Since the beginning of the Greek thought, the breakdown of the mythological thought opened the issue of the relationship between the being and the sign. The being was the 'everyday reality', with its becoming and change. It was the realm of space and time, its science was geometrical and evidence-driven, its (oral) language was a spoken social event, its numerical world was made up with the (duodecimal) submultiples of the units of measurement mirroring the motion of the sky and with the (decimal) counting grounded on the hand employment. The sign at the beginning was only a memory-tool, in its written linguistic and numerical form, but also a crucial part of the great universe of symbols, linked to magic and religion. These symbols were frequently interwoven with the oral language, a sort of 'cooperation' most of all in ritual activities, but there was not that tight link we set today in the idea of 'representation', which will be the core of the ''syntactic paradigm''. However, with the alphabetic revolution, (oral) language and (written) signs began to match. In the classic Greek civilisation the sign played a role more and more relevant, beginning its 'irresistible rise' toward the modern realm of syntax. Its function drifted toward the representation of being by discrete symbols, its science became axiomatic and proof-driven, arithmetical and rule-based. The flow of the speech broke in words, which were written and became objects. Peculiar objects, infinitely reproducible, and reproducing the being: the (written) words became forms, ideas. Also the continuous/discrete opposition and the concept of infinite belonged to the same framework. This last concept, initially reducible to a sort of 'undefinedness', became arithmetical for discrete magnitudes and entered the 'real' world with the idea of continuous in Aristotle's physics apparently to account for the potentially infinite subdivision of spatial and temporal quantities, required to deal with Zeno's paradoxes. These themes have been analysed in the second report. The 'dynamics' between the three worlds connected in the 'syntactic paradigm' has been so far the core of the western civilisation. And its terms have lasted substantially unchanged for more than two thousand years.
Almost all the logicians who studied the limitative theorems claimed their connection with the ''liar'' paradox, and it can be proved that each of the semantical antinomies can be trasformed into an incompleteness theorem. Almost all modern theoretical physicists advocated the kinship among quantum mechanics antinomies and ancient Zeno's paradoxes. Quine (QUINE 1953) in On what there is traced a parallel (''perhaps fortuitous'') between Russell's paradox and the wave/particle duality on the one hand, and between Goedel's ''incompleteness theorems'' and Heisenberg's ''indeterminacy principle'' on the other. Are these remarks to be ignored, or is there a bit of truth therein?
In this section we are going to find possible common and distinguishing features among old and new, logic and physical, mathematical and philosophical antinomies, employing the above outlined general framework. We stress that logic and physics deal with so different things, that it could be hardly believed a tight correspondence between their basic concepts, also 'negation' can not play the same role: in physical description it can not avoid to be linked to the idea of 'actual difference' rather than to its pure formal logic meaning. Anyway both disciplines are thoroughly framed in the 'syntactic paradigm', and then had to share some paradoxical features we suppose structural for the paradigm.
We already mentioned Ramsey's distinction between logical and semantic antinomies, according to the crucial involved concepts (truth, sentences, sets, proof, etc.). This distinction however did not account for the intuitive homogeneity among them. We could try to gain a better understanding by considering the general ideas of 'completeness' and 'correctness' of the representation, introduced at the beginning of the report. From this point of view, we can observe the analogy between the Greek discovery of ''incommensurability'' and Cantor's argument, curiously both including the 'diagonal' idea. Beyond this coincidence, we have to stress that both deal with the sign representation of continuous magnitudes. In both cases such an hypothesis yields a contradictory 'real' object, i.e. a square-angle equilateral triangle (of side 1 dot) in the first and a non-enumerated real number in the second. In both cases the object is 'syntactically' generated without any 'real' characterisation: this is obvious for the former. For the latter, we have only to observe that in the diagonal argument, if we represent the enumerated real numbers as segments, there is no geometrically meaningful construction to yield the 'pathologic' element, which has then got a purely 'syntactic' definition. In both cases we have an 'integer' coding in which an existing 'real' element cannot be expressed. An analogous antinomy arises in Goedel incompleteness theorems. Here, a sentence is made up, which is true in a metamathematical field, but unprovable in the formal number theory, that is accomplished in the semantic world and not accomplishable in the syntactic one. This kind of pattern < syntactic non-existence and semantic existence of the same element> is reproduced in the connected Tarski and II Goedel incompleteness theorems. The same characterisation could maybe hold for Zeno's Achilles and the tortoise paradox, and for the quantum mechanics 'incompleteness', as defined by Einstein with respect to Heisenberg's ''indeterminacy principle''. However, the opposite scheme <syntactic existence and semantic non-existence of the same element> can be argued in Russell's paradox, in which the existence of the pathological element is contradictory. The ''negative judgement paradox'' resembles the same scheme, claiming the non-existence of 'negative facts' corresponding to expressible 'negative sentences'. Also the ''liar'' could be read in the same scheme. In quantum mechanics we find both 'unintepreted' syntactic objects, as the wave-function y, and, as we saw above in the Einstein-Podolski-Rosen paradox, 'physically real' entities which cannot get a value in the formalization. In addition, the ''complementarity principle'' entails that we have real entities which cannot be simultaneously observed. That is, strongly related paradoxes seem to behave differently in our representation-based characterisation.
From a different point of view, we can try to find the common template of all these paradoxes, that we will call the ''never ending paradox''. A crucial feature of this template is that its 'purest' occurrence is in problems thoroughly deprived of any 'empirical' aspect. The reason will be enlighten in the conclusive section, and such feature entails that the 'best' form of the paradox is in logic and metamathematics, whereas its occurrence in quantum mechanics will be less sharp. To stress the connection between the different instances of the 'syntactic paradigm' in quantum mechanics and metamathematics, let us detail fig.3 (fig.8), drawing explicitly the asking (theorem/truth) aspects of the (formal/informal) theories. The vertical arrows give thus respectively the proof/satisfaction relations. The rectangle commuting property is the very idea of correct formalization or representation: in fact, we expect that a formal theorem (proved in the formal theory) corresponds to a true assertion (satisfied in the informal theory). ''Correctness'' and ''completeness'' of this coding in the usual logic meaning corresponds to the general definitions given at the beginning of the report for the ''syntactic paradigm''. We point out that, in Hilbert's approach, the metamathematical world is the informal arithmetic. The 'core' of the Goedel theorem ((MOSTOWSKI 1952),(KLEENE 1971), (SCHOENFIELD 1977)) states that the provability (i.e. the existence in the theorems world) of A(f) entails the truth of the 'decoded' proposition in the informal theory. If such proposition 'codes' the ''non-provability'' (which can be expressed in number-theoretical terms) of the formula with Goedel number f, the argument amounts to imply the non-provability of A(f), that is its non-existence in the theorem world. We point out that equivalent formulations in terms of ''(non-)weak representability'' (MOSTOWSKI 1966) or ''(non-)recursive enumerability''(KLEENE 1971) does not change the substance of the argument, but underline that here we deal with the 'formal' existence of concepts (representation, recursive enumerability) whose negation can not 'formally' exist.
As in the Einstein-Podolski-Rosen paradox, we can argue that the 'gap' in the commutativity of the diagram in fig.9 is due to the 'incompleteness' of the coding. Kleene pointed out that ''the incompleteness does not depend on the nature of the intuitive evidence'' of the postulates of the formal system, and the 'provability' predicate needs only to be ''effectively computable''. From the logic point of view, it is required the notion of ''the universal quantifier used constructively'' to formalise the intuitive theory of provability ((KLEENE 1971)). In fig.9 we show the theorem, as proved in a very general setting by Mostowsky (TARSKI 1953), in terms of non-simultaneous correct and complete recursive representability of the set of ''valid'' formulas, by a predicate ''Valid'', and of the ''diagonal function'' ''n''(n). By the hypothesis of representation of the ''diagonal function'' there exists a recursive predicate p such that p(n,v) <=> v=< ''n''(n)>, and let m = <not Valid(<''n''(n)>,x)>. Hence, for n=m, we have that |-''m''(m) <=> not Valid(<''m''(m)>). On the other hand, for the hypothesis of recursive representation of the ''provability'' predicate not(|-''m''(m)) implies not(Valid(<''m''(m)>)), and |- ''m''(m) implies Valid(<''m''(m)>). Thus, mixing the entailments, we show the inconsistency of the system. Thus, in a consistent system the ''provability'' predicate is not ''recursively representable''.
We point out that substituting in the above proof ''recursive representation'' by ''formal definition'', and ''validity'' by ''truth'', we get Tarski's theorem asserting that in a consistent system the ''truth'' predicate (for the same system) is not ''formally definable''. (fig.10)
We get a very general result:
these theorems are ... a metamathematical reconstruction and generalisation of arguments involved in various semantical antinomies. ... It applies to arbitrary formalised theories, and not only to those in which a comprehensive fragment of the arithmetic of natural numbers can be developed; to a large extent it is independent of the way in which the notion of validity has been defined for a given theory, and in particular it does not involve the notion of a formal proof within this theory; it does not use the apparatus of recursive functions...(Mostowski, in(TARSKI 1953))
In the proofs of metamathematical and logic theorems we employed at the beginning terms as 'formal system' or 'provability', which are normally connected with 'logic' and 'deduction'. Afterwards, we generalised the results to recursion theory and formal arithmetic. Now we realise that arithmetic is only the most relevant and the grounding aspect of a general 'syntactic representability', containing very few and simple elements of our logic and arithmetic language.
In the above generalised forms the essential 'ingredients' are: (i) the general scheme of the syntactic paradigm, with both an informal and a formal version of arithmetic. Among the entities of both versions, there are 'constant' and 'variable' numbers. We hypothesise an unbounded 'substitution' procedure of a variable with any constant, and the implicit relative universal quantification. (ii) among the entities of the two worlds, we can pick out (asking function) the subsets of the 'valid' elements, respectively 'true' and 'provable', and we suppose there are well-defined procedures for such selection. In the 'proof' process, there is full interchangeability between assumptions of the proof and antecedents of an implication (deduction theorem). (iii) we suppose that the entities and the selection procedures in the formal version can be somehow metamathematically represented as numbers and arithmetic computations in the informal version. This representation is supplied by the Goedel ''numbering''. To let the idea of 'provability' be not restricted to 'logic deduction', we need at least to hypothesise its arithmetic representation by a ''recursive'', and hence ''decidable'', predicate (this is the so called Church-Turing thesis). Furthermore metamathematics plays the role of a residual 'mental' world. (iv) To the formal representation we can apply the definition of 'correctness' and 'completeness' given at the beginning of the report. In general, the representation is supposed 'correct'. Otherwise the very idea of a 'syntax/semantics' pair would vanish. The second condition to accomplish the intuitive idea of a 'syntax/semantics' correspondence is that the inner rectangle 'commutes'. This substantially is a condition about the respective 'asking' functions. (v) we can employ a double 'not' (in ancient indo-european languages there were two negatives: in Greek, a 'subjective' me and an 'objective' ou). A 'semantic not' is used in the informal description of the formal version, and a 'syntactic not' is used as element in the formal entities. For both the ''non-contradiction principle'', i.e. that an entity and its negation cannot be simultaneously valid, and the ''third excluded principle'', i.e. that one between an entity and its negation must be valid, can hold. For the formal system, the ''non-contradiction'' is referred as ''consistency'', and the ''third excluded'' as ''decidability''.
Beyond these 'ingredients', the different proofs employ some crucial 'moves': <formally defining and informally representing a 'pathologic' entity>. First we have to define as unary predicate in x the ''non-provability of the diagonal function computed in x''. Its representation can then be achieved by the representability in the informal version of the 'syntactic not', the 'provability' relation and the 'diagonal function'. These are accomplished for Schoenfield's and Kleene's versions by a full use of hypothesis (iii), while for Mostowski's the first two are explicitly hypothesised, and the ''syntactic not'' is implicitly represented by the employment of the ''third excluded principle'' in the formal version. Then such number-theoretical predicate can be 'computed' in its own Goedel number, giving the 'pathological' entity expressing its own non-provability. This is always accomplished by a simple variable/constant substitution. < metamathematically representing the 'pathologic' entity>. Here, we have different cases in the different versions, in general expressing the incompatibility between consistency and completeness. In Mostowski's version the completeness of the representation allows us to 'read' the pathological entity as true if and only if it is provable. However its truth can be metamathematically 'read' as its non-provability, and thus the formal system is not consistent. In Kleene's and Schoenfield's we use the representability of the recursive sets by the 'Theor' relation and the diagonal argument to prove that the pathological entity can not be so represented.
We can also try to represent an analogous scheme of 'ingredients' for the EPR paradox: (i) the general scheme of the syntactic paradigm, with a 'classic mechanics' and a 'quantum mechanics'. We hypothesise a general formal representation in both worlds in form of 'physical law', so that the features of the representation can be applied to any entities. (ii) among the entities of the two theories, we can pick out ('asking' function) an element, the 'measurement' of an observable. In quantum mechanics, there is not a sharp border between the measuring subject and the measured system. (iii) Supposedly, we do not give an explicit theory to reduce a 'classic' measurement process to a pure 'quantum mechanic' description by the use of the Schroedinger equation. However, the idea of 'physical reality' must hold also in quantum mechanics, that is the 'mental' world had to appear in the 'physical reality' idea. (iv) The correspondence between classic and quantum mechanics is stated by the ''correspondence principle''. According to this principle, the correspondence is supposed 'correct' and 'complete', and the diagram presumably 'commutes'. (v) The 'negative' in itself does not ever appear in physical descriptions. However, the 'difference' between the discrete values of the observables holds in both systems, and the relative conservation laws are the syntactic tools to define relations between the value of an observable in two subsystem of a complex systems. The crucial 'moves' of the antinomy are the same: <formally defining and informally representing a 'pathologic' entity>. This is the 'spin measurement' in the EPR paradox: it has got both a formal quantum mechanic characterization through the wave-packet reduction and a more informal meaning in the classic measurement theory. <'mentally' representing the 'pathologic' entity>. It is in the physical model connected to the classic physics description, that the pathological measurement achieves its antinomical feature.
In the next two examples any trace of 'third world' disappear. The scheme can represent the ''liar paradox'' (see fig.11). Here, the diagram allows us to explicit the dichotomy between the language world and the real world, with the two different forms of 'truth' associated with our analysis of the 'negative judgement paradox'. In the real world, the 'truth' of a fact is recursively computed until the matching or non-matching of name (actor) and verb (action) in atomic propositions. In the language world the 'truth' of a sentence coincides with its assertion. If we claim the coincidence between non-assertion and denial, which is a form of ''third excluded principle'', then <not ''this sentence is false''> coincides with <''this sentence is true''>. The commutativity of the diagram is the ''truth as correspondence principle''.
To mirror the discussion about Goedel's incompleteness theorem, we underline the crucial 'ingredients' of the paradox: (i) the general scheme of the syntactic paradigm, with both a language and a real world. We hypothesise an unbounded formal representation in both worlds, so that the features of the representation can be applied without reference to the 'meaning' of the entities. (ii) among the entities of the two worlds, we can pick out an element, respectively a 'sentence' and a 'fact'. (iii) A sentence is also a fact in itself. And a sentence can be a meta-sentence, saying something about a sentence. Thus, a sentence can have a double 'meaning': a direct one, by the coding, and an indirect one, by saying something about the coding of a sentence. (iv) The correspondence between facts and sentences is the core of the syntactic paradigm. To this correspondence we can apply the definition of 'correctness' and 'completeness' given at the beginning of the report. The correspondence is supposed 'correct' and 'complete', and, following the Aristotelian and Tarskian 'truth as correspondence' principle, the diagram 'commutes'. (v) we can employ a double 'not'. A 'syntactic not' or 'denial' is used in the language world, and a 'semantic not' or 'falsity' is used in the real world. For both the ''non-contradiction principle'' and the ''third excluded principle'' can hold. In general, the ''non-contradiction principle'' is supposed to hold in both.
The same scheme can be reproduced for Russell's paradox (fig.12)
Here, we have to consider the correspondence between the 'syntactic' ''predicate calculus'' world and the 'semantic' ''set-theoretical'' world, defined by the relation:
x belongs to S <==> p(x), and S = { x | p(x) } where p is the predicate 'corresponding' to S. Let R = {x | x belongs to x } , then fig.12 illustrates an outline of the paradox., whose crucial features are:
(i) the general scheme of the syntactic paradigm, with both a formal and a real world. We hypothesise an unbounded formal representation in both worlds, so that the features of the representation can be applied without reference to the 'meaning' of the entities.
(ii) among the entities of the two worlds, we can pick out an element, respectively a 'proposition' and a 'set relation'. (iii) A proposition can use terms of set-theory. Thus, beyond its direct coding in set-theoretical terms, it can have an indirect meaning by the coding of its predicatively employed set-theoretical terms. (iv) The correspondence between sets and predicates is the core of this instance of the syntactic paradigm. To this correspondence we can apply the definition of 'correctness' and 'completeness' given at the beginning of the report. The correspondence is supposed 'correct' and 'complete', and thus the diagram 'commutes'. (v) we can employ a double 'not'. A 'semantic not' or 'not belonging' ( is used in the set theory, and a 'syntactic not' or 'negation' ''not'' is used in the predicate calculus world. For both the ''non-contradiction principle'' and the ''third excluded principle'' can hold. In general, the ''non-contradiction principle'' is supposed to hold in both. In the last two schemes, the antinomy appears directly in the formal development of the 'pathologic' entity, without any need of any 'mental' ('physical model' or 'metamathematics') world. These paradoxes show indeed the formal 'core' of the antinomy. It is worthwhile to remind that since its appearance, according to Frege's Grundgesetze (FREGE 1893), the paradox was linked to the unconstrained possibility of extending the conceptual coincidence to the extensional coincidence, i.e. the 'coding' in the figure.
These, and also other paradoxes, show a quite similar ground scheme centred on the peculiar characters of the syntactic paradigm and the connected 'correct representation' and 'diagram commutation' properties. Beyond the representation relation with a real entity, any formal entity is also a real entity and can be defined employing 'reality' terms. Such result can be obtained by the direct inclusion of the 'reality' terms in the 'language', or by the residual 'mental' world (metamathematics or physical models). Thus any formal entity has a double relation with the reality. This duplicity of the relation between 'language' and 'reality' was already revealed in the first report, with respect to Gorgias' antinomy. Another necessary feature for the production of the paradox is the existence of a couple of corresponding 'not''s, by which to build a 'pathologic' entity as result of the 'asking' function. This 'answered' pathologic entity roughly expresses the 'non-answerability' of its coded real entity. It admits thus a double 'coding': - a 'direct' one by the hypothesised coding relation, which, for the correctness of the representation is 'valid' and then 'answered', - an 'indirect' one by its embedding in the whole 'language' world, the following global coding of this embedding in the 'reality' world with the employed 'reality' terms left untranslated, and the final employment of the ''asking' function in this last world. Here, the 'pathological' entity, valid and hence 'answerable' for the first coding, asserts its own non-answerability. The 'commuting' property entails that these two codings had to coincide. On the contrary, for the 'pathologic' entity it does not happen: its coding has to be in the same time 'answerable' and 'non-answerable' (provable and unprovable, measurable and non-measurable, true and false, and so on in the different versions of the antinomy). This 'commutation gap' is the formal appearance of the paradox. We underline that out of the ''syntactic paradigm'' it would be impossible to utter these or similar paradoxes, and their effect is sharper in its 'strong' than in its 'weak' form.