SENSE, CAUSALITY AND PARADOXES

SENSE, CAUSALITY AND PARADOXES Jean-Paul BENTZ ABSTRACT The present article is a synopsis of the eponymous book by the same author, published by CreateSpace, and whose table of contents is reproduced at the end. Although the work carried out in particular by Bertrand Russell and Alfred Whitehead on the formalization of the foundation of mathematics in the beginning of the 20th century revealed some new paradoxes, most logical paradoxes still discussed or studied nowadays are known from ancient Greece. Such is notably the case of the Liar Paradox which, in its most concise version, is reduced to the assertion “I am lying”. If the author of this assertion is indeed lying, it should be concluded that he does exactly what he claims to do, and thus that he is not lying at all. But, if it is true that he is not lying, then he does the reverse of what he claims to do, which means that he is indeed lying. If a vast majority of logical paradoxes, as the one above, result from a few simple sentences expressed in natural language, their study today is frequently entrusted to mathematical theories which present both a much higher level of abstraction and a much higher complexity than the problems to be studied, and which, therefore, are much more likely than them to conceal a number of totally undetected or poorly understood semantic mechanisms. Contrarily, the phenomenological approach exposed hereafter consists in identifying the limits and logical constraints of natural language and in modelling natural language by means of a simple logical system complying with these limits and constraints, therefore efficiently applicable to the study of paradoxes. THE A PRIORI MODE OF ATTRIBUTION OF TRUTH VALUES IN NATURAL LANGUAGE It is possible, in any natural language, to express true propositions (ex: “Paris is the capital of France”) and, conversely, false propositions (ex: “Ireland is an island of the Dead Sea”; “My cousin is the Emperor of China”). Truth and falsehood are attributes called “truth values”. Every segment of speech called “proposition” or “statement” is allotted a truth value, whereas the other segments of speech, such as imperative (ex: go ahead!) or interrogative (ex: what time is it?) turns of phrase, are not. Why and how, then, is for instance the proposition “the sun is shining” spontaneously recognized as true by any person sunbathing on a beach, while it is spontaneously recognized as false by any person enjoying a midnight swim on the same spot? In fact, this spontaneous mode of attribution of a truth value to a statement, which will hereafter be referred to as the “a priori” mode of attribution, results from the unconscious implementation of a process, of linguistic nature and founded on training, through which each individual comes to master, among other things, his or her mother tongue. This is the reason why it is impossible, for instance, to a priori allot a meaningless proposition any truth value, as is typically the case of a proposition expressed in an unknown foreign language. Now, the mastery of a language results, for each individual, from the empirical apprehension of the supralinguistic relation conventionally established between each word of this language and what it designates, i.e. its extralinguistic referent. In the first of the stages required to learn a language, the young child initially assimilates the supralinguistic relation established between concrete referents, considered separately, and the words by which they are designated: “Mom”, “Dad”, “room”, “cat”, etc. Then, the child assimilates the meaning of more abstract words, in particular of verbs and linguistic connectors which enable him not only to invoke separated referents, but referents connected to each other by an 1

extralinguistic relation which these verbs and connectors encode in the language: “Mom is in the room”, “Dad is stroking the cat”, etc. It is at this level of linguistic acquisition that the process of attributing an utterance an a priori truth value appears. More specifically, the attribution of the truth value “true” to a statement referring to an objective, observable reality, in the a priori mode of attribution, results from a comparison between, on the one hand, the intralinguistic relation that this statement builds between the linguistic entities it relies on, and, on the other hand, the relation actually observed, the memory of the relation, or the knowledge of the relation, which the extralinguistic referents associated with these linguistic entities have between them in the real world, the de facto reference frame, when the terms of this comparison prove to be identical. Similarly, the attribution of the truth value “false” to a statement referring to an objective, observable reality, in the a priori mode of attribution, results from a comparison between, on the one hand the intralinguistic relation that this statement builds between the linguistic entities it relies on, and on the other hand, the relation really observed, the memory of the relation, or the knowledge of the relation, which the extralinguistic referents associated to these linguistic entities have between them in the real world, the de facto reference frame, when the terms of this comparison prove to be different. THE ABSOLUTE NEED FOR THE “INDETERMINATE” TRUTH VALUE IN NATURAL LANGUAGE The definitions of truth and falsehood, as given above, show that the a priori attribution of these truth values to propositions relating to the observable, concrete world can only result from an operation consisting in comparing one with the other a relation between linguistic entities on the one hand, and a relation between the concrete referents of these linguistic entities on the other hand. Any logical system that only relies on the sole truth values “true” and “false” implicitly rests on the idea that a comparison between two elements can only conclude either to the identity of these two elements or to the disparity between these two elements. Though a priori appealing, this idea is actually perfectly misleading insofar as the process of comparison itself is frequently made impracticable by the unobservable nature of the relation expressed between the extralinguistic referents. For example, though the sentence “One of the planets that man has not observed yet is inhabited” expresses an objective property of clearly identified extralinguistic entities, and thus undoubtedly constitutes an unambiguous and meaningful proposition, neither of the truth values “true” and “false” can a priori be allotted to it. Consequently, such propositions can only receive another truth value, which will hereafter be referred to as “indeterminate”. THE A POSTERIORI MODE OF ATTRIBUTION OF TRUTH VALUES IN NATURAL LANGUAGE As they expand the mastery of their mother tongue, children assimilate the meaning of words and phrases known as “logical operators”, whose function is to act on the truth values of the propositions to which they apply, or to make them interact. The logical operators most frequently used in natural language are: “not”, “and”, “or” (with the meaning of “and/or”), and “if… then”. For example: “Paris is not the capital of France” derives from the original proposition “Paris is the capital of France” by application of the logical operator “not”, which transforms the initial true proposition into a false proposition. “Ireland is not an island of the Dead Sea” derives from the original proposition “Ireland is an island of the Dead Sea” by application of the logical operator “not”, which transforms the initial false proposition into a true proposition. “Paris is the capital of France and my cousin is the Emperor of China” is a false proposition resulting from the combination, by the logical operator “and”, of a true proposition and a false proposition.

2

“My cousin is the Emperor of China or Paris is the capital of France” is a true proposition resulting from the combination, by the logical operator “or”, of a false proposition and a true proposition. “If Ireland is an island of the Dead Sea, then my cousin is the Emperor of China” is a true proposition resulting from the combination, by the logical operator “if… then”, of two false propositions. As shown in the examples above, each one of these logical operators transforms one or more original propositions into a final proposition whose truth value depends not only on the truth value of each original proposition, but also on the logical operator used. Insofar as the attribution of a truth value to the final proposition then involves the properties of at least one logical operator and not just a simple comparison between an intralinguistic relation and an extralinguistic relation, as is the case for the a priori mode of attribution of truth values, this other mode of attribution will be hereafter referred to as “a posteriori”. ABSTRACTION AND CAUSALITY Learning a mother tongue, a process which initially requires from the learner a growing awareness of his direct physical environment, the observation of this environment, and the memorizing of its rules and its evolutions, ends up offering the learner access to a more and more symbolic universe and thus, at least seemingly, to a universe less and less subjected to the constraints of the concrete world. For example, a schoolboy who learns how to count, and who is taught that two plus three is five, can always easily refer himself to the observable physical world, and validate this rule by materializing the numbers to be added on his fingers. By contrast, the high-school student who learns that the power two of the imaginary number i is equal to -1, or that the power zero of any number Z is equal to 1, touches on a field of higher level of abstraction in which the connection of these concepts to the concrete world, such as we perceive it, is very complex or even impossible. The attribution of the truth value “true” to the two mathematical rules previously mentioned can thus no longer result from a standard process of a priori attribution of a truth value, but requires the a priori attribution of the truth value “true” by a pure convention of the type of those which form the basis of all axiomatic theories. Yet, if the absence of any observable limit to the addition of successive layers of abstraction tends to obliterate the fact that any construction rests on a base, any construction, however abstract it may be, derives its meaning from the meaning attributed to each level of this construction, and notably to its most concrete base. Figuratively speaking, the meaning which may be attributed to any sense-conveying form, whatever its expression and degree of abstraction, can in no manner transcend the mechanisms through which the phenomenon of sense itself comes into existence, and whose most fundamental steps consist in the progressive apprehension of the concrete world in its physical reality, and also the intimate intellectual understanding of it. The consequences associated with this remark are twofold. First of all, as long as speech exclusively refers to the concrete physical world, the extralinguistic referents of the linguistic components of speech on the one hand, and these linguistic components on the other hand, coexist without ever being likely to generate the least confusion, as the former are of concrete nature whereas the latter are of abstract nature. By contrast, when speech refers to a purely abstract entity, as in the case of the imaginary number i in mathematics, the name given to this entity is the only concrete, though intangible, trace of its existence, which situation entails a high risk of occult and accidental confusion between the name of the abstract entity and the invisible extralinguistic referent constituted by this entity itself. In addition, insofar as the concrete world, such as revealed by our senses, appears to be subjected to a set of immutable rules, such as the law of irreversibility of time, the law of conservation of energy and the law of causality, and as the phenomenon of emergence of sense is basically rooted in our apprehension of the physical world, it would be particularly daring to hope that speech could retain all its meaning and yet disregard the rules that govern the physical world. For this reason, the law of causality appears to be of first importance. 3

This law, which universally applies to the concrete world as we perceive it, takes the form of three basic rules, namely (1) there is no effect without a cause, (2) the same set of causes always entails the same effects, and (3) an effect cannot constitute its own cause. The first rule notably reflects the fact that man cannot conceive the future without a past. The second rule expresses the constancy of the physical laws which govern the behaviour and the evolution of the physical world, as observed by man. And the third rule reflects in particular the experimental fact that the future cannot act on the past. The conviction according to which language cannot with impunity transgress the law of causality, and which constitutes one of the essential bases of this study on paradoxes, can easily be illustrated through a simple example. The proposition “the sun is a star”, which refers to the concrete world, and more precisely to the field of astrophysics, is easily translated into French by “le soleil est une étoile”, and the sentence resulting from the translation has the same truth value as the original one. By contrast, the correct literal translation into French of the proposition “the word sun is made of three letters” reads “le mot soleil comprend trois lettres”, but whereas the original English proposition is true, its translation is false, the word “soleil” being made of six letters and not three. Why does a faultless translation affect the truth value of this sentence? Because, by virtue of the law of causality, any constant cause should always lead to the same effect, whether this cause is the source of a physical effect in the concrete world or a source of emergence of sense in the field of language. Now, this principle is openly transgressed in the second of the two propositions above, since the form “sun”, which should normally have the effect of referring to the star of the day by virtue of a well-established metalinguistic convention, is used here to develop another effect, namely to refer to the form specifically used, in a given language, to designate the star of the day. MODALISATION OF LANGUAGE AND INSTANTIATION Despite the absolute need, as already emphasized above, to take into account the truth value “indeterminate” in addition to the truth values “true” and “false”, it should be recognized that, generally, speech does not leave much room to indeterminateness. Such an observation even provides enough justification for the huge success of binary logic. This phenomenon, which can probably be explained by the discomfort caused by uncertainty, results from a frequent operation, which will be referred to here as “modalisation”, by which uncertainty is generally transmuted into certainty. For instance, while the neutral assertion: “One of the planets that man has not observed yet is inhabited” is clearly indeterminate, the modalised proposition: “It is established that one of the planets that man has not observed yet is inhabited”, which consists in equating the original, non modalised, indeterminate proposition to an undeniable truth, is clearly false. The logical operator generating this modality will be called “guarantee” and denoted “g”. In contrast, the modalised proposition: “It is not excluded that one of the planets that man has not observed yet is inhabited”, which consists in denying that the falsehood of the non modalised original proposition has been established, and thus in accepting this proposition as a simple assumption, is true. The logical operator generating this modality will be called “hypothesis” and denoted “h”. Similarly, the modalised proposition: “Nobody knows whether one of the planets that man has not observed yet is inhabited”, which precisely consists in stating the indeterminate character of the non modalised original proposition, is equally true. The logical operator generating this modality will be called “ignorance” and denoted “i”. 4

Thus, when a proposition is indeterminate, i.e. allotted the truth value “indeterminate”, and when, consequently, the comparison between, on the one hand, the relation between the linguistic elements of this proposition and, on the other hand, the relation virtually established between the referents of these elements cannot be realized, instantiated, the application of one of the logical operators g, h, and i, here referred to as “modal operators”, leads to the replacement of this impracticable comparison by a comparison actually carried out, instantiated, between the truth value of this proposition and one of logical constants respectively represented by the universal truth, denoted by T, the universal antilogy (error or falsehood), denoted by E, and the universal indeterminateness, denoted by N (neutral). Thanks to the combined use of the truth value “indeterminate” and of the operators g, h, and i, it is thus possible, quite adequately, to make the distinction between the certainty of the erroneous character of a proposition and the uncertainty of its veracious nature, a distinction made in speech but completely ignored in binary logic. SCHEMA FOR THE DEVELOPMENT OF A NEW LOGICAL SYSTEM (CNL) For the sake of brevity, the following developments only provide a simplified version, therefore schematic and inevitably approximate, of the stages and operations necessary to the development of a formal logical system (referred to as CNL) meeting the constraints and limits of natural language. Therefore, the book mentioned in the Abstract is the only rigorous reference for these developments. In addition, as the notations used here are those of the book, their apparently unnecessary complexity results from the fact that all the logical operators and constants used here have been initially developed on the basis of a single original logical operator. Taking into account the observations formulated above, such a logical system must: - fulfil the law of causality; - include the truth value “indeterminate” in addition to the truth values “true” and “false”; - include the universal logical constants of antilogy E, indeterminateness N, and truth T; - include the modal logical operators of guarantee (g), hypothesis (h), and ignorance (i) in addition to the traditional logical operators, namely negation (NOT, denoted by ~), disjunction (OR, in the sense of and/or, denoted by ), conjunction (AND, denoted by ), implication (IF… THEN, denoted by →), and equivalence (denoted by ), the operators →, , g, h, and i being in addition conceived as generators of instantiation with regard to the indeterminate propositions, originally not instantiated by definition; and - explicate the a priori alloted truth values, namely to emulate the a priori mode of attribution of truth values which is spontaneously available in natural language. Let 0 be a language, referred to as “protolanguage”, in which linguistic forms, referred to as “primitive forms”, can be formulated prior to the attribution of any truth value and to the intervention of any logical operator. The primitive forms of this protolanguage 0 are built on the basis of a dictionary D0, use a specific alphabet A0, and comply with syntactic rules R0. These primitive forms, which are designed to become propositions, are suited to describe any chosen extralinguistic universe of reference, whether existing or to be built, and whether concrete or abstract. By a metalinguistic convention between this protolanguage 0 and the metalanguage used here, these primitive forms will be denoted here by x0, y0, z0 etc. Then let iP* be a logical language in which are imported, coming from the protolanguage 0, any number P of primitive forms, each of which is a priori alloted an explicit symbol of truth value freely selected among , , and , these symbols respectively representing truth, falsehood and indeterminateness. So that no given form may receive two different truth values in the language iP* (which would constitute an obvious violation of the law of causality), the P primitive forms imported in this language will all be supposed to be different one from the other. In addition, these P primitive forms will be supposed to be classified in an immutable - although a priori immaterial - order, for instance in alphabetic order. Under these conditions, the vector made up by the ordered sequence of the truth values successively allotted to the P primitive forms imported in the logical language iP* will be called “distribution of truth values” and denoted by iP*. 5

In fact, as there are 3 truth values and P primitive forms, there are 3P different distributions iP* of truth values, and therefore also 3P different logical languages iP*, the superscript number “i” appearing in “iP*” and “iP*” being precisely used to respectively designate, in a generic way, any of these 3P distributions and any of these 3P languages, and the superscript letter “P” being used to reflect the number of primitive forms imported. Insofar as these 3P logical languages iP* are all made up from the same P primitive forms ordered in the same way, and despite the fact that each of them differs from each other language by the truth value attributed to at least one of these P forms, they can be regarded as forming together a collection of languages which will be denoted by P*. Once assigned a truth value, every primitive form imported in each of the languages iP* becomes a proposition, this term generically including any propositional form. By a new metalinguistic convention posed between the languages iP* of the collection P* and the metalanguage used here, any proposition contained in any of these languages may be denoted here by a capital letter such as W, X, Y or Z. Each language iP* of the collection P* is developed as follows. First, each language iP* includes, as propositions, the universal antilogy (E), the universal indeterminateness (N), the universal truth (T) and the P propositions resulting from the P primitive forms. Moreover, if X and Y are any, possibly identical, propositions of any of the languages iP*, then this language also includes the propositions ~X, ~Y, g(X), g(Y), h(X), h(Y), i(X), i(Y), XY, YX, XY, YX, X→Y, Y→X, XY, and YX. The truth values allotted to these new propositions are defined in the table hereafter. More precisely, this table shows the different truth values which are assigned, in the different languages iP* of the collection P*, to the propositions of these languages which, in the metalanguage used here, are represented by E, N, T, ~X, g(X), h(X), i(X), XY, XY, X→Y, and XY, depending on the truth values which, in these same languages iP*, have been assigned to the propositions which are represented by X and Y in the metalanguage used here. X

Y

E

N

T

~X

g(X)

h(X)

i(X)

XY

XY

X→Y

XY











































































































































































































































6

Thus, each language iP* includes, on the one hand, P propositions, which will be referred to here as “primitive propositions”, directly constituted by the P primitive forms x0, y0, z0 etc. imported from the protolanguage 0 and each a priori assigned a truth value, and, on the other hand, an infinity of propositions, which will be referred to here as “complex propositions”, each a posteriori assigned a truth value resulting from the application of one or several of the operators ~, g, h, i, , , →, and  onto one or several primitive and/or complex propositions. THE PROPERTIES OF THE LOGICAL LANGUAGES iP* Each language iP* of the collection P* makes use of all the truth values, this property being readily apparent from the table above. Each language iP* of the collection P* is non-contradictory, this property being defined here by the fact that each of these languages is exempt from any pair of propositions that would only differ one from the other by their final symbol of truth value. Each language iP* of the collection P* is complete, this property being defined here by the fact that both the content and truth value of any proposition of each language iP* can be calculated, at least apart from any consideration of time. And each language iP* of the collection P* is decidable, this property being defined here by the existence of a general method which can be applied to any sequence of symbols to verify whether this sequence is, or not, a proposition of this language iP* and, if so, to check its truth value. TRUTHS - SCHEMAS - SCHEMAS OF UNIVERSAL TRUTHS (SUT) In what follows, “truth” shall designate any proposition of any language iP* which is assigned the truth value “true” ().“Universal truth” shall designate any proposition expressed in any language iP* and which is a truth in all the languages iP* of the collection P*. “Particular truth” of any language iP* shall designate any truth of this language iP* which fails to be universal. And “schema of truth” (or of antilogy or of indeterminateness) shall designate the transcription, into the present metalanguage, of a truth (or of an antilogy or of an indeterminateness), every schema being deprived of any symbol of truth value. It is worth noting that, insofar as every schema is deprived of any symbol of truth value, no schema of truth can be a priori distinguished, in the present metalanguage, from a schema of antilogy or from a schema of indeterminateness. If it is obvious that every truth consituted, in each of the languages iP* of the collection P*, by a primitive proposition can only be a particular truth of this language, in contrast, all these languages have in common an infinity of universal truths, whose expressions in the present metalanguage are thus “schemas of universal truths” (or SUT’s). Since the present CNL logical system fully encompasses classical binary logic, binary logic coinciding with the part of this system that is devoid of any indeterminate proposition, every tautology of classical binary logic is a schema of universal truth (see hereunder the examples * and a few examples of generalised SUT’s whose validity in binary logic, although not verified, may be presumed **). Conversely, however, the CNL system offers an infinity of schemas of universal truths necessarily unknown to classical binary logic insofar as it does not make use of the “indeterminate” truth value () nor of any of the modal logical operators g, h and i. A few SUT’s are exemplified hereafter: XX

~X~X

X~(~X)

~ET

~TE

~NN

g(X)(XT)

h(X)~(XE)

i(X)(XN)

i(X)i(~X)

h(X)(g(X)i(X))

h(X)~g(~X)

g(g(X))g(X)

g(h(X))h(X)

g(i(X))i(X) 7

h(g(X))g(X)

h(h(X))h(X)

h(i(X))i(X)

i(g(X))E

i(h(X))E

i(i(X))E

E→E*

E→N

E→T*

~(N→E)

~(N→N)

N→T

~(T→E)*

~(T→N)

T→T*

g(X)→X

X→h(X)

i(X)→h(X)

g(X)(~X→X)

h(X)(~X→i(X))

i(X)~(X→X)

(X→Y)(~Y→~X)*

(X→Y)g(~XY)

(X→Y)(~h(X)g(Y))

(X→Y)~h(X~Y)

(X→Y)(h(X)→Y)

(X→Y)(X→g(Y))

(X→Y)(h(X)→g(Y))

(g(~XY)g(XZ)) → g(YZ)

(g(~XY)g(~YZ)) → g(~XZ)

(Ei(Xi)→Ti(Yi))  (Ti(Xi→Yi)j≠i(Xj→Yi)j≠i(Xi→Yj))** (Ti(Xi→Yi)j≠i(Xj→Yi)j≠i(Xi→Yj)) → (Ti(Xi→Yi))** (Ti(Xi→Yi)) → (Ti(Xi)→Ti(Yi))**

(Ti(Xi→Yi)) → (Ei(Xi)→Ei(Yi))**

(Ti(Xi)→Ti(Yi)) → (Ei(Xi→Yi))**

(Ei(Xi)→Ei(Yi)) → (Ei(Xi→Yi))**

(Ei(Xi→Yi))  (Ti(Xi)→Ei(Yi)).** Thus, in particular, the implication X→Y may be interpreted in the CNL system either as the impossibility of X and/or the guarantee of Y, i.e. ~h (X)  g (Y), or as the impossibility for X to be correlated to non-Y, i.e. ~h (X  ~Y), or even as the guarantee of non-X and/or the guarantee of Y, i.e. g (~X)  g (Y), which may also read g (~X  Y). Besides, the statement “X implies Y” is equivalent to the statement “the assumption of X implies Y”, i.e. (h (X) → Y), as well as to the statement “X implies the guarantee of Y”, i.e. (X → g (Y)), and also to the statement “the assumption of X implies the guarantee of Y”, i.e. (h (X) → g (Y)). EMBEDMENT OF ANY AXIOMATIC THEORY INTO THE CNL SYSTEM Let kP* be a determined language, freely chosen among the languages iP* of the collection P*, and intended to be used as a model for the construction of an axiomatic theory. To this purpose, the P primitive forms imported into each of the languages iP* of the collection P* will of course be designed and formulated so as to describe the basic entities, the properties, and the rules of the axiomatic theory to be built. This axiomatic theory, although assumed to be totally and directly embedded into the metalanguage used here for the sake of simplificity, will exclusively contain: - a number A of particular truths of the language kP*, either primitive or complex, each of which will be referred to as “axiom” and be deprived of any symbol of truth value; - ad libitum, all the potentially useful schemas of universal truths (SUT’s), namely those that describe the most basic interactions between the operators E, T, N, ~, g, h, i, , , →, and ; - all the propositional forms that may be obtained from the axioms, whether directly or indirectly, by concatenation and/or substitution of forms originating from the dictionary D 0, under conditions and according to schemas set up by the syntactic rules R0 of the protolanguage 0; - a rule called “rule of coherence of interpretation” and denoted by “RCI”, according to which, in any language of the CNL system, and in particular in the axiomatic theory under construction, two propositional forms X and Y which cannot be distinguished one from the other (which will be denoted here by X # Y) are necessarily equivalent one to the other, i.e. bound by the logical relation X  Y; and 8

- a rule called “rule of logical transformation” and denoted by “RLT”, according to which, in any language of the CNL system and in particular in the axiomatic theory under construction, any propositional form of this language may, in any occurrence, be replaced by any other propositional form that is simultaneously equivalent to the first one in this same language, all the propositional forms whose content is modified by such a replacement being then added to the pre-existent propositional forms, with the truth values of the original propositional forms from which they result. The rule RCI derives from the law of causality, which excludes the possibility that a single form might entail two different results, in this case through the improper attribution of two different truth values. The rule RLT derives both from the law of causality and from the fact that the truth values of the different propositions are, with the constant causes constituted by the logical operators themselves, the only causes acting upon the a posteriori attribution of other truth values. The absence of reference to the standard rule of “modus ponens” in the presentation above does not result from an unfortunate oversight, but from the fact that the rule RLT, combined with all the desired SUT’s, provides an infinity of valid rules of inference, the “modus ponens” being only one inference rule among others and specifically resulting from the application of the rule RLT to the SUT: (X  (X → Y)) → Y. As all the propositional forms initially integrated into the axiomatic theory are schemas of truths, the syntactic rules R0 should be designed, as are, by construction, the rules RCI and RLT, so that all the new propositional forms built in this theory by application of these rules should also be schemas of truths, hereafter called “theorems”. In other words, the theorems obtained in the axiomatic theory, by application of the rules R0, RCI and RLT, must constitute schemas of truths that represent particular truths common to all the object languages iP* of the collection P* whose propositions corresponding to the axioms of the axiomatic theory are particular truths. As, by construction, these object languages include the “model” language kP*, they form, within the collection P*, a sub-collection which will be hereafter denoted by {kP*} and which, at the extreme, may be reduced to this model language itself in a particular (though worthless) case. The sequence consisting in the application of the various rules and resulting in the establishment of a theorem in the axiomatic theory shall be called “demonstration”. CAUSES OF INVALIDITY OF AXIOMATIC THEORIES IN THE CNL SYSTEM The step-by-step construction of an axiomatic theory in the CNL system, as outlined above, incidentally points to the possible causes of invalidity of such a theory. The first possible cause of invalidity of an axiomatic theory simply consists in choosing, for the development of this theory, one axiom or a set of axioms such that none of the languages iP* of the collection P* may accept this axiom or each axiom of this set of axioms as a truth. Indeed, in this case, the comparison, typically through the equivalence , of the flawed axiom or axioms with the universal truth T results in the universal antilogy E, which itself generates propositions affected with any possible truth value. In this case, obviously, it is no longer possible to guarantee that the application of the axioms exclusively produces schemas of truths. A second possible cause of invalidity of an axiomatic theory consists in altering the truth value “indeterminate” N by comparing it either with “true” T or with “false” E. In particular, the modal form ~ i (X) of an indeterminate statement X is then equated with the universal truth T, whereas it is equivalent to the universal antilogy E. A third possible cause of invalidity of an axiomatic theory in the CNL system consists in including, within the syntactic rules R0 of the protolanguage 0, rules of substitution that implicitly and/or potentially transgress the rule of coherence of interpretation RCI. For example, the adoption of a rule of substitution by virtue of which a form X may be replaced by a form Y, in a context where the form X is already defined as being equivalent to the form ~Y, typically constitutes another sure way of introducing the universal antilogy E into this theory. And a fourth possible cause of invalidity of an axiomatic theory in the CNL system consists in applying, typically implicitly, rules of attribution of truth values which are neither provided for nor attested in this theory. For instance, as long as it has not been demonstrated, Goldbach’s conjecture K according to which every even 9

number higher than two is the sum of two prime numbers appears to be an indeterminate statement. Nevertheless, it is not acceptable, in the frame of an axiomatic theory intended to study Goldbach’s conjecture K, to assign the truth value “indeterminate” to this conjecture. Indeed, first, such an a priori attribution of a truth value is not provided for at this stage in such a theory, whose construction rules have been exhaustively defined in the previous part. And second, the a priori assignment of the truth value “indeterminate” to the conjecture K would amount to introducing i (K), an additional, implicit, “clandestine” axiom, into the axiomatic theory. Now, this new axiom would annihilate any possibility of demonstrating in a coherent way that K is a schema of truth (i.e. g (K)), or a schema of antilogy (i.e. g (~K)), insofar as i (K)  g (K) on the one hand, and i (K)  g (~K) on the other hand, are both universal antilogies, whatever the truth value of K. THE LIAR PARADOX This paradox, known in ancient Greece in two slightly different versions respectively referred to as that of Epimenides of Knossos and that of Eubulides of Miletus, will be considered here in its most concise form, previously mentioned and expressed by the statement: “I am lying”, hereafter denoted by the variable X. If one assumes this statement is true, i.e. h(g(X)), then one should immediately, necessarily conclude, because of the meaning assigned to the verb “to lie”, that truth can only be expressed by the contrary statement, i.e. g(~X). But, if one assumes this statement is false, i.e. h(g(~X)), then one should immediately conclude, again because of the meaning assigned to the verb “to lie”, that the author of this statement is sincere when asserting that he is lying, and thus that this statement is true, i.e. g(X). Insofar as h(g(X))  g(X) and as h(g(~X))  g(~X), what the Liar Paradox comes down to is an axiomatic theory built on the basis of only two axioms, i.e. g(X) → g(~X), and g(~X) → g(X). Now, in the CNL system, these two axioms lead, without any contradition, to a single theorem expressed by i(X). In other words, the statement X, i.e. “I am lying”, is indeterminate, so that the guarantee g(X) of its truth and the guarantee g(~X) of its falsehood are both false and thus imply each other. It is worth noting that this conclusion is fully consistent with the absence of any available verification process which could be used to classify this statement as either true or false. Nevertheless, the two axioms of this theory, i.e. g(X) → g(~X) and g(~X) → g(X), are both true since they are each equivalent to the SUT: E → E, so that they correctly lead to the theorem i(X). Thus, the Liar Paradox only results from the illegitimate – though implicit – comparison of the indeterminate proposition X with a true or false proposition (which violates the law of causality and constitutes the second cause of invalidity of the axiomatic theories, as mentioned above), and from the intellectual discomfort generated by the necessity to accept the implication E → E as true. JOURDAIN’S PARADOX This paradox is an adaptation proposed in 1913 by British mathematician Philip E.B. Jourdain (1879-1919) of an ancient paradox with Plato and Socrates as protagonists. In its modern version, this paradox refers to a printed card, whose front side bears the statement: “what is written on the other side is true”, and whose back side bears the statement: “what is written on the other side is false”. If the mention written on the front side of the card is true, then one should trust what is written on the back side and eventually conclude that the mention written on the front side is false. If, conversely, the mention written on the front side of the card is false, then one should trust the opposite of what is written on the back side of the card and eventually conclude that the mention written on the front side is true. A similar line of reasoning conducted from the indication written on the back side of the card leads to a contradiction of the same type. Once again, the formal analysis of this paradox in the CNL system amounts to building, within this system, an axiomatic theory whose axioms are, in this case, easily and directly constituted by the mentions respectively printed on the front side and the back side of the card. Let X be the symbol of a variable used to represent the statement: “what is written on the other side is true”, and Y the symbol of a variable used to represent the statement: “what is written on the other side is false”. 10

Formally, the statement X cannot be distinguished from the assertion of the truth of the statement Y, while the statement Y cannot be distinguished from the assertion of the falsehood of the statement X. In other words: X # g (Y) and Y # g (~X), so that X  g (Y) and Y  g (~X) by virtue of the rule RCI. The possibility offered by the rule RLT to replace Y and ~X by equivalent expressions in each of these relations leads to: X  g (g (~X)) and Y  g (~g (Y)). Now, (X  g (g (~X)))  (X  g (~X)) and (Y  g (~g (Y)))  (Y  ~g (Y)) are two SUT’s in the CNL system, as well as (X  g (~X))  E and (Y  ~g (Y))  E. In the CNL system, Jourdain’s Paradox may thus be analysed as an axiomatic theory built on the basis of two contradictory axioms, i.e.: X  g (Y) and Y  g (~X), which, for this very reason, lead to two schemas of universal antilogies, i.e. X  g (~X) and Y  ~g (Y). In other words, none of the languages i2* of the collection 2* may contain, as truths, the propositions that the present metalanguage denotes by X and Y, and which would be linked one to the other by the relations that the present metalanguage expresses as X  g (Y) and Y  g (~X). In conclusion, the axiomatic theory at the root of Jourdain’s Paradox is invalid, owing to its defective construction (cf. supra the first cause of invalidity of axiomatic theories), and more specifically by violation of the law of causality. THE BARBER PARADOX This paradox, formulated by British mathematician and philosopher Bertrand A.W. Russell (1872-1970), was made famous by its last-minute insertion in the postscript of the work “The Foundations of Arithmetic” by German logician Gottlob Frege (1848-1925). In its informal version presented here, this paradox refers to a barber who is supposed to shave all the men of the city that do not shave themselves, and only those, the paradoxical question being then to know whether the barber should shave himself (i.e. should himself shave his own beard) or not. Indeed, if he shaves himself, he is in contradiction with his assignment which is limited to only shaving the men in the city who do not shave themselves. But, if he chooses not to shave himself, then again he is in contradiction with his assignment which compels him to shave all the men of the city that do not shave themselves. The axiomatic theory to be built in view of a formal analysis of this paradox in the CNL system involves both a non-logical operator and a larger number of variables than in the previous cases. Let “:” be the symbol of a non-logical operator standing for “has the property of being . . .”. Let M be the symbol of a variable used to represent any man of the city. Let X be the symbol of a variable used to represent any human being. Let Y be the symbol of a variable used to represent the statement: “X shaves himself”. Let Z be the symbol of a variable used to represent the statement: “X is shaved by the barber”. And let X 0, Y0, and Z0 be the instantiated values which the variables X, Y and Z take for the barber. The barber’s assignment is then defined by: T  ((X : M) → ((Y  Z)  ~ (Y  Z))), i.e.: the fact, for any human being, that he is a man of the city implies that this human being either shaves himself or is shaved by the barber, and that he cannot both shave himself and be shaved by the barber. Now, for the barber: Y0  Z0, which, by idempotence, leads to (Y0  Z0)  Y0 and ~ (Y0  Z0)  ~ Y0. Thus, the barber’s assignment unto himself is: (X0 : M) → (Y0  ~ Y0). Now, it is easy to show that: and that: so that: which can also read:

((X0 : M) → (Y0  ~ Y0))  ((X0 : M) → g (Y0  ~ Y0)), g (Y0  ~ Y0)  E, ((X0 : M) → E), ((X0 : M)  E).

Thus, according to his assignment, the barber should comply with the statement: ((X0 : M)  E).

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In other words, the barber cannot be a man of the city, which does not exclude that he may, for instance, be a young boy, or a man who goes to the city every morning to practise his trade there, though he is not a man of the city. Nevertheless, if it is explicitly stated in the formulation of the Barber Paradox that the barber is indeed a man of the city, i.e. (X0 : M)  T, then this paradox, as analysed in the CNL system, is reducible to an axiomatic theory founded on contradictory axioms, i.e. (X0 : M)  T, T  ((X0 : M) → ((Y0  Z0)  ~ (Y0  Z0))), and Y0  Z0, the last two axioms leading to the theorem (X0 : M)  E which is inconsistent with the first axiom (X0 : M)  T. However, one should notice that the theorem (X0 : M)  E only results from the effect of a syntactic rule which has degraded the expression ((Y  Z)  ~ (Y  Z)) by substituting to it (for the barber) the expression (Y0  ~ Y0), whose modalisation by the operator g leads to the universal antilogy E. The Barber Paradox is thus based on an axiomatic theory which is made invalid by a syntactic rule which violates the law of causality by breaking the rule RCI of coherence of interpretation (cf. supra, the third cause of invalidity of the axiomatic theories). BROUWER’S PARADOX This paradox, made famous in its modern form by Luitzen Egbertus Jan Brouwer (1881-1966), the founder of intuitionistic mathematical philosophy, was in fact already known to the philosophers of ancient Greece and is also mentioned, at least in an elementary form, in Cervantes’ Don Quixote. A shipwreck survivor reaches an island whose inhabitants sentence him to death as a punishment for his intrusion, and invite him to say a few last words, first informing him that they will behead him if he tells the truth, and hang him if he tells lies. The castaway eventually manages to escape both beheading and hanging, simply choosing to assert: “You will hang me”. In these conditions, the natives cannot hang him, since hanging is the penalty for liars and the castaway would then indeed have told the truth if they actually hanged him. But the natives cannot behead him either, since beheading is the penalty for those who have told the truth and the man would then indeed have lied if they actually beheaded him. The axiomatic theory necessary to the formal analysis of this paradox in the CNL system requires three variables only. Let H be the symbol of a variable used to represent the statement “the prisoner shall be hanged”. Let B be the symbol of a variable used to represent the statement “the prisoner shall be beheaded”. And let X be the symbol of a variable used to represent the declaration of the castaway himself. Insofar as it is at least implicit that hanging and beheading are not cumulative, the rules set up by the potential executioners of the castaway are as follows. A false assertion leads to hanging: g (~X) → H. A true assertion leads to beheading: g (X) → B. Beheading amounts to excluding hanging: B  ~H, and hanging amounts to excluding beheading: H  ~B. As the declaration of the prisoner is expressed by X # H, which leads to X  H by virtue of the rule RCI, the statements g (~X) → H and g (X) → B respectively lead to g (~H) → H and g (H) → B by replacing X by the equivalent statement H, and the statement g (H) → B becomes g (H) → ~H by replacing B by the equivalent statement ~H. Thus, the conjunction of the two laws sealing the fate of the castaway reads (g (~H) → H)  (g (H) → ~H). Now, whatever H, the equivalence (g (~H) → H)  (g (H) → ~H)  i (H) is a SUT in the CNL system. In other words, the axioms (g (~H) → H) and (g (H) → ~H) of the relevant axiomatic theory lead to the theorem i (H), the statement “you will hang me” made by the castaway being thus indeterminate. Once again, it is worth noting that this conclusion is fully consistent with the absence, at the time of the prisonner’s declaration, of any available verification process which could be used to classify this declaration as either true or false. 12

The impossibility to apply any death sentence correctly thus results from the fact that, as hanging and beheading only apply to true or false propositions, they do not apply to the indeterminate statement made by the castaway. Brouwer’s Paradox thus only derives from the unfounded attempt, in the course of its development, to compare the indeterminate statement made by the castaway with a true or false proposition. An equivalent analysis would easily show that the assertion “You will not behead me”, i.e. ~B, would of course lead to a similar result, i.e. (g (~ ~B) → ~B)  (g (~B) → ~ ~B)  i (~B), by application of the same general pattern. THE SURPRISE EXAMINATION PARADOX This paradox, first identified, so it seems, by Swedish mathematician Lennart Ekbom during World War II, was published in 1948 in the review “Mind”. The paradox is said to originate from an announcement presumably made on the Swedish radio, that a civil defence exercise was going to take place during a predetermined week. The precise date of the exercise was not specified in advance, though, in order to evaluate the degree of preparation of the civil defence forces in a realistic way, thanks to the element of surprise. In the most widespread version of the paradox, a professor informs his students that a surprise examination will take place on any day of the following week, from next Monday to next Friday, the element of surprise being of course the specific date chosen for the examination. The students then make the following reasoning: the surprise examination will take place either on any day of the following week from next Monday to next Thursday, or next Friday. In the first case, it is certain that it will not take place on Friday as the professor has announced only one surprise examination. But then, the students notice that, in the second case, a “surprise” examination could not take place on Friday either, insofar as the absence of any examination before Thursday evening would, in this case, ascertain the choice of Friday as a date for the examination. Thus, in all circumstances, Friday is excluded as a possible date for a “surprise” examination. Friday being thus excluded, the time range available for a surprise examination is necessarily reduced to the period from next Monday to next Thursday. Finally, applying the same reasoning to increasingly shorter periods, the last day being each time excluded, the students ultimately rule out the possibility of any surprise examination altogether. Yet, on Wednesday morning, let us say, as the class is about to start, the professor informs the students that he is now going to proceed to the examination, indeed a surprise, as promised. What was wrong, then, with the students’ reasoning? Some authors have claimed that the error in the students’ reasoning proceeded from the fact that the professor was not sincere when announcing that the examination would be a surprise, or else from the ambiguous nature of the definition of the word “surprise”. Yet, the argument is easily dismissed if one clarifies the definition of this word. For example, the professor could have (i) offered the students a chance to guess the date of the surprise examination (provided, of course, this was a one-off offer, not to be renewed every day), (ii) promised not to carry out any examination at a date predicted by the students and announced to him in the evening of the day before the predicted date, and nevertheless (iii) kept all his promises, without giving up his project to carry out a surprise examination provided it took place on a date not predicted by the students. Other authors have considered that the students’ reasoning was fallacious simply because it is utterly impossible to guess what a third party secretly thinks. However, the fact that such a prowess is indeed inconceivable is completely foreign to the reasoning constructed by the students.

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The fact is that the Surprise Examination Paradox, as each of the paradoxes studied above, is generated by a reasoning which is absolutely faultless in binary logic. Therefore, its erroneous conclusion necessarily results from an intrinsic deficiency of binary logic itself, an obvious indication that no theory founded on binary logic could safely and legitimately be relied on to analyse these paradoxes. More specifically, as in the Liar Paradox and in the Brouwer Paradox, the dysfunction results here from a degenerative process which consists in representing a universe necessarily requiring three truth values by means of a model which only uses two truth values. The axiomatic theory necessary to the formal analysis of this paradox in the CNL system only requires three variables if one decides, to cut the analysis short without any loss of general information, that the surprise examination is announced only on the Tuesday of the week it is supposed to take place. Let We be the symbol of a variable used to represent the statement: “The surprise examination will take place on Wednesday”. Let Th be the symbol of a variable used to represent the statement: “The surprise examination will take place on Thursday”. And let Fr be the symbol of a variable used to represent the statement: “The surprise examination will take place on Friday”. The announcement of the surprise examination, made on Tuesday by the professor, can thus be expressed by h (We  Th  Fr)  T. At this stage, this statement is indeed a mere assumption, as no verification process of We  Th  Fr is yet available on Tuesday. Since the examination, as presented by the professor, will occur only once during the period concerned, it is legitimate to consider that, if this examination were to take place on Wednesday or Thursday, it would then be established that it would not take place on Friday. In other words: h (g (We  Th)) → g (~Fr), which relation may also be written as: g (We  Th) → g (~Fr). Again, since the examination is presented by the professor as a “surprise” examination, assuming it has been established that it will neither take place on Wednesday nor on Thursday, it is then legitimate to consider that it has also been established that it could not, as such, take place on Friday. Thus: h (g ~(We  Th)) → g (~Fr), which relation may also be written as: g ~(We  Th) → g (~Fr). The students’ reasoning which leads them, with absolute certainty, to exclude Friday as a possible date for a surprise examination is thus expressed by: ((g (We  Th) → g (~Fr))  (g ~(We  Th) → g (~Fr))) → g (~Fr). The students’ mistake lies in their unfounded conviction that this relation is true whatever the truth values of We, Th and Fr, whereas it is false for (We  Th)  N and Fr  N, i.e. precisely for the truth values which the statements We, Th and Fr have on Tuesday. Indeed, the expression ((g (We  Th) → g (~Fr))  (g ~(We  Th) → g (~Fr))) → g (~Fr) is then equivalent to ((E → E)  (E → E)) → (E), thus to (T  T) → (E), and finally to T → E, which is a schema of universal antilogy. In fact, the only expression which could have been accepted as a basis for a valid reasoning, which is a schema of universal truth, and which is therefore true whatever the truth values of the variables We, Th and Fr, reads ((g (We  Th) → g (~Fr))  (~g (We  Th) → g (~Fr))) → g (~Fr). In short, although the students’ reasoning is “almost” correct as it “only” confuses g ~(We  Th) with ~g (We  Th), it proves to be completely erroneous in practice, due to the confusion of these two terms. The Surprise Examination Paradox thus draws its origin from the illegitimate equation of these two statements, i.e. from the confusion of the truth value “indeterminate” with the truth values “true” and “false”. In other words, the Surprise Examination Paradox exemplifies the second cause of invalidity of axiomatic theories, as identified supra.

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TABLE OF CONTENTS FOREWORD ......................................................................................................... 9 FIRST PART ................................................... 11 CHAPTER I .............................................................................................. 13 EXPRESSIONS AND CHARACTERISTICS OF SENSE 1. Sense and learning ............................................................................................ 13 2. Man and the computer .................................................................................... 19 3. The semantic continuum................................................................................. 22 4. Towards a relative objectification of sense .................................................. 25 CHAPTER II............................................................................................. 29 DEVELOPMENT AND VULNERABILITY OF SENSE 1. The construction of intralinguistic relations ............................................... 29 2. The a priori attribution of truth values ........................................................ 30 3. Erroneous referents ......................................................................................... 32 4. The unavailability of the comparison process ............................................. 33 CHAPTER III ........................................................................................... 35 THE PYRAMID OF ABSTRACTIONS 1. The modalisation of language ........................................................................ 35 2. The a posteriori attribution of truth values ................................................. 35 3. The Physicist and the Mathematician ........................................................... 37 4. Formal systems and semantic reference frames .......................................... 38 SECOND PART ................................................ 43 CHAPTER IV ........................................................................................... 45 CONTENT AND REQUIREMENTS OF THE LAW OF CAUSALITY 1. Time, sense and causality ................................................................................ 45 2. Justifying axiomatic theories: a delicate issue ............................................... 48 3. Brouwer and intuitionistic logic ..................................................................... 51 4. The theory of the blue bears (an overview) ................................................ 52 CHAPTER V............................................................................................. 55 A FEW SEEDS OF PERPLEXITY 1. Gödel’s Incompleteness Theorem (a presentation) .................................... 55 2. The cavers’ bet (a presentation) ..................................................................... 57 3.The Surprise Examination Paradox (a presentation) ................................... 57 CHAPTER VI ........................................................................................... 61 INTRODUCTION OF THE BASIC CONCEPTS OF A NEW LOGICAL SYSTEM (CNL) 1. A presentation of the objective ..................................................................... 61 2. Definitions......................................................................................................... 62 3. Schema for the development of a new logical system ............................... 65 4. The collection P of the logical languages iP ............................................ 66 5. The logical languages iP* of the collection P* ....................................... 67 15

6. The schemas of universal truths of the logical languages iP*................. 69 7. Indeterminateness and indiscernibleness ..................................................... 69 CHAPTER VII ......................................................................................... 71 INTERPRETATION OF THE BASIC NOTIONS OF THE CNL SYSTEM 1. The truth values of the CNL system ............................................................ 71 2. The logical operators of the CNL system.................................................... 71 3. A few remarks on the operators of the CNL system................................. 74 4. A Survey of the schemas of universal truths (SUT’S) ............................... 75 5. Reflections on the interpretation of the CNL system ............................... 79 6. Construction of an axiomatic theory in the CNL system ......................... 81 7. Degradation of an axiomatic theory in the CNL system .......................... 82 THIRD PART .................................................. 85 CHAPTER VIII ........................................................................................ 87 LOGICAL PARADOXES 1. Introduction ...................................................................................................... 87 2. The Liar Paradox .............................................................................................. 88 3. Jourdain’s Paradox ............................................................................................ 90 4. Richard’s Paradox ............................................................................................. 92 5. The Barber Paradox ......................................................................................... 94 6. The Surprise Examination Paradox (An analysis) ....................................... 96 7. Brouwer’s Paradox ............................................................................................ 98 8. Grelling’s Paradox........................................................................................... 100 9. Berry’s Paradox ............................................................................................... 103 10. Curry’s Paradox............................................................................................. 107 11. Pinocchio’s Paradox ..................................................................................... 108 12. Russell’s Paradox and the set theory ......................................................... 108 13. The theory of the blue bears (An analysis) .............................................. 110 14. The cavers’ bet (an analysis) ....................................................................... 112 15. The Yeti and the “overmodalisation” of language ................................. 114 16. Generation and generalization of paradoxes ........................................... 115 CHAPTER IX ......................................................................................... 119 THE ARGUMENTS BASED ON CONTRADICTIONS 1. Gödel’s Incompleteness Theorem (an analysis) ........................................ 119 2. The reductio ad absurdum arguments ........................................................ 126 3. Cantor’s diagonal argument .......................................................................... 127 4. The existence of non-Turing-computable functions ............................... 131 CHAPTER X........................................................................................... 137 THE PARALOGICAL ARGUMENTS 1. Newcomb’s Paradox ...................................................................................... 137 2. Hempel’s conjecture....................................................................................... 139 3. The Doomsday argument ............................................................................. 142 FOURTH PART.................................................. 147 CHAPTER XI ......................................................................................... 149 FORMALIZATION OF THE CNL SYSTEM 16

1. Construction of a logical language iP ........................................................ 149 2. Construction of the propositions of the language iP............................. 152 3. The collection P of the languages iP ....................................................... 153 4. The properties of the logical languages iP................................................ 154 4.1 - Each language iP uses all the truth values ...................................... 154 4.2 - Each language iP is non-contradictory ........................................... 154 4.3 - Each language iP is a complete logical language ........................... 155 4.4 - Each language iP is decidable ........................................................... 155 5. Definitions....................................................................................................... 156 6. A selection of noteworthy propositions of the languages iP ................ 157 7. The properties of the logical languages iP* of the collection P* ....... 163 8. The Schemas of Universal Truths of the logical languages iP* ............ 163 CHAPTER XII ....................................................................................... 171 APPLICATIONS OF THE CNL SYSTEM 1. Embedding an axiomatic theory in the CNL system ............................... 171 2. The rules of transformation of the CNL system ..................................... 172 3. The infinite – Persistent causes –Recurrence ............................................ 176 4. Completeness of the axiomatic theories in the CNL system.................. 179 5. Quantifiers and models of interpretation .................................................. 186 6. Relations between quantifiers ....................................................................... 192 7. The handling of quantifiers .......................................................................... 192 8. Levels of language in the CNL system ....................................................... 193 9. Generalized SUT’s .......................................................................................... 194 CONCLUSION ................................................................................................. 197 BIBLIOGRAPHY ............................................................................................. 199 DEFINITIONS AND SYMBOLS ................................................................. 205 ABOUT THE AUTHOR ................................................................................. 209

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