It would be difficult to give a more formal setting of our interpretation of the paradoxes, and also useless, when considering that our aim indeed is measuring the real extension of the realm of syntax. However, if we accept that there are strong hints of a common background among the old and new analysed antinomies, centred on the gap in the commutativity of the correct representation diagram, and that this is connected to the very nature of the syntactic paradigm as framework of the western thought since Parmenides, then we can think that our phenomenological enquiry about the paradoxical structure of the syntactic paradigm is complete. However this analysis has left us with a well-defined structure of the paradigm and of the paradox. Thus, we can not avoid to set further questions: Is there in our language the possibility of characterising the kind of connection between knowledge and reality the syntactic paradigm entails? Why has the paradox that peculiar structure? How does it develop in the human knowledge?
The most conventional interpretation of the above analysed antinomies are: (i) platonic, advocating for example the existence of non-standard models of arithmetic to explain its incompleteness, (ii) skeptical, claiming the irreducibility of semantic to syntax and, consequently, of mind to formal representations, to explain the contradictions appearing in the limitative theorems, (iii) positivistic, underlining the limits of our natural language and common sense to account for the breakdown of our intuition in the elementary particles world. We cannot really introduce these approaches in our analysis, for (i) we do not believe in the 'real' existence of the standard model of arithmetic, out of the tight borders of the syntactic paradigm, and then we can not even consider the nature of 'non-standard' models, (ii) we consider the appearance of those contradictions not as a sort of 'halt' to a theoretical enquiry, but only as a necessary consequence of 'moves' in the sign form of knowledge, for example connected to the formal employment of an unified negation, (iii) we do believe that no formal or scientific language can judge about the relation between reality and natural language.
To try a first answer to the above questions, we observe that the 'commutativity' property corresponds to the 'ideal' setting for a formal science, depicted as a chain of 'worlds' connecting reality and language, linked by a possibly thorough isomorphism. This framework was claimed by Parmenides, with reference to the first theoretic natural language, and by Wittgenstein, with reference to the first formal logic language. In both cases this hypothesis was embedded in a cultural and scientific evolution leading to a great season of antinomies. I think that there is no loss of generality if we focus our attention to a simple language/reality correspondence, beyond the details of the different versions of the paradigm shown in the above sections.
Our idea is that those ancient metaphysical terms, being, truth, negation, etc. must appear twofold in 'the syntactic paradigm', first as their formal translations in specific theories, linguistic or metalinguistic, subjective or objective, etc., and second as structural properties of the whole paradigm, that is, as consistency, completeness, non-contradiction, diagram-commutativity, and so on. That is, the 'negation' admits its different symbolic forms, but its real structural meaning in the paradigm is in the non-contradiction and third-excluded principles, and in the isomorphism between the above different formal occurrences warranted by the commutativity of the correspondence diagram. Analogously, we can use the terms 'true', 'provable', 'real', 'assertion' etc. in different versions of the paradox, but the structural meaning of 'truth' and 'being' in the paradigm is given by the correctness and completeness properties of the coding. And its consistency gives us the structural link between 'truth', 'being' and 'negation'. The Aristotelian and Tarskian truth as correspondence idea, reminded at the beginning of the report, defines the relation between 'truth', 'being', 'negation' and 'assertion'. Thus, the occurrence of incompatibilities among these properties is not the 'halt' command to the theoretical enquiry, but only the 'blurring' of those metaphysical and structural definitions, when taken beyond the range of their functionality. More precisely, it is in the too close connection between being, truth and negation in absence of other more empirical terms, that we find the background of the antinomies concerning consistency, completeness and non-contradiction. The tight connection between these terms is a 'singularity point' for the paradigm. Not different from the North Pole for the polar representation of the earth on a planar map: when we approach that singular point the representation becomes useless, and inconsistent in the Pole. In his 'Notes on logic' (1913), L. Wittgenstein, in his approach heavily based on a tight connection between language and reality, displayed in an analogy between real images and retinal images, cannot avoid to compare negative sentences to the 'blind point' of the retina.
The crucial role of the negative in establishing the basic non-empirical concepts was already stressed, when dealing with the idea of infinite in the second report. But it can be revealed also in the basic Euclidean definitions (point, line, etc.), as underlined by Proclus ((PROCLUS 1970), 94):
Negative definitions are appropriate to first principles, as Parmenides teaches us in setting forth the first and ultimate cause by means of negations alone. For every first principle is constituted by a different essence from that of the things dependent on it, and to deny the latter makes evident to us the peculiar property of the principle.
Looking closely at the ancient metaphysical grounds of our paradoxes, we point out that the core of the rupture inside the syntactic paradigm is in the non-empirical nature of those crucial words as being, negation, truth, sameness ( or equality, as something different from both identity and similarity), rule etc., together with the Eleatic attempt of a tight matching between language and reality. The difficulty in dealing with the negatives in a tight connection between reality and language is also present in Wittgenstein 'Notebooks' (WITTGENSTEIN 1961), where it is possible to recognise the traces of the 'negative judgement paradox', when the author remarks the impossibility of denying an image (26-11-14). It is worthwhile to remind also Frege's remarks against formalism, when he addresses that ''No science is completely formal'', and that logic concepts, as ''negation, identity, subsumption, subordination of concepts'', which cannot be 'interpreted' otherwise (109), must be accepted as 'primitives', and in another point (96) underlines that also the concept of ''rule'' could be subjected to the same doubt in a coherent application of formalists' philosophy. In fact, for those words we cannot show any correspondence with the reality more or less ostensibly, as we can do instead for almost all the words, as normal adjectives, names and verbs. A crucial role is played by the negative. We already analysed the great difficulties in Greek culture in dealing with lying and falsehood. Snell (SNELL 1982) pointed out that already in archaic societies 'to lie' or 'the incoherence between what is said and what is done' is forbidden, and this maybe is the core of the never ending paradox.
A crucial hint to this hypothesis could come from the analysis of the first linguistic child development. (CRISTAL 1987). Here, we find that the greatest part of the earliest vocabulary is made with persons, objects, actions, features of the real world surrounding the child. Other words are related to social interaction: greetings ('bye-bye'), responses ('no'), deictics ('my'). To be is not used until late development, and its existential usage is achieved by single-word utterances ('daddy'), its copula or locative usage by two-word utterances ('there teddy', 'she cold'). Further development shows the beginning of more complex structures. Except for the negation, the nuclear concepts we are enquiring are of late development, and always developed in non-empirical 'linguistic games'. Negation, rule and truth are most of all learnt in games of prohibition, lying and punishment. Being and sameness are likely linked to the beginning of asking questions, knowledge acquisition, and Self awareness, as it has been continuously restated (''cogito ergo sum'') in modern philosophy. That is, these words are learnt in the most primitive admittance rites that the human babies have to participate for the reproduction of the social group. And their making finds its roots in the deepest links between Self, language and reality (VER EECKE 1970),(FREUD 1920), (FREUD 1925).
Thus, we could say that the whole enterprise of western thought since its beginning has been that of establishing a tight matching between language and reality, grounded on strictly social-reproduction founded primitives. If this is true, formal knowledge cannot ever be a picture of the world in sign form, for its primitives are not empirical. Knowledge is rather the secret core of our whole social praxis, that, in the realm of syntax, has got a purely 'formal' shape. But also in this form it can not conceal its roots, which can be found in those metaphysical simplest non-empirical elements of the framework in which modern science frames the whole reality. So much that, if we try to give even to those social-praxis-rooted primitives an empirical rationale, we cannot escape the 'blurring' of the terms in the 'never ending paradox'. This ancient social root of the knowledge is the deep link between science and society. Thus, the relation between science and way of production is not one of dominance, as in scholastic Marxism, but is grounded in their common connection with the making of the syntactic paradigm.
If the framework of our knowledge enterprise is in a set of social-praxis-rooted primitives, we must also observe that their inner architecture is 'imprinted' in the human language in the earliest forms of social interaction of the children. Thus, the atomic reproduction of the human social group in the earliest concepts and words formation is the ground of the human cognitive social praxis. These first connections between language and unconscious shape both the Self and the Tribe. Freud underlined as the career of the negation begins as defined by the pleasure principle and then develops its new form as defined by the reality principle:
The study of judgement afford us, perhaps for the first time, an insight into the origin of an intellectual function from the interplay of the primary instinctual impulses. Judging is a continuation, along lines of expediency, of the original process by which the ego took things into itself or expelled them from itself, according to the pleasure principle. The polarity of judgement appears to correspond to the opposition of the two groups of instincts which we have supposed to exist. Affirmation - as substitute for uniting - belongs to Eros; negation - the successor to expulsion - belongs to the instinct of destruction. ... But the performance of the function of judgement is not made possible until the creation of the symbol of negation has endowed thinking with a first measure of freedom from the consequences of repression and, with it, from the compulsion of the pleasure principle. (FREUD 1920)
Freud in 'Beyond the pleasure principle' sets the negation at the beginning of the symbolisation process. And in its double form, it draws the unification and the fight of the Self with reality, creating that set of metaphysical primitives, which are the framework of the human praxis-grounded knowledge. In the earliest language acquisition, negation is one of the first word to be uttered, and accomplishes, in negative, those meaning roles that being, in positive, can not achieve: from a first role as pure rejection ('no', 'no drink'), ''not'' develops also non-existence ('no car') and denial ('not mine') meanings. Whereas there is no trace of a positive corresponding particle (neither 'car is' nor 'yes car').
We can also return to the formal ''contradiction principle''. We saw as it was deeply established in scientific Western thought, since Parmenides, and as it was fuzzy, if any, in Chinese scientific thought. However we saw as it was common instead in any known civilization in its concrete form. Such form substantially appears to be the non-employment in common sense reasoning of contemporary opposite assertions (''something is white, and at the same time the same thing is not white'') by the same person, and a critical situation when such assertions are made by different people. In Greek civilization, on a codified laws environment, such critical situation could cause lawsuits (between persons) or also war (without states), in Chinese civilization the 'correlative' thinking looked for more dynamical solutions. The way Greek thought developed the formal contradiction principle was traced by two great changes: i) the idea of 'formal' (without reference to the meaning) manipulation of the dialectic reasoning, as appears in the being/not-being issue in Parmenides, Zenonian and Sophist paradoxes, Platonic dialectics and Aristotelian 'organon'; ii) its application in an unstructured world of 'equal' subjects (as with the codified laws in the poliV) or in an homogeneous world of 'ideal' signs (as in geometry or in logic). It is not hard to realise that, without this background, the formal contradiction principle could not play any role at all. Also the connection between enigmata, riddles and antinomies - established in the Greek culture since the first pre-Socratic philosophers - help us to reveal the deep unconscious dwelling of that ancient dynamics of the paradoxes.
The interesting process of condensation accompanied by the formation of a substitute, which we have recognised as the core of the technique of verbal jokes, point towards the formation of dreams, in the mechanism of which the same physical process have been discovered. This is equally true, however, of the techniques of the conceptual jokes - displacement, faulty reasoning, absurdity, indirect representation, representation by the opposite - which reappear one and all in the technique of the dream-work. (FREUD 1905)
The whole outfit of the paradox can thus be found in dreams and jokes as well. It is a trace of the entrance of the child in the social group.