Little

How many littles of little bits AUTHOR (Note 1)

Abstract: In this paper, we immerse the Sorites Problem in contextualism. Aiming maximum dissemination, we present things in a non-orthodox way. Our major concern is proving that a piece of discourse, be it a word or a set of words, may need much more than one paragraph of text to become suitable, or prepared, for scientific analysis. Key-words: Sorites, paradox, contextualism, puzzle, language, communication. 1.Introduction Four pieces of writing form our introduction: 1.1) Explanations on the Sorites Problem; 1.1.1) How to generate problems containing the essence of the Sorites Problem; 1.2) Explanations on contextualism; 1.3) Organization of the paper. In what follows the introduction, we try to apply the above concepts to the scientific theory (underlying logic) the vast majority of the language professionals in the world believe to be attached to the ethos of translation. 1.1 The Sorites Problem The name Sorites derives from the Greek word for heap, which is soros, and originally referred to a puzzle known as The Heap (not to a paradox) (AUTHOR 2006a): “Would you describe a single grain of sand as a heap? No. Would you describe two grains of sand as a heap? No... . You must admit the presence of a heap sooner or later, so where do you draw the line?” (HYDE 1997) The expression Sorites Paradox is generally used in association with any variation of the puzzle above. The Heap, for instance, which is the original puzzle, starts with a clear nonheap of sand and one grain of sand is added, in each step, to the updated amount of sand until the audience agrees that the updated amount of sand may be called a heap. The actual emerging issue is that of telling where one starts having a heap (determining where the line (if there is any), which separates non-heaps from heaps, is located). The puzzle has been astonishing people for millennia because it is evident that there must be a line splitting non-heaps from heaps: One has, at the beginning of the puzzle, a clear non-heap and, at the end of the puzzle, a clear heap. Admitting that there is no line to be drawn, that things are and are not, at the same time, as Paraconsistency wants to defend, would be stating that it is the case that either what you see at the beginning of the puzzle is not

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what you see, once you clearly see a non-heap, or what you see at the end of the puzzle is not what you see, once it is clearly a heap. Also, as for the intermediary possible heaps, it seems completely unnatural to admit that something that subject X, for instance, would easily classify as a non-heap is actually a heap as well. Therefore, it must be the case that there is a moment when the non-heap stops being a non-heap instead of us having, all the way through, both a non-heap and a heap (of sand), as this seems to be the only possible way of proposing a paraconsistent sound solution for this puzzle (see our work on Paraconsistency and the Sorites Problem for a deeper discussion on this issue). We have defended the point of view that The Heap is actually a problem, rather than a paradox, in our previous papers, providing argumentation that we believe to be soundest in those regards, and we have also defended the point of view that the correct scientific proposal of the problem is the converse of the original proposal, that is: If willing to exhibit something that looks like a universal paradox, we had better propose we hold a heap instead of a non-heap, like a world reference deserving the sigmatoid heap as its pair in the lexicon, considering the opinion of everyone on Earth who is considered able to decide on language matches with us, as for departure mark. This way, in this paper, each and every time we refer to The Heap, or to the Sorites Problem, we actually mean a soritical sequence departing from a universal heap (our universe being the people we choose to decide with us on that) and ending on zero grains of sand accompanied by the paragraph of text mentioned at the beginning of this item (1.1). We also have defended our standing as people to actually have solved the Sorites Problem for good. Thus, we believe we have addressed all the issues, which have been mentioned in the scientific literature (until 2000) as issues, historically found associated with the Sorites Problem (see (Author 2006a; 2006b), for instance). 1.1.1

How to generate problems containing the essence of the Sorites Problem

The basic ingredients of a Sorites (Note 2) are: The fixed amount to be added to some specificity of any of the soritical sequence terms (for instance brightness in the case of colours) in order for us to have the next term in the soritical sequence; the soritical keysigmatoid (for instance bald or heap), which may be chosen from several options (any vague term), and a specific favourable order so that people are led to believe that there is no way to tell where the line is (increasing, or decreasing, as for the specificity: If one changes the order of the sequence placing, for instance, the final element from the sequence beside the first one, there is the danger that one identifies at least one sharp cutoff in the sequence immediately, and that will make the problem lose its value). More clearly, a soritical sequence contains, as most basic elements, the following: a) A soritical key-sigmatoid with some specificity that may be generating the overall feeling of vagueness if both varied by fixed increments and considered in what regards a progressive sequence of world references, which will differ from each other, being them next to each other, by that fixed incremental amount only (decreasing (Note 3)); b) The right amount (increment) to be added to each specificity of each soritical sequence term am, in its turn, in order to generate the next element in the sequence, am+1; c) A first element for the sequence (our a1);

d) A clear opposition between the first and the last elements (say element a1 and element an), with the number n of elements (Note 4) allowed to be infinity (either the attribute may be given to the first element, but not to the last element, or vice-versa). 1.2

Contextualism

The contextualist conception of communication is the idea that the mere literal meaning of a sentence does not correspond to a complete proposition with truth conditions that determines the meaning of each of the sentences utterances. In a communicative situation, in order to obtain an appropriate linguistic understanding of the uttered sentence, a series of contextual and background knowledge must be invoked to assign a determinate meaning to the uttered words, this will give us what is said. Further inferential work can be needed in order to sort out what was implicated by what was said, i.e., what was communicated. (SAAGUA 2006) The extract above gives us a good idea of what contextualism is, the level of perfection in the writing, as for grammar and syntax, being irrelevant for the purposes of understanding the concept. In a more objective way, contextualism defends the study of what we could call environmental conditions (context) of the assertions for the purpose of transforming the contents of the assertions into viable logical entries. 1.3 Organization of the paper This paper is written in an exotic way, when compared to most of the scientific literature in Philosophy, because we believe this is the best way to convey the intended message. The titles for the sections in this paper follow: 2Prologue; 3Piece of writing 1: John and Mary; 4Piece of writing 2: John, Mary, and their issues (or our reading of their possible issues); 5Uniqueness issue; 6Time issue; 7Boundary issue; 8Personal issue; 9Personal x universal issue; 10Pattern/Uniqueness issue or stamp issue; 11Pointers issue; 12Personal issue; 13Communication issue or dualistic issue; 14Uniformity issue; 15Vagueness issue; 16Extras; 17Piece of writing 3: You are now invited to get in the apartment;

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18References. 2.Prologue We ask our readers to imagine they are watching a play on Piece of writing 1. On Piece of writing 2, the readers are asked to imagine that they have been taken to a special room, where their teacher will make skilled remarks on the play that has just been presented to them. On Piece of writing 3, the readers are requested to focus on our proposed main issue, which is expected to have emerged in their minds naturally during the reading of the Pieces of writing 1 and 2. 3. Piece of writing 1: John and Mary. At first, there was emptiness… . A woman came and dropped a bit of sand in the emptiness. Light has been made and the man noticed that emptiness had become a surface with a light layer of sand over it. The woman turned to him and asked: -John, is it such that they call heap? (Uniqueness issue) The man replied: -No, Mary, I am sure that heap is not such, it is something else! The woman, unhappy with the response heard, brought another little bit of sand and dropped it over the layer already sitting there. She stared at the man in an inquisitive manner. -No, Mary, not yet a heap! The woman rushes to grab another little bit of sand pouring it all over the rest. -No, Mary, it is not yet a heap, but it is almost becoming! The woman then gets another little bit of sand, keeps it in a bag, sits before the surface, stares deeply at the man’s eyes and starts adding one grain of sand at a time. (Time issue) (…) -Hey, Mary, what are you doing now? -You have told me that it was almost a heap: If I add one grain of sand at a time it is going to be easier not to miss the exact point where it all starts being a heap… . (Boundary issue) -Are you insane? -I need to know how much sand will precisely tell us that we have a heap. -How in a hell can I possibly know? -If I add one grain at a time, you will know.

(Personal issue) -Mary, Mary, you are killing me! I don’t have the slightest idea on Earth about how many grains of sand are needed so that we can call that a heap and, even if I had, I am sure that another person could disagree. (Personal x universal issue) -John, there must be a definition in the dictionary telling us what a heap actually is. (Pattern/Uniqueness issue or stamp issue) -There is, Mary, but the existing lexicon definition for heap does not provide us with enough logical reasons to believe our personal verdict on the adequacy of the match (this is a heap of sand; world reference) may be considered a universal verdict, rather providing us with logical reasons to believe that we would be playing irrational if stating to someone else this is a heap of sand, like we can at most state this is what I take to be a heap of sand instead. -And how can that possibly be called a definition? (Pointers issue) -It is a definition in a sense: Heap is a vague term, not a precise term, what implies that the meaning of heap actually depends on who is making use of the word (relative, or personal, or even private, meaning). The definition of heap, found in any ordinary lexicon of the English language, serves us just as a piece of information regarding the historical use of the sigmatoid heap, that is, research made on the application of heap to world references by the date in which the lexicon has been formed returned those particular world references as being found frequently associated to the sigmatoid heap. (Personal issue) -In this case, John, I cannot understand how it is possible that we apply the term ever. I think I also do not understand how actual communication may happen between any number of people greater than, or equal to, two. (Communication issue or dualistic issue) -Very simple: If someone applies the term heap to whatever, we all accept that that whatever is a heap! (Uniformity issue) -But, in this case, it is very hard to tell whether this is a heap is true or false when the sentence is found attached to a world reference.

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-Exactly; it is so. Vague terms were made to be free from universal paradigms! (Vagueness issue) 4. Piece of writing 2: John, Mary, and their issues (or our reading of their possible issues) The dialogue above described illustrates, in detail, the problem with creation, which is associated, in the literature, to Eubulides of Miletus, dating from four centuries before Christ (see (Hyde 1997)). Philosophers from all over the world have been discussing the problem for several years without ever reaching a definite solution. The problem has even been the target of monetary prizes of relevance. It has been named paradox by the non-scientific audience (the soritical paradox or The Sorites). One of the main experts in the problem is in Australia (Dr. Dominic Hyde (UQ)) and he has a share of the Stanford Encyclopedia on the problem, material that may easily be found online. We believe to have proved that The Sorites is actually an allurement: There to make a statement on how rich, artistic, and creative, human language is, in all senses. Expressed with the best Classical Logic lingo available, or most appropriate, the problem would read: Premises: q: Adding one grain of sand cannot change a non-heap into a heap. Definitions: A(n, p): A set with n grains of sand and the property p. Properties: p: Being a heap. Sorites Problem in the negative direction, (~ p): A ( n, ~ p ), q { A ( n, ~ p ), q }

=> A ( n+1, ~ p )

{ A ( n+1, ~ p ), q } => A ( n+2, ~ p ) … …

=> A ( m+1, ~ p ), any m ∈ N

{ A ( m, ~ p ), q }

∴

A ( ∞, ~ p ) ( -> <- )

Contradiction because it is impossible that an infinite number of grains of sand will not form, finally, a heap! (Universal paradigm) Sorites paradox in the positive direction (p): Premises: q: Taking away one grain of sand cannot change a heap into a non-heap. Definitions: A(n, p): A set with n grains of sand and the property p. Properties: p: Being a heap. A ( n, p ) { A ( n, p ), q }

=> A ( n-1, p )

{ A ( n-1, p ), q } => A ( n-2, p ) … … { A ( 1, p ), q }

∴

=> A ( 0, p )

A ( 0, p )

( -> <- ) Contradiction because it is impossible that zero grains of sand still form a heap. (Absolute fallacy) 7/2 3

5. Uniqueness issue Perfect, and unique, matches between world references and pointers are amongst the most sought after items in what regards human communication. Were we ever able to, each time we point to the same world reference, state the same word, not mattering what - all of us, the world would be a place of no mistake in communication, that is, in the transmission of mental images. However, the formation of the human speech is the most polluted (noisiest) process on Earth: No one can be sure about what goes on inside of a person’s body before they utter a word. This way, a baby may stare at their mother and state daddy. The baby might not know, might have heard from someone else who has pointed their finger at their mum and said daddy, or they are simply trying any random sound on any target, or they might be so spiritually clever as to actually know that their mum is both mum and daddy: Who knows? The sentence there is far more between Earth and Heavens than any of us may possibly guess seems to tell all about what goes on between what is expressed by a human being and what is intended by them, or even what is seen inside of a human being’s mind before they try to express something to anyone else. It is obviously the case that all we have to base ourselves at when assessing a person, who does not hold any intimacy with us, is their discourse. Basically, scientists are life watchers when they write. How good they are watching is how good they are living. Because only a few scientists get to experience full contact with people, establishing actual relationships, the probability that any of them may write something of scientific use about those people is almost null. As the abstraction increases, that is, people write based on previous papers and books, rather than based on direct observation of other people and their actions, the mistakes of description can only grow in a fully unbearable way. Every word needs a human being: The day we dissociate discourse from humans, and this is the intention of Mary’s incursions in the language world, we are left with emptiness, true emptiness. This emptiness might be so complete that we end up stuck with the meaning of words such as God (who is God? What is God?). In the case of the sigmatoid God, for instance, once we believe in God, and prove to ourselves it works, why would any of those questions be relevant? Once we use the sigmatoid, and we are understood by someone else, or think that we are, why would the analysis of it make any sense? With a whole text, the scientist may have a better go, once it is easier to work out what is being singled out from the world if we are given more references, but with one sigmatoid only, it all depends on who is writing, what context, for language exists because human beings do, not being God to exist on their own: A sigmatoid, with no context, is absolutely nothing. One could argue that there is a context in the case with Mary, once there is a world reference and a question on that world reference. Notwithstanding, nobody can tell another person what a word is for real: They can tell them what the word has been said to be that far only, and never in relation to an object of discourse, which is to be chosen as deserving the word by the person pointing at it by the time they are pointing at it, not by

anyone else. The definitions in the dictionary are usually based on pairs of the sort (sigmatoid; world reference), pairs formed by those who have already made use of that specific sigmatoid in a context, not the other way around (apart from very special cases, such as those with the purely scientific words (starants, for instance)): One obviously starts with the application of the word. This way, Mary should be asking herself: Do I wish to baptize such a world reference with the word heap (Yes or No)? Mary, for the sake of belonging to the human race, may do whatever she likes, even pointing to the sky and uttering heap. Anyone listening to her will have their go on the interpretation of whatever comes out of her mouth. Trivially, passing a message, as it is, to someone else, is an almost impossible-to-achieveobjective: Nobody will ever know, even if the reader is repeating the same, with the same sort of expression, in all senses, as the one the writer had when writing the sentence x, whether the reader really got it or not (basically, we all pretend to communicate with each other, but what we hold is occupation of our time with trials of communicating with others instead). There is some standard meaning in the dictionary and it is true that if Mary had learned that meaning at school, that is, Mary had learned that (heap; world reference y) was a sound match, assuming that this world reference appears expressed by means of a picture, for instance, then, perhaps, Mary would not have asked that question (to John). Assuming that Mary learns through pictorial description and supposing, as well, that Mary has got the best pictorial memory on Earth and that Mary has been able to accurately guess (because she has not counted grain by grain herself) how many grains formed her school heap, for instance, we may end up being driven to also assume that Mary thinks that she knows what a heap of sand is. The same way, we could assume that John has learned his match from someone else: A match containing the same sigmatoid, learned from a person who presented a picture of a heap of sand when instructing John on the match, but such a heap contained half of the amount of sand that Mary's school heap contained, that becoming John's perfect, and unique, match containing the word heap. Things being so, John’s match differs from Mary's: Now John and Mary hold a problem. But, hang on, is it a communication problem or simply waste of our times? Why would they be worried? Is that relevant to anything they wish to do? In terms of communication, what matters is that the least amount of things as possible comes on the way for whatever people want to achieve together: Why would Mary be worried about a universal picture of what a heap is if such a picture, first of all, does not exist? There is obviously a multitude of possible matches between the pointers set and the world references set. Mary, get a life! would be the best way to address her problem. In short, we may even have, as we indeed have, a collection of pairs in the dictionary P, which hold sigmatoid x as a member. Thus, we find a dictionary entry, say entry number 500, which reads: E500 = { (sigmatoid x; world reference w); (sigmatoid x; world reference y); (sigmatoid x; world reference z)}, for instance.

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Notwithstanding, we also hold instantaneous matches, which are made each time a new context appears associated with the sigmatoid x, these matches belonging to either the human discourse (in written, oral, or pictorial, form) or the human mind (personal interpretation). It might be the case that a particular instantaneous match clearly does not coincide with any of the matches that are currently found in E500. If the instantaneous match does not coincide with any of the matches in E500 then it should be considered as a candidate for inclusion in the next edition of the dictionary P. Some personal interpretation matches (instantaneous matches) are made sometimes long before the interval of time during which the person actually interprets the word x when finding it in a piece of text they read, for instance. These matches (instantaneous or pre-recorded matches) hold little likelihood of coinciding with any number of the matches that are already found in the dictionary: Personal matches involve several factors, which may be mechanically controlled and limited, but also include several factors which may not be mechanically controlled and limited. The beauty in human kind is definitely in its creative power, and such a power can only increase proportionally to the interest of human beings in human beings, what may only appear with diversity, especially in communication. There is a never ending list of reasons for us not to think of universality in what regards personal matches in language, yet the personal character of the use of the language besets any questioning on the precise meaning of words, changing the set of Mary's issues into waste of time unless the concern is with how lexicons are written, that is, unless the focus of the problem is on what brings decrease of noise in communication, like what could make communication be more machine-effective (the concern is then about the possible noise during communication trials, not about forcing people to match the sigmatoid x, for instance, with the world reference y each time they make use of such a sigmatoid). 6. Time issue The time issue has to do with the adverb when: Mary worries about what precise moment in time, during her pouring of sand, John (and it all can only be attached to John as an individual for what we have just explained) thinks we have a heap. That is possibly fine, for it is possible that Mary works for John, John has told Mary to produce five heaps of sand, Mary is a good employee, and she then wants to get it right in her first go. There are several ways to approach Mary's issues: One of them is expressing the soritical sequence idea by means of mathematical symbols (there could be a specific amount of time for each individual who tries to deal with Mary's issues, for instance, amount of time regarding when the heap of sand stops being a heap of sand, and this when may even be the last instant of time, when no grains are left). This way, n(x1), for instance, would determine the line between heap of sand and nonheap of sand for the individual X. However, individual Y would have n(x2) as their point in time to determine the line. Such a reasoning, that we have just described, considering each individual's limiting line as a valid line, has led to both the supervaluationist and the subvaluationist solutions to the Sorites Problem. The mistake in these solutions is the complete negligence given to the humane character

of language, for trivially n(xm) may vary from zero to infinity and, instead of neglecting or fixing those lines somehow, we should be looking for making accurate notes attaching those lines to each individual, once language is personal and, at the same time, it is a universal communication tool (because it is personal, all those lines should be considered acceptable; because it is universal, we should look for a decision theory in order to register a countable, but representative, number of those lines in the lexicon in association with the sigmatoid they refer to). To explain things in a simpler way, the tools to produce a painting are always the same for each and every painter. However, no two paintings are alike and everyone on Earth appreciates the diversity of the paintings: Like nobody would prefer to have a copy of Van Gogh's painting W instead of having a Van Gogh's painting that has not been reproduced by any other painter in the world, it all being about how different, or exotic, the painting is. Some painters will exhibit their paintings together and believe they have in common, some will proceed on their own for the rest of their lives, and both positions are equally accepted by society. We may easily compare language with the tools for painting: It is always available to each one of us so that we try to communicate our minds, or even souls, to others, but each one of us will communicate their own way (or will paint their own thing), unable to mimic precisely, to detail, anyone else, with all of us knowing that every person who plagiarizes the discourse of another person (or copies the original art of some other painter) gets caught as soon as an adequate expert examines their discourse (painting). Basically, it is all harmonious this way, and it should all be acceptable: People with the same communication style, just like the painters who feel they have something in common, when using words instead of the painters' tools, will eventually believe they are a group and will exhibit their texts together; others will proceed on their own, diversity being what we all praise and believe to be fun, the rarest pieces of language being those that apparently cannot be re-written in any other way that may be seen as having more impact on the readers' souls, so to write, than the original piece of language had. A painter may exhibit a totally clean canvas (brand-new canvas sort of look) and call that painting All. Another painter may exhibit a canvas, with all colors and shapes he may think of, and call that painting Nothing. A person, judging their works, may think they are illogical and the first painting should be called Nothing whilst the second painting should be called All: Who is right? Apparently, there is a common understanding in human race as to agree both matches are perfect and harmless to society. At the same time, we all apparently feel more grateful in experiencing seeing things from varied perspectives instead of from just the painters' perspective, so that the examination of all those possible matches of the type (sigmatoid; world reference) is actually bringing wealth and joy to our lives, rather than boredom and poverty. The canvasses, presented this way, could easily be replaced by the world references from the lexicon for the sigmatoids All and Nothing, respectively, with us then having pictures, which are not internal to the human body, to associate to those sigmatoids as references, instead of the usual pieces of text, which are not always so easy to be seen by everyone else as the same thing as the mental image that the linguist (lexicon) had when writing those pieces of text. In the case with the painting, people apparently usually get to draw in their minds the mental picture intended by the artists,

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sometimes after conversing with them. Therefore, we should be accepting both the impossibility of thinking of a line to define all, once all could easily be nothing on a canvas, and the possibility of having the line anywhere we like, once there is no limit for how many objects may be seen by people observing a canvas individually, experimentation having already being made in those regards (with humans and perception). All the previous lines considered, is it reasonable to ask when all starts or when all ends (the same question could be asked replacing all with nothing)? And notice that we can do that with almost any lexicon word, apart, perhaps, from the purely scientific terms of the language. It is perfectly plausible, for instance, to state that white is all colors of the spectrum together, therefore that canvas, with apparently no painting, means all. The same way, it is plausible to state that, in terms of writing, a blank piece of paper is nothing. Thus, the plain canvas means nothing. There could be easily a lexicon that mentions both senses we have just described for all and nothing, or even mentions all of them, like the world references for the two mentioned terms (all and nothing) regarding every human being alive right now. The problem is that it would take us a volume as big as the current most complete lexicon of the English language to report all current world senses for the sigmatoids all and nothing. A researcher, when making a survey on all that has been published on S-convexity, for instance, so far (2009), will obviously never even get access (or be able to state they had) to all that has been written on the topic. What the researcher does then is relying on the most popular sources available to them or on what they have personal privileged access to for their survey writing. Even so, they still think deeply about all sources, compare, and refine, to then report their results in the most useful, and objective, way as possible (obviously, once more, with utility and objectivity being personal (or private) concepts). The lexicon writers are considered researchers, so that their linguistic labor does equate the labor of the researcher producing a survey in the case just mentioned. When we do not belong to the researcher's métier, or we have not studied their topic with their depth, we usually simply accept their results as sound, otherwise discussing them scientifically with the own researcher, seeking personal belief on their theory. 3. Boundary issue Here, Mary is worried about the limit: When is it that heaps appear? Mathematically, one would immediately think of real intervals and their boundaries. However, how can humans make their minds fit inside of mathematical intervals (all of them? Majority?)? Such is obviously not possible: The words are an inaccurate translation of a human mind; if the human mind does not fit in real intervals how can what is a poor trial of expressing them fit? Just for starters, if all we are doing is expressing a world reference, the ontological dimension of that reference will have to be reflected in our terms of discourse... . We have already proved that a heap of sand (for other heaps exist) would be something between zero and an infinite amount of grains of sand. Besides, sand is used to designate more than one object nowadays: We have, for instance, sand-color and we may be referring to the color, rather than the beach sand, by the time we use the word. We would

then have to state heap of sand of the beach, but if we write this way, the beach might be an artificial beach from a game, appearing on the screen of a computer, so that the sand is actually rectangles, or pitches, on a screen, rather than grains. Then, let’s say it is the actual beach from nature. In this case, we could think of referring to our heap like this is a heap of sand of the natural beach. Notwithstanding, the beach might contain a heap of sand in its pictorial description and still be the picture of a natural beach, rather than a natural beach. We must then state the heap of sand of the natural beach, which is an object of nature that I am observing from my window at the moment, for instance, to make sure that the person reading our words will not be vulnerable to any sort of noise (memories of their own, references of their own, and etc.) or to give a more accurate idea of what we refer to. As a result, for us to be able to actually progress in any discussion, which refers to human discourse, we must use a whole set of references, which seem now to form at least one good paragraph, for perhaps each trio of sigmatoids constituting our discussion matter. Once more, if ever trying to intuitively match Mathematics with sigmatoids and their world references, one could, for instance, for the sake of progressing, omit all needed singlers to point to the heap Mary has intended, and write that heap of sand is something between zero grains of sand and an infinite number of them. So is a non-heap of sand... . One may then identify, by now, common places between our discourse and the explosion law from Classical Logic: Anything will do… (a situation in which the systems crash). Were the robot a classical robot, it would be unable to go for the intended heap having a sequence of possible heaps before it: It simply would break/stop or short-circuit (everything is a heap and everything is a non-heap at the same time). As mentioned before, a non-heap, if we ever wished to define one mathematically, that is, if we tried, would be something between zero and an infinite number of grains of sand, for how large is infinity? Infinity may certainly fit inside of infinity. This is just an allurement for the obvious statement: Mathematics lingo is an inadequate tool for us to write about language with the level of detail needed by Science. And what tool would be suitable for that job then? The obvious response to this question is that everything that cannot describe a human mental picture with precision to someone else (anyone else) is not good enough to be used in written discourse for scientific purposes. Even if we use pictures, together with sigmatoids, and there is closeness to the readership (level of memory intimacy) by the part of the writer, that is, even if all the elements for the best go of teaching something to someone else are present, we will not be able to guarantee that another person gets the idea precisely the way we, writers, have intended they got it. However, those tools together would represent our best odds in terms of achieving successful transmission of the writer's mental image to the mind of a random able reader. Notice that we are actually stating that what we have just written in mathematical lingo is true, yet it is not acceptable that we include heap in a lexicon like that. It is very easy to understand that before the kid is taught that five means five units instead of four, they may call the world reference for five both five and four. So, even though the lexicon is irrelevant for the purposes of instantaneous communication, for the purposes of civilization, for continuous communication with the same people, the lexicon meaning is a reference. In short, what is in the lexicon should not be used to judge how correct the

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application of a sigmatoid to a reference is, but should be used to improve both the efficacy and the speed of exchange of information between human beings, unless we are writing about the learning and teaching processes having the own lexicon as subject. 8. Personal issue If words are based in personal, and internal, abstractions, such as physical experiences are, they would have to allow for more effective communication (they need less context to be understood, with noise zero, by someone who reads them from a text, for instance). Why? Because there is no external observation involved, so that there are several factors attached to the interface between the person and the environment, which will not be present as possible noise creators. The person might be blind and still know what a palpitation in their hearts is, for instance, and that concept bears far more likelihood to coincide, as a pointer, with what another person sees in their head whilst uttering palpitation than other concepts, of concrete nature, bear. Technical terms, therefore, are obviously far better understood than other terms that may escape Science. A human being, however, is a very complex entity (perhaps, if all elements, which are needed to describe uniquely a human being are listed, say by means of a vector, the dimension of this vector could end up being infinity). We start: feelings, memories, physical body, karma (do we include it or not?), spirit (do we include it or not?), senses, …. Suppose we are able to decide, with no hesitation, along with the vast majority of the people on Earth, or even only with the vast majority of the scientists, whether we will include things like karma or not: Now, how are we going to refer to senses and their accuracy, for instance, which will vary from individual to individual in human kind? It is all soritical material, and it all comes back to the same question: How do we describe specific human characteristics by means of logical lingo? And if such is possible, how can we guarantee we hold no mistakes in our observation/classification at a point of daring stating it is such to another person? Who, on Earth, is able to safely utter they hold the best eyes available to measure something in others, say beauty, for instance? It is obviously ridiculous that people even try: That is like uttering I am God. Therefore, each and every thing a human being utters, or hears, or reads, or writes, taking away purely scientific discourse, has to be considered always in association with that individual somehow, that is: It is all a matter for the private logic or personal logic of the individual, rather than a matter for logical systems, as in Computer 'Science'. We have used the expression private logic before: What we mean by private logic is the human equivalent to computers logic, that is, the systems involved in a particular human being's processes. 9. Personal x universal issue The Bible reads: Thou shall not judge. If we took the Bible to be a book of all wisdom, which has made of quite a few people happy beings - equally happy - by the time they lived, we would simply obey it and the world would not have so many conflicts as it has. How can anyone think that pointers to the world references are passive of discussion? If a person dares stating Mary, this is not a heap then they should be writing the dictionary.

However, if they go and try, they will not be hired, for the dictionary is formed, in the vast majority of its entries, from the frequent application of the word to a certain world reference, as explained before, not from arbitrary imposition (it all depends on when the sigmatoid has been applied of course: It might be perceived as an imposition by the vast majority of the people on Earth if the sigmatoid, for instance, has never been thought of by anyone outside of the Club X of people (if the word has only been applied by two scientists, for instance, who happen to have solved a major world problem, so that they are relevant, their definition will probably be perceived as an imposition to others), but that is one of the only ways it ever will). Truth is that our emission of judgment over someone else's discourse should hold no logical value, unless it regards the grammar of their discourse rather than the semantics of it because grammar is fully tied to severely detailed and clear rules, and these rules are part of a finite list. Semantics, however, is always going to be personal. So, say Andre wrote the house I lived at was blue, but he meant the boat I lived at was blue instead, as for our own understanding. Can we tell Andre: Andre, you are wrong, you lived in a boat, and you should have said the boat I lived at was blue instead? Well, if he person says that, there is a generalized agreement the person is socially unbearable, and is obstructing communication, interfering negatively with the speed of exchange of information, for they actually know what Andre has meant. On the other hand, if third parties are present, and they do not know that Andre refers to a boat, stopping Andre, and saying that, might actually mean facilitating communication and speeding up information exchange both at that very moment and in the future. As a third case scenario, if the entire village of Andre calls boats houses and he is giving a speech only to his village people, we are back to the first analysis results, when telling Andre of his mistake is the worst thing to do, even because it is not a mistake. Now, we believe things are becoming clearer: It is not the case that the lexicon is irrelevant in all... . The lexicon is a reference for conversations in the cases involving third parties, as in the case in which Andre's mate never interrupts his speech and there is a third party with them. Yet, if Andre and his village hold a private meaning for house, as explained before, the lexicon is actually on the way to the targeted noiseless reception by the third party instead. Is there an incorrect and a correct, a universal verdict, which is acceptable, over the question Is this still a house? in the case involving Andre (like when does boat start and house ends or when is a boat a non-house, for instance)? The example provided by us seems to clarify any possible doubt, as for our intents with our piece of theory this far. 10. Pattern/Uniqueness issue or stamp issue What Mary looks for is a sort of stamp business: She wishes for finding out how to stamp envelopes as if she were working in a post office shop and asking for instruction. Were her items envelopes, Mary would eventually succeed, but they are objects, which are not made by people usually: Heaps of sand seem to exist independently of human beings. If an abstract entity has been created by scientists (starants (a new sort of graph (Note 5)), for example) then the scientists, who have coined the name, will definitely be defining it

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to maximum detail and that will increase enormously the probability of such a sigmatoid having a perfect match with some world reference, forming a bijective speech function pair with it. Whatever has been human-created, and is passive of description by pictures that are clear, is more likely to serve a sigmatoid in bijection than what has not been created by human beings. See: - Mary points at a certain amount of sand and utters heap. Mary is then trying to mimic nature and manufacture a heap, but heap has been the baptism name of that particular object, created by nature, and, therefore, should not serve properly as a pointer for the object manufactured by the hands of Mary! However, suppose Mary has pointed at a certain amount of sand and has uttered heap before the subject X. Next, someone says heap and the subject X sees in their minds, as a mental picture, the world reference Mary had pointed at, earlier on, when stating heap. The story above explains the miracle called bijection: The only sort of relationship between sigmatoids and world references that allows for null noise in communication. This is obviously what scientists, as for the vast majority of them, seem to take for granted to be a rule in language regarding either the vast majority of the sigmatoids or all of them when they propose the Sorites allurement to the beauty, or art, in language, as a puzzle or paradox. With our examples, we believe we have proven that they could not be more mistaken in those regards. 11. Pointers issue We write lexicons because we wish to refer to sigmatoids in written and spoken discourse without having to describe all logical correspondence going on between the sigmatoids and their world references (even because we are, most of the time, unable to do it) each time we use them: It is the will of being together with others and conversing about the same world references that motivates us to try. We have created our symbols, which are attached to world references, each culture having theirs (for instance, the Japanese people have their own alphabet, which can only be read by us, without us learning their symbols, if found in its Romanized version), in a trial of passing history, knowledge, and wisdom, onwards. Human beings try, with the search for univocal pointers in language, since the start of everything, to overcome God, or equate themselves to God, who is assumed to be so powerful that He can read our souls always knowing, with no mistake, what we are thinking, or seeing as our mental image, as we apply any term of the language. If they did not do such, there would not be any progress, or there would be far less progress, in their knowledge and understanding of all that there is. Language is definitely human beings and human beings are, in part, language or, even better: Communication is human beings and human beings are, in part, communication.

Pointers, from an individual's perspective, are personal. On the other hand, from a collectivity's perspective, they are universally determined. The relationship formed between both perspectives is what makes of communication one of the most fascinating human areas of study. 12. Personal issue There cannot be any writing on universal applications: Even a single sound, that is, a phonetic emission, not necessarily holding a lexicon entry for it (the lexicon entries usually being formed from phonetic emissions with some meaning), may mean far more than one thing… . As a side allurement, suppose that John says Ah. Now Mary would ask, once she has proceeded this way in the past: John, what does Ah, sound emitted by you, just one second ago, mean? John does not know what to say, even though he is the one who has emitted that sound: He is not sure himself about what he meant, and he is now thinking of why he did that because of Mary's question to him. Oh, Mary, give me a break! - John could then say, given the way he dealt with the previous situation, which was very similar to this one. Truth is: Who cares? Basically, if that sound is not listed in any entry of the lexicon and it is impossible, even for John, who has emitted that sound, to uniquely associate that sound with a world reference, who is going to be able to state Ah means such without being considered insane (by insane we will always mean irrational user of the language)? However, we are all sure that Mary will insist: John, hey, John, come on… , you are not cooperating with me, you do not like me anymore? Oh, well, poor Mary! 13. Communication issue Some Arts person said very well: Hell is the others. Of course: If a person never had to communicate, never felt the need to, what problems would that person have, of what nature? How many problems does an autistic person have? Is it them or us? Truth is that communication will never happen for real: It is all about illusion and how happy we are with such. It is obviously true that the less linguistic communication a couple needs (and this is the simplest communication scenario involving two people, as for communication amongst members of the human race that we may find), the better they will live (for instance, compare the time marriages amongst less linguistically educated members of society last for with the time regarding couples in the extreme opposite situation, that is, maximum refinement in linguistic education that is possible to be found on Earth, and remember that the more education one has, the bigger one's linguistic communication needs will be). The increase in the amount of linguistic communication adds positively to the complexity of the relationships formed in human race, even in the simplest case scenario, that between a couple of humans; yet, if there is maturity in all parts involved, there is gain for the relationships, which is proportional to that increment, so that the decrease of noise

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in linguistic communication would be seen as a synonym for happiness increment in these particular human relationships (where maturity is present). Even though it is true that the decrease of noise in human linguistic communication will lead to our species having less harmful practical actions, at least in what regards particular social groups, if we worry about mechanicism (that is, precise boundaries of definition for sigmatoids for instance) instead of worrying about exercising our minds in the highest levels of Bloom's taxonomy, therefore developing our rationality, we will be finishing with all that distinguishes our species from the other species on Earth (power of evolving and adapting, inventive power, and etc.) and that cannot be seen as a positive outcome for the human race in general. 14. Uniformity issue First of all, it is not true that everyone accepts the use made by others of a sigmatoid: Most of us simply do not realize that the others' usage and our usage of a certain sigmatoid hold different intervals of meaning when we pretend that we have communicated with them, most of the time truly believing we have done so. Second, nothing may make less sense than stating there is an absolute truth, or falsity, to a certain isolated application of a sigmatoid (say to a sentence): Everything is true, yet everything is false. Examples are found everywhere: If I am a weather scientist and state tomorrow it will rain heavily while on duty, and state it to one of my clients, that should attract some level of responsibility over me, perhaps even legal, therefore implying my utterance has got logical value. However, if I am a comedian and state tomorrow it will rain heavily during my comedy show, that should obviously not attract any level of responsibility over me, therefore implying my statement has got no logical value, possibly deserving the label logical waste. That, once more, testifies on the absence of plausibility of stating tomorrow it will rain heavily, on its own, without information about the society members affected by that sentence, or about the context, or even about the person to make use of that sentence, is true or false. It is clearly the case that any language sentence, which is not computational, and the term computational involves Classical Logic and Mathematics, will depend heavily on contexts when it comes to possible logical evaluations. Mathematically writing, suppose the sigmatoid T holds meaning U, which has been made popular by means of the most sold dictionary of the English language in Australia. Person V, when making use of the sigmatoid T in their discourse, would, despite the dictionary meaning, think of the meaning U' when applying the sigmatoid T in their writings. Person W, who reads the discourse of V, would think of U'', however, as a world reference for T. If U = U' = U'' then we have a perfect communication ring, with noise zero, which should be called ideal. If U' differs from U, for instance, even in case person V has imagined U when writing T, which is perceived as U' instead (say by experts in V's language), then we have high likelihood of holding noise in our communication ring, what may imply even effectiveness zero in our now trial of communication, rather than communication ring,

that is, may imply that our dreamed communication ring will now be inside of our wasted linguistic token basket. If U'' = U, but U'' ≠ U', then we should have a non-ideal communication ring, once communication is always a personal thing, not supposed to be tied to the lexicon of the language, usually things being the other way around, as already explained. Notwithstanding, if V has intended to pass the token U instead of U', U' being what an expert in the language of V would grasp, but U being what W would grasp, we then should have an ideal communication ring, as for the perspectives of both W and V. It could, however, even so, be a wasted linguistic token for the expert who reads T, written by V, but sees U' instead of U there and knows that V has intended to pass the token U instead of the token U'. The expert would, in this case, adequately label V's trial of communication a linguistic mistake of V. Notice that we have just written about things considering the perspective of the writer (person V), having ended up with waste whenever the receptor does not get the image intended by V when the receptor reads V's writing O containing T. Notwithstanding, if we write based on the perspective of a language expert, therefore expert in V's language, then we may end up with an ideal communication ring for the same sequence of events. The perspective of the person W, however, must be the same as that of the person V, things becoming waste when W does not get the same mental image as that held by person V by the instant of time during which V writes T in the text O that W reads. Notice, as well, that it is not necessarily the case that any of the people involved, in our case V or W, is aware of the difference between their mental pictures, and it is perhaps impossible, for any third party, to tell whether what went on between V and W, in that special instant of time in which W reads the text O, written by V, was waste or an ideal communication ring, as for W's or V's perspective. The most an expert in V's language may do, provided W writes enough, as to allow the expert to grasp what W seems to think of V's text, is producing statements at the linguistic level on the sequence of events, never at the communication level, scientifically writing. Contrary to the expected, we here prove that the more sigmatoids we use in our trial of communication of small tokens of information, the more chances of holding a ring instead of waste we have if referring to the non-purely scientific language. It is worth mentioning, as well, that in purely scientific language, it is not necessarily true that the smaller the amount of symbols we use, being those enough to convey the intended message, the more likelihood of having an ideal communication ring, rather than waste, we hold. What is true, regarding purely scientific language, is that the smaller the amount of symbols we use to convey the intended message, being those enough, the more likelihood we hold to contribute to the increase of the speed of the mind processes involved in understanding the intended message, as for the receptor's share of the story, what does not necessarily mean diminishment in noise levels, but may mean increase in the amount of applications over time regarding that specific message (receptor's), therefore broader and quicker scientific progress for all of us. 15. Vagueness issue Almost every word in human language is a trial to reduce a mental picture, which is never universal, to a written expression. To escape the surrealistic thought that such a

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thing is possible, assume we reduce one's mental picture to a possible common subset in the mental pictures of everyone on Earth (is that possible?), or of the vast majority of the people on Earth (is that possible?), then, so that we keep on going. The major obstruction to the execution of such a task, of dantesque proportions, is the fact that neither the intended mental picture can be seen the same way by everyone on Earth, or by the vast majority of the people, or even by two people, when they have contact with its supposed written expression, and that is the minimum requirement for us to be able to state the word is precise, not vague, nor part of that intended mental picture can. Try, for instance, the mental picture associated with cardinality one in the case of John. Mary goes, stares at John, and utters: One! John is watching soccer. John then replies: No, Mary, two! Mary stares at John and tries two sigmatoids: One beer! John is still watching soccer and replies: No, Mary, vodka! Mary stares at John and tries three: One beer, fire! John stares at her: Now? Mary stares at John and starts picking up all she can pick up in a rush… . John is a bit astonished, for he had started thinking of the day they had sex with beer, and etc., but Mary looks like a mad bird and is picking things up, apparently to get out of the building… . John asks her: Mary, when you said One beer, fire!, were you thinking of having sex outside of the building, close to the fireplace, as we have done in the past? Mary is now with the worst face on Earth: John, if you do not go with me, you will die! John still does not get it, but goes. As he is out, he watches their apartment on fire, all of it being demolished by the fire, and realizes that if he had remained inside of the apartment he would, indeed, have died! Now, who would think that we can argue on what a sigmatoid, singled out of an environment, that is, put out of a context, considered apart from either writers or readers, could possibly mean, universally? Notice that one corresponds to 1 in their dialogue (world reference), therefore it bears, in principle, a unique popular mathematical reference. Yet, in our context, 1 is associated with the natural language, rather than confined to the universe of Mathematics, different from the 1 appearing in 1+1 = 2 ∧ 2+2 = 4

∴

3+3 = 2+4 = 6, which inhabits the purely mathematical environment. When 1 leaves the mathematical environment, it may, or may not, have its meaning confined to the mathematical universe, therefore it is possible that it holds another world reference, different from the popular mathematical reference for 1, as its match, the mentioned mathematical reference being one of the available candidates to world reference for that one (we also use 1 to express the neutral element in Algebra, for instance, so that even inside of the own Mathematics, we have more than one possible world reference for that 1. However, any person from outside of Mathematics will have only the cardinality reference in their minds when they read, or write, or utter, one with sense 1).

In short, even the purely mathematical entities, when considered outside of the space in which Mathematics fits precisely, will frequently represent world references that differ from the so well determined popular lexicon references for those entities. Notice that even though Mary utters one, she is actually holding an entire picture of their apartment under fire inside of her head so that her one has actually meant that entire picture by the moment it gets out of her mouth, as for her intended message. 16. Extras (Universal paradigm issue (Note 6)) It is obviously the case that if people are free to apply any term of language - absolutely any, for that is even a human right, then there is no clear fallacy in the Sorites proposal that goes in the negative direction: It is definitely true that anyone on Earth may think it is simply fun to keep on pressing the button Non-Heap in a TV show that demands a choice between the buttons Heap and Non-Heap (of sand, for instance), for example. If things are this way, are people who do this wrong or it is acceptable that they do whatever they wish? It is obviously acceptable that people do whatever they wish and there cannot be any right or wrong, for it is actually true, in the negative direction, that the person may always state that their heap is bigger, starts later, not mattering how many grains have already been added to the previous set of grains of sand, like: It is simply not my heap! (Absolute fallacy) When there are no grains of sand left, what previously was a set of grains of sand could then be referred to as a heap of nothing: Still a heap, but not the heap the problem has originally referred to. Therefore, if ever trying to make it look as if the application of the word heap is inadequate in all then there is only one hope: The positive direction. 17. Piece of writing 3: You are now invited to get in the apartment. This is an invitation to the reader: Go further on the insights inspired by the pieces of writing 1 and 2! If universality of pointers, or even universality of intervals of meaning, is not worth arguing about, for it is impossible to progress coherently in those issues, should we be writing theories about paragraphs instead, or even pieces of human interaction? It seems that this was the line of reasoning supported by the Bible, for instance, one of the most read and understood books of all times! In the Bible, everything comes by means of stories, attached to contexts and people, apart from the commandments. Even so, there are several real life stories to tell how to apply those commandments in real life situations, all of them attached to individuals. God is also mentioned as God of Abraham, Isaac, and Jacob, that is, in the Bible, not even God is allowed to exist without clear connection to contexts and people, actually connection to real life situations, which are all described in chapters, rather than

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paragraphs. The issue we would like to start discussing here, therefore, regards contexts: How small the description of a context may be in order to allow us to scientifically discuss semantics in written human communication? If wishing to speak soritical: How many littles of little bits should we have in order to scientifically guarantee that we hold a universal semantic human unit in terms of written discourse? We believe that our sub-titles may bring inspiration that allows for insight in terms of answering this question in the most correct way as possible. 13. Conclusion Human language is definitely a place that is as messy as the human thought. Such a fact is no surprise, for the human language is a trial to draw whatever is seen inside of the head of the writer, in the most accurate way as possible, in the reader's head, so that the reader sees, mentally, precisely the picture that the writer had in their heads when writing the text by the time they read the writer's text. We believe to have proven that, in what regards semantics, and possible scientific issues with it, one should refer to sigmatoids via chunks of text, or via contexts, rather than refer to the sigmatoids in isolation as if they could mean something on their own that could be considered a logical entity. In order to scientifically deal with sigmatoids, we should first be able to scientifically deal with the context they appear immersed at, and each context must hold a standard, or universal, minimum size, or minimum number of world references of non-zero weight (weight attached to the contribution of that reference to the precise, and sole, world address of that sigmatoid), to be considered suitable for scientific analysis. A sigmatoid must, from now on, be seen as a tent: Unless we hold both enough rope and points of firm connection to the soil, it will serve for nothing and nobody in practical life, taking into account its main purpose, as intended by its creator (in the case with words, effective communication, that is, transmission of mental images without any noise in the process). We have also introduced, in the recent past, the name sigmatoid in order to refer to part of what a word means. A word should be seen as a phonetic emission with a sensible world reference, that is: Word = (sigmatoid; world reference). We also get to propose a new research topic, which may be dealt with by us (or not) in the future: Deciding on how many words are necessary for us to be able to deal with a specific sigmatoid in a universal way (scientific). It is worth stating that the underlying reasoning for such concerns is that Science must be completely detached from the scientist in order for it to add to human progress: This means that any person, say person W, who has been adequately introduced to Science, must be able to draw the same mental pictures (precisely the same, or close enough, like with almost zero mistake) as those that the original scientist, who has written the paper X, had in their minds when writing the paper X by the time W reads such a paper. In this paper, we have, as well, reinforced our magic assertions, which formed our solution to the Sorites Problem, solution that we strongly believe to be final.

14. Acknowledgements We take this opportunity to publicly thank, once more, Professor Hyde, for his generosity of instructing us on the Sorites Problem with so much accuracy and detail.

Notes: Note 1: Po Box 12396, A'Beckett st, Melbourne, Victoria, Australia. E-mail: xxx Note 2: By Sorites we mean soritical problem from now onwards. Note 3: We regard the opposite direction, in which the Sorites could go, as being logically redundant. Note 4: Elements of the sequence. Note 5: See (Author 2007). Note 6: Observe that whilst before this section our sub-titles appear in between brackets and after the chunk of text they referred to, we here do the opposite, placing those subtitles before the chunks of text they refer to.

References HYDE, DOMINIC (1997). (Sorites Paradox) STANFORD ENCYCLOPEDIA OF PHILOSOPHY. Found online at Http://plato.stanford.edu/about.html. ISSN 1095-5054 (accessed 31 October 2000). AUTHOR (2006a). AUTHOR (2006b). AUTHOR (2007). SAAGUA, JOAO (2006). "Contextualism, Semantic Minimalism, and Communication" Paper presented at the annual meeting of the International Communication Association, Dresden International Congress Centre, Dresden, Germany, Jun 16, 2006. 2008-09-13 http://www.allacademic.com/meta/p68715_index.html (accessed 3 October 2008).

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