it as being as ubiquitous in cognition. There are whole philosophical and linguistic schools dealing with vagueness as their favorite subject. Heather Burnett (UCLA) recently provided a rather comprehensive overview  about the various approaches, including own proposals to solve some puzzles of vagueness in language, particularly related to relative and absolute adjectives and their context-dependency. In the domain of scientific linguistics, vagueness is characterized by three related properties: being fuzzy, being borderline, or being susceptible to the sorites (heap) paradox. A lot of rather different proposals for a solution have been suggested so far [1,2], most of them technically quite demanding; yet, none has been generally accepted as a convincing one.
The mere fact that there are many incommensurable theories, models and attitudes about vagueness we take as a clear indication for a still unrecognized framing problem. Actually, in the end we will see that the problem of vagueness in language does not “exist” at all. We will provide a sound solution that does not refer just to the methodological level. If we replace vagueness by the more appropriate term of indeterminacy we readily recognize that we can’t speak about vague and indeterminate things without implicitly talking about logics. In other words, the issue of (non-linguistic) vagueness triggers the question about the relation between logics and world. This topic we will investigate elsewhere.
Regarding vagueness, let us consider just two examples. The first one is about Peter Unger’s famous example regarding clouds . Where does a cloud end? This question can’t be answered. Close inspection and accurate measurement does not help. It seems as if the vagueness is a property of the “phenomenon” that we call “cloud.” If we conceive it as a particular type of object, we may attest it a resemblance to what is called an “open set” in mathematical topology, or the integral on asymptotic functions. Bertrand Russell, however, would have called this the fallacy of verbalism [4, p.85].
Vagueness and precision alike are characteristics which can only belong to a representation, of which language is an example. […] Apart from representation, whether cognitive or mechanical, there can be no such thing as vagueness or precision;
For Russell, objects can’t be ascribed properties, e.g. vague. Vague is a property of the representation, not of the object. Thus, when Unger concludes that there are no ordinary things, he gets trapped even by several misunderstandings, as we will see. We could add that open sets, i.e. sets without definable border, are not vague at all.
As the second example we take an abundant habit in linguistics when addressing the problem of vagueness, e.g. supervaluationism. This system has the consequence that borderline cases of vague terms yield statements that are neither true, nor false. Despite there is a truth-value gap induced by that model, it nevertheless keeps the idea of truth values fully intact. All linguistic models about vagueness assume that it is appropriate to apply the idea of truth values, predicates and predicate logics to language.
As far as I can tell from all the sources I have inspected, any approach in linguistics about vagueness is taking place within two very strong assumptions. The first basic assumption is that (1) the concept of “predicates” can be applied to an analysis of language. From that basic assumption, three other more secondary derive. (1.1) Language is a means to transfer clear statements. (1.2) It is possible to use language in a way that no vagueness appears. (1.3) Words are items that can be used to build predicates.
Besides of this first assumption of “predicativity” of language, linguistics further assumes that words could be definite and non-ambiguous. Yet, that is not a basic assumption. The basic second assumption of that is that (2) the purpose of language is to transfer meaning unambiguously. Yet, all three aspects of that assumption are questionable, being a purpose, serving as a tool or even a medium to transfer meaning, and to do so unambiguously.
So we summarize: Linguistics employs two strong assumptions:
- (1) The concept of apriori determinable “predicates” can be applied to an analysis of language.
- (2) The purpose of language is to transfer meaning unambiguously.
Our position is that both assumptions are deeply inappropriate. The second one we already dealt with elsewhere, so we focus on the first one here. We will see that the “problematics of vagueness” is non-existent. We do not claim that there is no vagueness, but we refute that it is a problem. There are also no serious threats from linguistic paradoxes, because these paradoxes are simply a consequence from “silly” behavior.
We will provide several examples to that, but the structure of it is the following. The problematics consists of a performative contradiction to the rules one has set up before. One should not pretend to play a particular game by fixing the rules upon one’s own interests, only to violate those rules a moment later. Of course, one could create a play / game from this, too. Lewis Carroll wrote two books about the bizarre consequences of such a setting. Let us again listen to Russell’s arguments, now to his objection against the “paradoxicity” of “baldness,” which is usually subsumed to the sorites (heap) paradox.
It is supposed that at first he was not bald, that he lost his hairs one by one, and that in the end he was bald; therefore, it is argued, there must have been one hair the loss of which converted him into a bald man. This, of course, is absurd. Baldness is a vague conception; some men are certainly bald, some are certainly not bald, while between them there are men of whom it is not true to say they must either be bald or not bald. The law of excluded middle is true when precise symbols are employed, but it is not true when symbols are vague, as, in fact, all symbols are.
Now, describing the heap (Greek: sorites) or the hair of “balding” men by referring to countable parts of the whole, i.e. either sand particles or singularized hairs, contradicts the conception of baldness. Confronting both in a direct manner (removing hair by hair) mixes two different games. Mixing soccer and tennis is “silly,” especially after the participants have declared that they intend to play soccer, mixing vagueness and counting is silly, too, for the same reason.
This should make clear why the application of the concept of “predicates” to vague concepts, i.e. concepts that are apriori defined as to be vague, is simply absurd. Remember, even a highly innovative philosopher as Russell, co-author of an extremely abstract work as the Principia Mathematica is, needed several years to accept Wittgenstein’s analysis that the usage of symbols in the Principia is self-contradictory, because actualized symbols are never free of semantics.
Words are Non-Analytic Entities
First I would recall an observation first, or at least popularly, expressed by Augustinus. His concern was the notion of time. I’ll give a sketch of it in my words. As long as he simply uses the word, he perfectly knows what time is. Yet, as soon as he starts to think about time, trying to get an analytic grip onto it, he increasingly looses touch and understanding, until he does not know anything about it at all.
This phenomenon is not limited to the analysis of a concept like time, which some conceive even as a transcendental “something.” The phenomenon of disappearance by close inspection is not unknown. We meet it in Carroll’s character of the Cheshire cat, and we meet it in Quantum physics. Let us call this phenomenon the CQ-phenomenon.
Ultimately, the CQ-phenomenon is a consequence of the self-referentiality of language and self-referentiality of the investigation of language. It is not possible to apply a scale to itself without getting into some serious troubles like fundamental paradoxicity. The language game of “scale” implies a separation of observer and observed that can’t be maintained in the cases of the cat, the quantum, or language. Of course, there are ways to avoid such difficulties, but only to high costs. For instance, a strong regulations or very strict conventions can be imposed to the investigation of such areas ad the application of self-referential scales, to which one may count linguistics, sociology, cognitive sciences, and of course quantum physics. Actually, positivism is nothing else than such a “strong convention”. Yet, even with such strong conventions being applied, the results of such investigations are surprising and arbitrary, far from being a consequence of rationalist research, because self-referential system are always immanently creative.
It is more than salient that linguists create models about vagueness that are subsumed to language. This position is deeply non-sensical and does not only purport ontological relevance for language, it implicitly also claims a certain “immediacy” for the linkage between language and empirical aspects of the world.
Our position is strongly different from that: models are entities that are “completely” outside of language. Of course, they are not separable from each other. We will deal elsewhere with this mutual dependency in more details and a more appropriate framing. Regardless how modeling and language are related, they definitely can not be related in the way linguistics implicitly assumes. It is impossible to use language to transfer meaning, because it is in principle not possible to transfer meaning at all. Of course, this opens the question what then is going to be “transferred.”
This brings us to the next objection against the presumed predicativity of language, namely its role in social intercourse, from which the CQ-phenomenon can’t be completely separated from.
Language: What can be Said
Many things and thoughts are not explicable. Many things also can be just demonstrated, but not expressed in any kind of language. Yet, despite these two severe constraints, we may use language not only to explicitly speak about such things, but also to create what only can be demonstrated.
Robert Brandom’s work  may be well regarded as a further leap forward in the understanding of language and its practitioning. He proposes the inferentialist position, to which our positioning of the model is completely compatible. According to Brandom, we always have to infer a lot of things from received words during a discourse. We even have to signal that we expect those things to be inferred. The only thing what we can try in a language-based interaction is to increasingly confine the degrees of freedom of possible models that are created in the interactees’ minds. Yet, achieving a certain state of resonance, or feeling that one understands each other, does NOT imply that the models are the identical. All what could be said is that the resonating models in the two interacting minds allow a certain successful prediction of the further course of the interaction. Here, we should be very clear about our understanding of the concept of model. You will find it in the chapters about the generalized model and the formal status of models (as a category).
Since Austin  it is well-known that language is not equal to the series of graphical of phonic signals. The reason for this simply being that language is a social activity, both structural as well as performative. An illocutionary act is part of any utterance and any piece of text in a natural language, sometimes even in the case of a formal language. Yet, it is impossible to speak about that dimension in language.
A text is even more than a “series” of Austinian or Searlean speech acts. The reason for this is a certain aspect of embodiment: Only entities stuffed with memory can use language. Now, receiving a series of words immediately establishes a more or less volatile immaterial network in the “mind” of the receiving entity as well as in the “sending” entity. This network owns properties for which it is absolutely impossible to speak about, despite the fact that these networks represent somehow the ultimate purpose, or “essence”, of natural language. We can’t speak about that, we can’t explicate it, and we simply commit a categorical mistake if we apply logics and tools from logics like predicates in the attempt to understand it.
Logics and Language
These phenomena clearly proof that logics and language are different things. They are deeply incommensurable, despite the fact that they can’t be separated completely from each other, much like modeling and language. The structure of the world shows up in the structure of logics, as Wittgenstein mentioned. There are good reasons to take Wittgenstein serious on that. According to the Tractatus, the coupling between world and logics can’t be a direct one .
In contrast to the world, logics is not productive. “Novelty” is not a logical entity. Pure logics is a transcendental system about usage of symbols, precisely because any usage already would require interpretation. Logical predicates are nothing that need to be interpreted. These games are simply different games.
In his talk to the Jowett Society, Oxford, in 1923, Bertrand Russell, exhibiting an attitude quite different to that in the Principia and following much the line drawn by Wittgenstein, writes [p.88]:
Words such as “or” and “not” might seem at first sight, to have a perfectly precise meaning: “p or q'” is true when p is true, true when q is true, and false when both are false. But the trouble is that this involves the notions of “true” and “false”; and it will be found, I think, that all the concepts of logic involve these notions, directly or indirectly. Now “‘true” and “false” can only have a precise meaning when the symbols employed—words, perceptions, images, or what not—are themselves precise. We have seen that, in practice, this is not the case. It follows that every proposition that can be framed in practice has a certain degree of vagueness; that is to say, there is not one definite fact necessary and sufficient for its truth, but a certain region of possible facts, any one of which would make it true. And this region is itself ill-defined: we cannot assign to it a definite boundary.
This is exactly what we meant before: “Precision” concerning logical propositions is not achievable as soon as we refer to symbols that we use. Only symbols that can’t be used are precise. There is only one sort of such symbols: transcendental symbols.
Mapping logics to language, as it happens so frequently and probably even as an acknowledged practice in linguistics in the treatment of vagueness, means to reduce language to logics. One changes the frame of reference, much like Zenon does in his self-generated pseudo-problems, much like Cantor1  and his fellow Banach2  did (in contrast to Dedekind3 ), or what Taylor4 did . 3-dimensionality produces paradoxes in a 2-dimensional world, not only faulty projections. It is not really surprising that through the positivistic reduction of language to logics awkward paradoxes appear. Positivism implies violence, not only in the case linguistics.
We now can understand why it is almost silly to apply a truth-value-methodology to the analysis of language. The problem of vagueness is not a problem, it is deeply in the blueprint of “language” itself. It is almost trivial to make remarks as Russell did [3, p.87]:
The fact is that all words are attributable without doubt over a certain area, but become questionable within a penumbra, outside which they are again certainly not attributable.
And it really should be superfluous to cite this 90-year old piece. Quite remarkably it is not.
Language as a Practice
Wittgenstein emphasized repeatedly that language is a practice. Language is not a structure, so it is neither equivalent to logics nor to grammar, or even grammatology. In practices we need models for prediction or diagnosis, and we need rules, we frequently apply habits, which even may get symbolized.
Thus, we again may ask what is happening when we talk to each other. First, we exclude those models of which we now understand that they are not appropriate.
- – Logics is incommensurable with language.
- – Language, as well as any of its constituents, can’t be made “precise.”
As a consequence, language (and all of its constituents) is something that can’t be completely explicated. Large parts of language can only be demonstrated. Of course, we do not deny the proposal that a discourse reflects “propositional content,” as Brandom calls it ( chp. 8.6.2.). This propositional or conceptual content is given by the various kinds of models appearing in a discourse, models that are being built, inferred, refined, symbolized and finally externalized. As soon as we externalize a model, however, it is not language any more. We will investigate the dynamical route between concepts, logics and models in another chapter. Here and for the time being we may state that applying logics as a tool to language mistakes propositional content as propositional structure.
Again: What happens if I point to the white area up in the air before the blue background that we call sky, calling then “Oh, look a cloud!” ? Do I mean that there is an object called “cloud”? Even an object at all? No, definitely not. Claiming that there are “cloud-constituters,” that we do not measure exactly enough, that there is no proper thing we could call “cloud” (Unger), that our language has a defect etc., any purported “solution” of the problem [for an overview see 11] does not help to the slightest extent.
Anybody having made a mountain hike knows the fog in high altitudes. From lower regions, however, the same actual phenomenon is seen as a cloud. This provides us a hint, that the language game “cloud” also comprises information about the physical relational properties (position, speed, altitude) of the speaker.
What is going to happen by this utterance is that I invite my partner in discourse to interpret a particular, presumably shared sensory input and to interpret me and my interpretations as well. We may infer that the language game “cloud” contains a marker that is both linked to the structure and the semantics of the word, indicating that (1) there is an “object” without sharp borders, (2) no precise measurement should be performed. The symbolic value of “cloud” is such that there is no space for a different interpretation. Not the “object” is indicated by the word “cloud,” but a particular procedure, or class of procedures, that I as the primary speaker suggest when saying “Oh, there is a cloud.” By means of such procedures a particular style of modeling will be “induced” in my discourse partner, a particular way to actualize an operationalization, leading to such a representation of the signals from the external world that both partners are able to increase their mutual “understanding.” Yet, even “understanding” is not directed to the proposed object either. This scheme transparently describes the inner structure of what Charles S. Peirce called a “sign situation.” Neither accuracy, nor precision or vagueness are relevant dimensions in such kinds of mutually induced “activities,” which we may call a Peircean “sign.” They are completely secondary, a symptom of the use and of the openness.
Russell correctly proposes that all words in a language are vague. Yet, we would like to extend his proposal, by drawing on our image of thought that we explicate throughout all of our writings here. Elsewhere we already cited the Lagrangian trick in abstraction. Lagrange got aware about the power of a particular replacement operation: In a proposal or proposition, constants always can be replaced by appropriate procedures plus further constants. This increases generality and abstractness of the representation. Our proposal that is extending Russell’s insight is aligned to this scheme:
Different words are characterised (among other factors) by different procedures to select a particular class (mode) of interpretation.
Such procedures are precisely given as kind of models that are necessary besides those models implied in performing the interpretation of the actual phenomenon. The mode of interpretation comprises the selection of the scale employed in the operationalization, viz. measurement. Coarser scales imply a more profound underdetermination, a larger variety of possible and acceptable models, and a stronger feeling of vagueness.
Note that all these models are outside of language. To our opinion it does not make much sense to instantiate the model inside of language and then claiming a necessarily quite opaque “interpretation function,” as Burnett extensively demonstrates (if I understood her correctly). Our proposal is also more general (and more abstract) than Burnett’s, since we emphasize the procedural selection of interpretation models (note that models are not functions!). The necessary models for words like “taller,” “balder” or “cloudy” are not part of language and can’t be defined in terms of linguistic concepts. I would not call that a “cognitivist” stance, yet. We conceive it just as a consequence of the transcendental status of models. This proposal is linked to two further issues. First, it implies the acceptance of the necessity of models as a condition. In turn, we have to clarify our attitude towards the philosophical concept of the condition. Second, it implies the necessity of an instantiation, the actualization of it as the move from the transcendental to the applicable, which in turn invokes further transcendental concepts, as we will argue and describe here.
Saying this we could add that models are not confined to “epistemological” affairs. As the relation between language (as a practice) and the “generalized” model shows, there is more in it than a kind of “generalized epistemology.” The generalization of epistemology can’t be conceived as a kind of epistemology at all, as we will argue in the chapter about the choreosteme. The particular relation between language and model as we have outlined it should also make clear that “models” are not limited to the categorization of observables in the outer world. It also applies—now in more classic terms—to the roots of what we can know without observation (e.g. Strawson, p.112 in ). It is not possible to act, to think, or to know without implying models, because it is not possible to act, to think or to know without transformation. This gives rise to model as a category and to the question of the ultimate conditionability of language, actions, or knowing. In our opinion, and in contrast to Strawson’s distinction, it is not appropriate to separate “knowledge from observations” and “knowledge without observation.” Insisting on such a separation immediately would also drop the insight about mutual dependency of models, concepts, symbols and signs, among many other things. In short, we would fall back directly into the mystic variant of idealism (cf. Frege’s hyper-platonism), implying also some “direct” link between language and idea. We rate such a disrespect of the body, matter and mediating associativity as inappropriate and of little value.
It would be quite interesting to conduct a comparative investigation of the conceptual life cycle of pictorial information in contrast to textual information along the line opened by such a “processual indicative.” Our guess is that the textual “word” may have a quite interesting visual counterpart. But we have to work on this later and elsewhere.
Our extension also leads to the conclusion that “vague” is not a logical “opposite” of “accurate,” or of “precise” either. Here we differ (not only) from Bertrand Russell’s position. So to speak, the vagueness of language applies here too. In our perspective, “accurate” simply symbolizes the indicative to choose a particular class of models that a speaker suggests to the partner in discourse to use. Nothing more, but also nothing less. Models can not be the “opposite” of other models. Words (or concepts) like “vague” or “accurate” just explicate the necessity of such a choice. Most of the words in a language refer only implicitly to that choice. Adjectives, whether absolute or relative, are bivalent with respect to the explicity or impliciteness of the choice of the procedure, just depending on the context.
For us it feels quite nice to discover a completely new property of words as they occur in natural languages. We call it “processual indicative.” A “word” without such a processual indicative on the structural level would not be a “word” any more. Either it reduces to a symbol, or even an index, or the context degenerates from a “natural” language (spoken and practiced in a community) into a formal language. The “processual indicative” of the language game “word” is a grammatical property (grammar here as philosophical grammar).
Nuisance, Flaws, and other Improprieties
Charles S. Peirce once mentioned, in a letter around 1908, that is well after his major works, and answering a question about the position or status of his own work, that he tends to label it as idealistic materialism. Notably, Peirce founded what is known today as American pragmatism. The idealistic note, as well as the reference to materialism, have to be taken extremely abstract in order to justify such. Of course, Peirce himself has been able for handling such abstract levels.
Usually, however, idealism and pragmatism are in a strong contradiction to each other. This is especially true when it comes to engineering, or more generally, to the problematics of the deviation, or the problematics posed by the deviation, if you prefer.
Obviously, linguistics is blind or even self-deceptive against their domain-specific “flaw,” the vagueness. Linguists are treating vagueness as a kind of flaw, or nuisance, at least as a kind of annoyance that needs to be overcome. As we already mentioned, there are many incommensurable proposals how to overcome it, except one: checking if it is a flaw at all, and which conditions or assumptions lead to the proposal that vagueness is indeed a flaw.
Taking only 1 step behind, it is quite obvious that logical positivism and its inheritance is the cause for the flaw. The problem “appeared” in the early 1960ies, when positivism was prevailing. Dropping the assumptions of positivism also removes the annoyance of vagueness.
Engineering a new device is a demanding task. Yet, there are two fundamentally different approaches. The first one, more idealistic in character, starts with an analytic representation, that is, a formula, or more likely, a system of formulas. Any influence that is not covered by that formula is either shifted into the premises, or into the so-called noise: influences, about nothing “could” be known, that drive the system under modeling into an unpredictable direction. Since this approach starts with a formula, that is, an analytic representation, we also can say that it starts under the assumption of representability, or identity. In fact, whenever you find designers, engineers or politicians experience to speak about “disturbances,” it is more than obvious that they follow the idealistic approach, which in turn follows a philosophy of identity.
The second approach is very different from the first one, since it does not start with identity. Instead, it starts with the acknowledgement of difference. Pragmatic engineering does not work instead of nuisances, it works precisely within and along nuisances. Thus, there is no such thing as a nuisance, a flaw, an annoyance, etc. There is just fluctuation. Instead of assuming the structural constant labeled as “ignorance,” as represented by the concept of noise, there is a procedure that is able to digest any fluctuation. A “disturbance” is nothing that can be observed as such. Quite in contrast, it is just and only a consequence of a particular selection of a purpose. Thus, pragmatic engineering leads to completely different structure that would be generated under idealistic assumptions. The difference between both remains largely invisible in all cases where the information part is neglectable (which actually is never the case), but it is vital to consider it in any context where formalization is dealing with information, whether it is linguistics or machine-learning.
The issue relates to “cognition” too, understood here as the naively and phenomenologically observable precipitation of epistemic conditions. From everyday experience, but also as a researcher in “cognitive sciences”, we know, i.e. we could agree on the proposal that cognition is something that is astonishing stable. The traditional structuralist view, as Smith & Jones call it , takes this stability as a starting point and as the target of the theory. The natural consequence is that this theory rests on the apriori assumption of a strict identifiability of observable items and of the result of cognitive acts, which are usually called concepts and knowledge. In other words, the idea that knowledge is about identifiable items is nothing else than a petitio principii: Since it serves as the underlying assumption it is no surprise that the result in the end exhibits the same quality. Yet, there is a (not so) little problem, as Smith & Jones correctly identified (p.184/185):
The structural approach pays less attention to variability (indeed, under a traditional approach, we design experiments to minimize variability) and not surprisingly, it does a poor job explaining the variability and context sensitivity of individual cognitive acts. This is a crucial flaw. […]
Herein lies our discontent: If structures control what is constant about cognition, but if individual cognitive acts are smartly unique and adaptive to the context, structures cannot be the cause of the adaptiveness of individual cognitions. Why, then, are structures so theoretically important? If the intelligence-and the cause of real-time individual cognitive acts-is outside the constant structures, what is the value of postulating such structures?
The consequence the authors draw is to conceive cognition as process. They cite the work of Freeman  about the cognition of smelling
They found that different inhalants did not map to any single neuron or even group of neurons but rather to the spatial pattern of the amplitude of waves across the entire olfactory bulb.
The heir of being affected by naive phenomenology (phenomenology is always naive) and its main pillar of “identifiability of X as X” obviously leads to conclusions that are disastrous for the traditional theory. It vanishes.
Given these difficulties, positivists are trying to adapt. Yet, people still dream of semantic disambiguation as a mechanical technique, or likewise, dream (as Fregean worshipers) of eradicate vagueness from language by trying to explain it away.
One of the paradoxes dealt with over and over again is the already mentioned Sorites (Greek for “heap”) paradox. When is a heap a heap? Closely related to it are constructions like Wang’s Paradox : If n is small, then n+1 is also small. Hence there is no number that s not small. How to deal with that?
Certainly, it does not help to invoke the famous “context dependency” as a potential cure. Jaegher and Rooij recently wrote :
“If, as suggested by the Sorites paradox, ne-grainedness is important, then a vague language should not be used. Once vague language is used in an appropriate context, standard axioms of rational behaviour are no longer violated.”
Yet, what could appropriate mean? Actually, for an endeavor as Jaegher and Rooij have been starting the appropriateness needs to be determined by some means that could not be affected by vagueness. But how to do that for language items? They continue:
“The rationale for vagueness here is that vague predicates allow players to express their valuations, without necessarily uttering the context, so that the advantage of vague predicates is that they can be expressed across contexts.”
At first sight, this seems plausible. Now, any part of language can be used in any context, so all the language is vague. The unfavorable consequence for Jaegher & Rooij being that their attempt is not even a self-disorganizing argument, it has the unique power of being self-vanishing, their endeavor of expelling vagueness is doomed to fail before they even started. Their main failure is, however, that they take the apriori assumption for granted that vagueness and crispness are “real” entities that are somehow existing before any perception, such that language could be “infected” or affected with it. Note that this is not a statement about linguistics, it is one about philosophical grammar.
It also does not help to insist on “tolerance”. Rooij  recently mentioned that “vagueness is crucially related with tolerant interpretation”. Rooij desperately tries to hide his problem, the expression “tolerant interpretation” is almost completely empty. What should it mean to interpret something tolerantly as X? Not as X? Also a bit as Y? How then would we exchange ideas and how could it be that we agree exactly on something? The problem is just move around a corner, but not addressed in any reasonable manner. Yet, there is a second objection to “tolerant interpretation”.
Interpretation of vague terms by a single entity must always fail. What is needed are TWO interpretations that are played as negotiation in language games. Two entities, whether humans or machines, have to agree, i.e. they also have to be able to perform the act of agreeing, in order to resolve vagueness of items in language. It is better to drop vagueness all together and simply to say that at least two entities are necessarily be “present” to play a language game. This “presence” is , of course, an abstract semiotic one. It is given in any Peircean sign situation. Since signs refer only and always just to other signs vagueness is, in other words, not a difficulty that need to be “tolerated”.
Dummett  spent more than 20 pages for the examination of the problem of vagueness. Up to date it is one of the most thorough ones, but unfortunately not received or recognized by contemporary linguistics. There is still a debate about it, but no further development of it. Dummett essentially proofs that vagueness is a not a defect of language, it is a “design feature”. First, he proposes a new logical operator “definitely” in order to deal with the particular quality of indeterminateness of language. Yet, it does not remove vagueness or its problematic, “that is, the boundaries between which acceptable sharpenings of a statement or a predicate range are themselves indefinite.” (p.311)
He concludes that “vague predicates are indispensable”, they are not eliminable in principle without loosing language itself. Tolerance does not help as much selecting “appropriate contexts” fails to do, both proposed to get rid of a problem. What linguists propose (at least those adhering to positivism, i.e. nowadays nearly all of them) is to “carry around a colour-chart, as Wittgenstein suggested in one
of his example” (Dummett). This would turn observational terms into legitimated ones by definition. Of course, the “problem” of vagueness would vanish, but along with it also any possibility to speak and to live. (Any apparent similarity to real persons, politicians or organizations such like the E.U. is indeed intended.)
Linguistics, and cognitive sciences as well, will fail to provide any valuable contribution as long as they apply the basic condition of the positivist attitude: that subjects could be separated from each other in order to understand the whole. The whole here is the Lebensform working underneath, or beyond (Foucault’s field of proposals, Deleuze’s sediments), connected cognitions. It is almost ridicule to try to explain anything regarding language within the assumption of identifiability and applicability of logics.
Smith and Jones close their valuable contribution with the following statement, abandoning the naive realism-idealism that has been exhibited so eloquently by Rooij and his co-workers nearly 20 years later:
On a second level, we questioned the theoretical framework-the founding assumptions-that underlie the attempt to define what “concepts really are.” We believe that the data on developing novel word interpretations-data showing the creative intelligence of dynamic cognition-seriously challenge the view of cognition as represented knowledge structures. These results suggest that perception always matters in a deep win: Perception always matters because cognition is always adaptive to the here-and-now, and perception is our only means of contact with the here-and now reality.
There are a number of interesting corollaries here, which we will not follow here. For instance, it would be a categorical mistake to talk about noise in complex systems. Another consequence is that engineering, linguistics or philosophy that is based on the apriori concept of identity is not able to make reasonable proposals about evolving and developing systems, quite in contrast to a philosophy that starts with difference (as a transcendental category, see Deleuze’s work, particularly ).
We now can understand that idealistic engineering is imposing its adjudgements ways too early. Consequently, idealistic engineering is committing the naturalistic fallacy in the same way as many linguistics is committing it, at least as far as the latter starts with the positivistic assumption of the possibility of positive assumptions such as identifiability. The conclusion for the engineering of machine-based episteme is quite obvious: we could not start with identified or even identifiable items, and where it seems that we meet them, as in the case of words, we have to take their identifiability as a delusion or illusion. We also could say that the only feasible input for a machine that is supposed to “learn” is made from vague items for which there is only a probabilistic description. Even more radical, we can see that without fundamentally embracing vagueness no learning is possible at all. That’s now the real reason for the failure of “strong” or “symbolic” AI.
Conclusions for Machine-based Epistemology
We started with a close inspection and a critique of the concept of vagueness and ended up in a contribution to the theory of language. Once again we see that language is not just about words, symbols and grammar. There is much more in it and about it that we must understand to bring language into contact with (brain) matter.
Our results clearly indicate, against the mainstream in linguistics and large parts of (mainly analytic) philosophy, that words can’t be conceived as parts of predicates, i.e. clear proposals, and language can’t be used as a vehicle for the latter. This again justifies an initial probabilistic representation of those grouped graphemes (phonemes) as they can be taken from a text, and which we call “words.” Of course, the transition from a probabilistic representation to the illusion of propositions is not a trivial one. Yet, it is not words that we can see in the text, it is just graphemes. We will investigate the role and nature of words at some later point in time (“Waves, Words, and Images”, forthcoming).
Secondly we discovered a novel property or constituent of words, which is a selection function (or a class thereof) which indicates the style of interpretation regarding the implied style of presumed measurement. We called it processual indicative. Such a selection results in the invoking of clear-cut relations or boundaries, or indeterminable ones. Implementing the understanding of language necessarily has to implement such a property for all of the words. In any of the approaches known so far, this function is non-existent, leading to serious paradoxes and disabilities.
A quite nice corollary of these results is that words never could be taken as a reference. It is perhaps more appropriate to conceive of words as symbols for procedural packages, recipes and prescription on how to arrange certain groups of models. Taken such, van Fraassen’s question on how words acquire reference is itself based on a drastic misunderstanding, deeply informed by positivism (remember that it was van Fraassen who invented this weird thing called supervaluationism). There is no such “reference.” Instead, we propose to conceive of words as units consisting from (visible) symbols and a “Lagrangean” differential part. This new conception of words remains completely compatible with Wittgenstein’s view on language as a communal practice; yet, it avoids some difficulties, Wittgenstein has struggled with throughout his life. The core of it may be found in PI §201, describing the paradox of rule following. For us, this paradox simply vanishes. Our model of words as symbolic carriers of “processual indicatives” also sheds light to what Charles S. Peirce called a “sign situation,” being not able to elucidate the structure of “signs” any further. Our inferentialist scheme lucidly describes the role of the symbolic as a quasi-material anchor, from which we can proceed via models as targets of the “processual indicative” to the meaning as a mutually ascribed resonance.
The introduction of the “processual indicative” also allows to understand the phenomenon that despite the vagueness of words and concepts it is possible to achieve very precise descriptions. The precision, however, is just a “feeling” as it is the case for “vagueness,” dependent on a particular discursive situation. Larger amounts of “social” rules that can be invoked to satisfy the “processual indicative” allow for more precise statements. If, however, these rules are indeterminate by themselves quite often more or less funny situation may occur (or disastrous misunderstandings as well).
The main conclusion, finally, is referring to the social aspect of a discourse. It is largely unknown how two “epistemic machines” will perceive, conceive of and act upon each other. Early experiments by Luc Steels involved mini robots that have been far too primitive to draw any valuable conclusion for our endeavor. And Stanislav Lem’s short story “Personetics” does not contain any hint about implementational issues… Thus, we first have to implement it…
1. One of Cantor’s paradoxes claims that a 2-dimensional space can be mapped entirely onto a 1 dimensional space without projection errors, or overlaps. All of Cantor’s work is “absurd,” since it mixes two games that apriori have been separated: countability and non-countability. The dimensions paradox appears because Cantor conceives of real numbers as determinable, hence countable entities. However, by his own definition via the Cantor triangle, real numbers are supra-countable infinite. Real numbers are not determinable, hence they can’t be “re-ordered,” or put along a 1-dimensional line. Its a “silly” contradiction. We conclude that such paradoxes are pseudo-paradoxes.
2. The Banach-Tarski (BT) pseudo-paradox is of the same structure as the dimensional pseudo-paradox of Cantor. The surface of a sphere is broken apart into a finite number of “individual” pieces; yet , those pieces are not of determinate shape. Then BT proof that from the pieces of 1 sphere 2 spheres can be created. No surprise at all: the pieces are not of determinate shape, they are complicated: they are not usual solids but infinite scatterings of points. It is “silly” first to speak about pieces of a sphere, but then to dissolve those pieces into Cantor dust. Countability and incountability collide. Thus there is no coherence, so they can be any. The BT paradox is even wrong: from such premises an infinite number of balls could be created from a single ball, not just a second one.
4. Taylor’s paradox brings scales into conflict. A switch is toggled repeatedly after a decreasing period of time, such that the next period is just half of the size of the current one. After n toggling events (n>>), what is the state of the switch? Mathematically, it is not defined (1 AND 0), statistically it is 1/2. Again, countability, which implies a physical act, ultimately limited by the speed of light, is contrasted by infinitely small quantities, i.e. incountability. According to Gödel’s incompleteness, for any formal system it is possible to construct paradoxes by putting up “silly” games, which do not obey to the self-imposed apriori assumptions.
This article has been created on Dec 29th, 2011, and has been republished in a considerably revised form on March 23th, 2012.
-  Heather Burnett, The Puzzle(s) of Absolute Adjectives – On Vagueness, Comparison, and the Origin of Scale Structure. Denis Paperno (ed). “UCLA Working Papers in Semantics,” 2011; version referred to is from 20.12.2011. available online.
-  Brian Weatherson (2009), The Problem of the Many. Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. available online, last access 28.12.2011.
-  Peter Unger (1980), The Problem of the Many.
-  Bertrand Russell (1923): Vagueness, Australasian Journal of Psychology and Philosophy, 1(2), 84-92.
-  Robert Brandom, Making it Explicit. 1994.
-  John Austin. Speech act Theory.
-  Colin Johnston (2009). Tractarian objects and logical categories. Synthese 167: 145-161.
-  Cantor
-  Banach
-  Dedekind
-  Taylor
-  Peter Strawson, Individuals: An Essay in Descriptive Metaphysics. Methuen, London 1959.
-  Linda B. Smith, Susan S. Jones (1993). Cognition Without Concepts. Cognitive Development, 8, 181-188. available here.
-  Freeman, W.J. (1991). The physiology of perception. Scientific American. 264. 78-85.
-  Michael Dummett, Wang’s Paradox (1975). Synthese 30 (1975) 301-324. available here.
-  Kris De Jaegher, Robert van Rooij (2011). Strategic Vagueness, and appropriate contexts. Language, Games, and Evolution, Lecture Notes in Computer Science, 2011, Volume 6207/2011, 40-59, DOI: 10.1007/978-3-642-18006-4_3
-  Robert van Rooij (2011). Vagueness, tolerance and non-transitive entailment in Understanding Vagueness – Logical, Philosophical and Linguistic Perspectives, Petr Cintula, Christian Fermuller, Lluis Godo, Petr Hajek (eds.), College Publications, 2011.
-  Gilles Deleuze, Difference and Repetition.
-  Stanislav Lem, Personetics. reprinted in: Douglas Hofstadter, The Minds I.