There is a complete series of paradoxes somehow linked to self-reference, the diagonal method and the 'liar' paradox, which have been the core of the foundational debate in the last century (See (BARTLETT 1992), (SAINSBURY 1988), (SIMMONS 1993), (BARWISE 1987), (MARTIN 1984 )). In these paradoxes a crucial role is played by the new 'infinite' ideas connected with Cantor's theory. The ''diagonal method'' allows us to face different infinities, starting from the 'natural' hypothesis that the cardinalities of two sets are equal if it is possible to put them in a bijective relation. This way, we can proof that for infinite sets, the 'whole' is not necessarily greater than its 'parts'. For example, we can confront the sets of integer and even numbers and discover that they have the same cardinality, though the latter is a proper subset of the former. Analogously the set of pairs of integers (and then the set of rational numbers) has the same cardinality that the set N of integers. Calling any set having the same cardinality of N an ''denumerable'' or ''countable'' set, we can substitute in our propositions the term N with the term denumerable set. Cantor was thus able to prove analogously that the set of n-ples of integers, for any integer n, was denumerable. And hence, also the set of the finite sequences of members of an denumerable set, the set of the algebraic equations and the set of the algebraic numbers were denumerable. A crucial result was that the set of real numbers R was non-denumerable and had a greater cardinality than N. Another example of non-innumerable set is the ''set of all the sets of integers''. The abyss between denumerable and not-denumerable is characterised by the possibility of a finite expression (by a denumerable alphabet) for the elements of the set. Thus, the set of the integer functions (which not always admit a finite expression) is not-denumerable, whereas the set of integer functions represented by an expression is denumerable, the set of real numbers is not-denumerable, but the set of the real numbers computable by an algorithm is denumerable.
The same procedure allows Cantor to prove that, for any set S, its cardinality is strictly less than the cardinality of its power set P(S). If now we apply this result to the ''set of all sets'', we have the ''Cantor paradox''. In fact the power set of the ''set of all sets'' is a set of sets and then a subset of ''the set of all sets'', and hence this power-set must have a lesser or equal cardinality than the set, in spite of the above theorem.
Maybe the most important modern paradox was the Russell paradox, discovered at the beginning of our century and fatal for the Fregean logic foundation of arithmetic. In such theory it was possible to define S, the ''set of all sets which are not members of themselves''. The problem is: S belonging to S ? If the answer is yes, from the definition of S, it follows that S is not member of itself, and then S does not belong to S. If the answer is not, for the same reason, we have that S belongs to S. The argument can be set in the form of the ''diagonal'' argument, putting member(T,T') = 1 iff T belongs to T', 0 otherwise. Then we define S as build by the sets T such that member(T,T) = 0. It is evident that member(S,S) = 0 ==> member(S,S) =1, and vice versa member(S,S) = 1 ==> member(S,S) =0. This form of Russell's paradox can be put in a linguistic form, obtaining Grelling's paradox: let an adjective be called 'heterologic' if it is not true of itself. Hence 'long' is heterologic and 'short' is not. Now, is 'heterologic' heterologic? It is clear that we drop in a paradox, which has the same features of Russell's.
The range of application of the diagonal argument is very large, and comprehends also many results concerning the limit of computability in theoretical computer science. For example, within a finite alphabet the set of all Turing machines (and hence the set of all recursively enumerable languages) and the set of all context-sensitive languages are denumerable and can be enumerated by a suitable coding. We can build by the infinite matrices, whose columns are such languages and rows their coding numbers, respectively, a non-recursively-denumerable and a recursive non-context-sensitive language. Same way it is possible to prove the ''halting theorem'', that is to prove the non-existence of a Turing machine, which has as input the coding of an algorithm and its input data, and always halts answering the question whether the algorithm with those data halts or not.(HOPCROFT 1979) It is noteworthy that all these applications show a peculiar mutual matching and support, as in a sort of jigsaw, so that they seem to refer to a common deep root. To find this root, we must underline that all these paradoxes have a 'family resemblance' with the old Liar paradox: ''This sentence is false'', of presumably megaric origin. This is perhaps the most famous and it will deserve a further discussion in the following. A crucial aspect of Cantor's theory is that it allows to reduce to integer numbers any 'written' language, i.e. any set of finite strings built from a discrete alphabet (arithmetization of syntax), and this entails that the ''strong syntactic paradigm'', with the new centrality of the signs it outlines, has to set its base in arithmetic.
A crucial feature of these paradoxes is the occurrence of an ''impredicative definition'', i.e. the definition of a set containing an element whose definition depends on this set. Poincare' ascribed to these definitions the source of the paradoxes, and Russell characterized them as 'a vicious circle', proposing his ''ramified theory of types'' as a solution. In this theory the ''individuals'' are assigned to ''type 0'', their properties to ''type 1'', their properties of properties to ''type 2'' and so on. Moreover, types beyond 0 are divided into orders: properties (type 1) defined without any reference to any totality belong to order 0, and, recursively, properties defined employing totality of properties of order s belong to order s+1. However, this approach, though it works, is nevertheless problematic, for the presence in mathematical analysis of those same impredicative definitions it gets rid of. For example, the definition of the ''least upper bound'' of a set of real numbers, according to Dedekind theory, employs this kind of definition. To avoid this difficulty, Russell had to introduce the ''axiom of reducibility'', asserting that for any property of order s>0, there is a coextensive property of order 0. This solution is sharply ad hoc and is a doubtful answer to the questions entailed by the paradoxes. Ramsey proposed to distinguish between logical or set-theoretical antinomies, as Russell's or Cantor's, and linguistic or semantic antinomies, as ''the liar''. According to Ramsey, Russell's proposal could be used to solve the 'logical' paradoxes, but it would be useless for the 'linguistic' ones. To solve these, the most famous approach is the Tarskian distinction between ''language'' and ''metalanguage''. In Tarski, as in Goedel, the ''liar'' paradox is not actually 'something to solve', as rather the hint of a deep problem in natural as in formal languages. In fact, it arises in any possible formalization of a given language (consistent and including arithmetic), in which the 'truth' of sentences of the language is expressed as a predicate of the Goedel numbers of the sentences in the same language. The natural languages, according to Tarski, are not ''semantically universal'', for they can not contain their own 'truth' predicates. To allow the expression of semantical concepts, the truth of sentences of the ''language'' has to be expressed in a richer ''metalanguage''. It is nevertheless difficult to accept Ramsey's distinction, for the close connection between the two kinds of paradoxes, enlightened also by the similarity between Russell's ''ramified theory of types'' and Tarski's ''truth theory'': Kneale (KNEALE 1962) pointed out that Tarski's theory can be derived by the ''theory of types'', and Church (CHURCH 1984) underlined that Russell's resolution is a special case of Tarski's.
Tarski solution's 'negative' aspect has been however felt insufficient by many authors, and in recent years there have been other proposals to solve the paradox, trying somehow to unify all the Tarski hierarchical languages in an unified one. The most famous is in Kripke's 'Outline of a theory of truth' (KRIPKE 1984). Here, to avoid the introduction of metalanguages, charged to be not corresponding to the reality of natural languages, Kripke defines, for any i, two monotone class of sentences Si and Ti. In the first class there are the ''true'' sentences, and in the second the ''false'' ones. Such classes are inductively defined, and, for any i, Si and Ti are obviously disjoint (for the non-contradiction principle), but they are not exhaustive (the middle excluded principle is not true) and the complement of their union is a set of sentences which are neither true nor false (the so called ''truth gap''). The smallest fixed point of these two sequences is ''the most natural model for the intuitive concept of truth'' in natural languages. Other authors addressed theories in which the truth value of the ''liar'' sentence was not stable, or variable depending on the context, or not a proposition at all. However, although it is not our aim to deal with the problem of 'solving' the paradox, these proposals are weak for two kinds of reasons. First, for they give answers in formal models which are far from being suitable to deal with natural languages reality, but only with a caricature of it. Second, for they do not seem to be able to cope with other reinforced versions of the paradox, as those given for example by the so called ''revenge liar'', which can be defined for each of these proposals. So, for example, for the ''truth gap'' proposal the ''revenge liar'' is the sentence: ''This sentence is false or neither true nor false'', for the solutions unifying the Tarskian hierarchy, it is: ''This sentence is not true at any level of the hierarchy'', and so on (SAINSBURY 1988). It is noteworthy that these two reasons of weakness are 'synergetic', for it is the very idea of introducing new formal terms (metalanguages, truth gaps, fixed points, etc.) in the natural languages to solve the paradox saving the semantical universality, that allows the formulation of the 'revenge' form. It somehow reminds the property of Goedel's 'pathologic' sentence to be redefined whenever we introduce its earlier form as an axiom. Then the 'liar' seems to be 'essentially' paradoxical.
In Poincare' (POINCARE' 1914) the criticism involves then the whole Cantorian construction: '' I think...that the important thing is never to introduce any entities, but such as can be completely defined in a finite number of words.''(45) For him we cannot employ those impredicative definitions, for they employ an actual infinity, which cannot actually exist. In Poincare' the core of the mathematical infinite is in the principle of complete induction and in the intuition of the potential infinite it gives us. It is irreducible to logic, which ''sometimes breeds monsters'' ((POINCARE' 1914), 125). To understand these remarks, we must not forget that Russell had to introduce as an hypothesis (the ''axiom of infinity'') the existence of the series of the natural numbers.
In these different points of view we can recognize the differences between logicism (Frege, Russell, Whitehead) and intuitionism (Poincare', Brouwer, Heyting, Weyl). The former tried to reduce mathematics to a branch of logic, and the latter advocated that logic, conversely, was abstracted from the mathematics of finite sets. This 'supremacy' dispute entailed relevant differences in crucial points of mathematics and logic: the above problem about the infinite is an example. A fundamental logic principle as the law of the excluded middle (and the related double negation elimination principle) was not accepted by Brouwer when applied to infinite sets. In general, the intuitionistic methods are constructive. So, if we want to prove the existence of a natural number 'n' satisfying P, it is not sufficient to show that the assumption that ''for all numbers n, P(n) is not true'' entails a contradiction. We have rather to give an explicit method by which to find such a number 'n'. For them, logic is simply a set of general properties of mathematical origin, which have received a special status for their generality, but that cannot work beyond the limits of the mathematical intuition. Beyond the role of logic and some proof techniques linked to the existential quantifier and the negation, the dispute between intuitionists and logicists was centered on the role of arithmetic and the potential nature of the infinite, and it is difficult to deny that the flourishing of paradoxes and limitative theorems supported their good reasons. However, despite of the substantial support their theses received in the development of modern logic, there remains a weakness in their program, in its purely 'restrictive' dealing with the rival theses regarded only as 'unlawful moves', and the consequent effort of proving the same results with longer proofs. To find at most the same results with greater efforts was not so appealing for most of the mathematicians who found in Cantor, Russell and Hilbert, in formal rigor and axiomatization, often in a purely formal activity, their optimal background, despite of the fertility of the intuitionists' ideas.
Logicism found the sense-presupposition of the logic-mathematics formalism, thus grounding their 'meaning' and 'being', in set theory, and then it is not strange that, notwithstanding Russell's ''type theory'', it did not actually survive in its pure form to the set-theoretical paradoxes. However, somehow it 'met' with platonic and set-theoretic approaches in a sort of 'common sense' of mathematical practice, which answered for the quiet requirement towards the paradoxes with the original Parmenidean solution: to find something fixed, and then 'contradiction-free', as the root of being, and then the warrant for the mathematical work. In the Introduction of his Foundations of arithmetic, Frege (FREGE 1884) underlines that the absence of something fixed in the uninterrupted flux of all the things, would destroy any chance of knowing the world and everything would fall in a great confusion. However, the real philosophical debate, at least in my opinion, during the period between the two world wars, was between the intuitionists and the formalists (Hilbert).
Formalism tried to save the new mathematics of Cantor and Dedekind from the above paradoxes and the intuitionistic criticism as well. The most common interpretation of formalism is a sort of 'reductionism' of the whole mathematics to pure formal deductions from any set of axioms, maintaining that for this approach there is nothing to say beyond the set of the employed axioms: a simple 'proof machine'. A more correct reading of Hilbert's works ((HILBERT 1925), (HILBERT 1927)) shows that his approach agreed with the requirement of an 'intuitive' finitistic core for the mathematics enterprise, but that he believed possible and necessary the extension of mathematics to those 'ideal' elements, axiomatically and consistently definable, whose formal existence gave the mathematics an elegance and a perspicuity unattainable in the pure 'intuitive' framework. His examples most of all concerned with the ''actual infinite'': e.g. the addition of the ''limit points'' to the Euclidean geometry allowed to create the Projective Geometry, whose formal elegance and often also perspicuity was far greater than the original 'intuitive' Euclidean core. Hilbert can be counted in the short list of thinkers believing that the mere existence of 'something formal' in the realm of signs warrants some sort of 'real' existence for such 'something'. Another classic example of mathematical ''actual infinite'' could be found in set theory, which then had to be studied in a purely axiomatic framework (Zermelo). This way, the only accepted sets are those built according with the set-theoretical axioms, and such axioms can be tailored to avoid the existing paradoxes. Obviously, it would be also necessary to warrant that such axioms can not produce any other inconsistency. The definition of a model, by which to interpret an axiomatic theory in another axiomatic theory, can only reduce the question of the consistency of the first theory to the same question for the second, and so underline the problem of the consistency for the most fundamental theories, most of all arithmetic. There is, however, another way Hilbert suggested possible to follow: to prove that some obviously false proposition (e.g. 0 not equal to 0) can not be derived in the 'object theory'. For, in an inconsistent theory any proposition can be proved, and then, if there exists an unprovable proposition, the theory cannot be inconsistent. For any axiomatization this consistency proof had to be developed in the 'intuitive' mathematical core. To this aim Hilbert needed a theory employing only ''finitary'' methods, i.e. employing ''only intuitively conceivable objects and performable processes'' ((KLEENE 1971), 63), about proofs in axiomatic theories, the so called ''proof theory'' or ''metamathematics''. The full extent of Hilbert's program was vanished by Goedel's theorems, but it opened a large field of researches about the foundations of mathematics and logic. We could say that Hilbert's mistake, as Colombo's, was to believe to have found a new road for an old world, where instead he had discovered a brand new one.
In his work we can find the thoroughest outline of the evolution of the ''syntactic paradigm'' from its 'weak' to its 'strong' form, which deserves an extended quotation:
No more than any other science can mathematics be founded by logic alone; rather as a condition for the use of logical inferences and the performance of logical operators, something must already be given to us in our faculty of representation, certain extralogical concrete objects that are intuitively present as immediate experience prior to all thought. If logical inference is to be reliable, it must be possible to survey these objects completely in all their parts, and the fact that they occur, that they differ from one another, and that they follow each other, or are concatenated, is immediately given intuitively, together with the objects, as something that neither can be reduced to anything else nor requires reduction. This is the basic philosophic position that I regard as requisite for mathematics and, in general, for all scientific thinking, understanding and communication. And in mathematics, in particular, what we consider is the concrete signs themselves, whose shape, according to the conception we have adopted, is immediately clear and recognisable. This is the very least that must be presupposed; no scientific thinker can dispense with it, and therefore everyone must maintain it, consciously or not.(464-465) What the physicist demands precisely of a theory is that particular propositions be derived from laws of nature or hypotheses solely by inferences, hence on the basis of a pure formula game, without extraneous considerations being adduced. Only certain combinations and consequences of the physical laws can be checked by experiment -just as in my proof theory only the real propositions are directly capable of verification. ((HILBERT 1927), 475)
This text is noteworthy, for here we see the 'mental' world, that in the 'intuitionist' view remained in the form of the intuitive 'Kantian a-priori', i.e. number theory, as progressively shifting in the finitistic metamathematical function and in the pure 'syntactic' skill of sign manipulation. For Hilbert
...mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincare', or the primal intuition of Brouwer, or finally, as do Russell and Whitehead, axioms of infinity, reducibility or completeness, which in fact are actual, contentual assumptions that can not be compensated for by consistency proofs. ((HILBERT 1927))
The signs are for the mathematicians 'workshop tools' indeed, and their role is enhanced by their intersubjectivity: thus the 'numbers-signs', and not the 'numbers-ideal objects' or the 'numbers-mind acts', can ground arithmetic. The paradigm is extended also to physics, which Hilbert knew very well, and in the above quotation we find also the parallel, in their 'asking' role, between ''theorems'' in mathematics and ''measurements'' in physics, on which we shall return in the following. Before ''Goedel's incompleteness theorems'', Hilbert thought his proof theory as a 'silver bullet', suitable to ground finally the mathematical enterprise: ''in der Mathematik gibt es kein Ignorabimus'' (28-71 18), which reminds Viete's ''Nullum non problema solvere'' (KLEIN, 185). Arithmetic is for both the real ground of any formalization, probably grounded on an instinctive notion of ''an indefinite repetition of the conception of entities'' and with a set of connected simple properties (among them the idea of ''equality'' between natural numbers is credibly the first). And this 'built in' structure is thoroughly intersubjective, and thus suitable to ground the human scientific enterprise. Thus Heyting ((HEYTING 1971),4) can ''see the difference between formalists and intuitionists mainly as one of taste'', being the intuitionists simply not interested ''in the formal side of mathematics''. In Weyl's characterization of the constructive-symbolic knowledge:
a) The result of given operations on a fact (Gegeben), ...which is generally accomplishable, is, insofar as it is univocally determined by the fact, set as a sign (Merkmal) concerning in itself the fact....b) By the introduction of signs (Zeichen) an analytical translation (Aufspaltung) of the judgment (Urteil) is accomplished ... and a part of the operations, independent from the fact and its persistence, is made by the signs manipulation (Verschiebung)....g) The signs are not born (hergestellt) only for the present fact, but are set (entnommen) as potential supplies (Vorrat) of an ordered variety of signs (Zeichen) bearable by fixed procedures and open toward the infinite. ...d) It remains however unforeseeable which formulas can be derived in the 'proof playing' (Beweisspiel): we do not get the truth, it must be found by hand (durch Handeln) in each single case. It depends on that by the syllogism from two formulas a third one, which is shorter of one of the antecedents, can be derived (hergeleitet), and hence in the proofs expansions and contractions of formulas alternate.(28-30)
only the d) issue could not be endorsed by a formalist. However other differences occur in the theoretical apparatus: thus, the existence of mathematical entities is simply 'to be constructed' in intuitionists' approach, and 'to be consistent' in formalists' one. The 'place' of such existence is respectively ''in the mind, as mental constructions'' and ''on the paper, as symbols''. The last source for the scientific truth is respectively in the ''evidence'' and in the ''coherence'' (and, at the same time, before Goedel's results, in the ''proof''). Brouwer pointed out in his 'On the significance of the principle of excluded middle in mathematics' (1923) that formalist critique started from the repeated contradictions encountered in mathematics, aiming to subject ''the language accompanying the mathematical mental activity ... to a mathematical examination. To such an examination the laws of theoretical Logic present themselves as operators acting on primitive formulas or axioms'', and we must transform the axioms so that ''...the linguistic effect of the operators mentioned (which are themselves retained unchanged) can no longer be disturbed by the appearance of the linguistic feature of a contradiction'', but ''...an incorrect theory, even if it can not be inhibited by any contradiction that would refute it, is nonetheless incorrect'', whereas in his On the foundations of logic and arithmetic (1904) Hilbert advocates that, in developing a theory, ''a further proposition is true as soon as we recognise that no contradiction results if it is added as an axiom to the propositions previously found true''. The difference exactly amounts to the distance between the pure syntactic signs-manipulation and the thoroughly semantic numbers-manipulation, and, as pointed out by Weyl, in his Comments to the aforementioned lecture of Hilbert (HILBERT 1927), the opposition between the 'purely phenomenological' spirit of intuitionism, and a reduction of the evolution of the mathematical performances to the practical or historical success advocated by the formalists.
In 'On what there is' (QUINE 1953) Quine established a comparison between these three philosophies and analogous Middle Ages schools. Logicism could play the role of ancient realism, which advocated the existence of universal and abstract entities independently of the mind, founding mathematics on logic. We could also underline the presence of a tradition, from Leibniz to logic-positivism, of which logicism is a part. In Leibniz, for the first time, there was the centrality of a 'formal' science, characteristica universalis, as the core of the natural science. This science of combinations, ars combinatoria, revealed the inner structure of mathematics as science of signs, and not only quantity, founded on the non-contradiction and identity principles. Hence, mathematics truth was analytic. Its connection with the world was warranted by a ''preestablished harmony'' between ideal and real world. We could say that Leibniz claimed the logical structure of the world and the syntactic nature of the scientific reason as necessary for the human knowledge, whereas God could directly see the truth without any syntactic tool.
In the Quinean analogy intuitionism recalls conceptualism, according to which the ''universals'' exist but are mind-made. However, the most important reference for the intuitionists is Kant. In fact, for Kant arithmetic and geometric truths are a-priori and synthetic. That is, such propositions neither can be derived by the experience nor can find in their form, as the logic principles, their reason of truth. Arithmetic and geometry cannot then be founded on something else, for they belong to the apriori conditions of our knowledge. Arithmetic is the scheme of our relationship with the time (inner sense), geometry analogously with the space (outer sense). However, arithmetic and geometry do not play in Kant a symmetric role, and the geometrical representation appears somehow central in the apriori criticist foundation of science. This feature accounts for the underevaluation of the symbolic in Kant. The intuitionists in fact will underline only the arithmetic apriori: this change in the XIX century from the geometric to the numeric centrality, in Kantism and in mathematics as well, reminds the analogous one among Pythagoreans from Philolaus to Archytas, from the geometric non-syntactic proof procedures to the Euclidean Elements. In our century the Kantian legacy has been also renewed by Cassirer(CASSIRER 1923), who underlined the apriori role of the symbolizing human activity, which does not come from the world, but constitutes it.
Finally, formalism corresponds to nominalism, refusing of admitting abstract entities at all, beyond the signs. It can be also connected to an old conventionalist and empiricist tradition, and, for this approach, it is the thoroughly artificial nature of the signs which explains their effectiveness. Berkeley underlined that without signs, arithmetic, as a pure verbal science, would be completely useless. However, the a-priori role of signs manipulation, at least in Hilbert, seems to depict a brand new perception of mathematics which recalls the general linguistic turn of the XX century. Weyl in his ''Philosophie'' sets out a similar analogy: ''set theory'' is a sort of ''naive realism'', ''intuitionism'' a sort of ''idealism'', and ''formalism'' a sort of ''transcendental realism''
As told before, the modern paradoxes refer to the megaric ''liar'' paradox, and in its analysis we can recognize the two different aspects of the idea of truth we pointed out in our analysis of the ''negative judgment paradox''. A better understanding of this connection will be got in the following. It is useful however to give now an outline of the argument. In fact the truth appears both as something that can be said, and functionally analyzed, about an argument ('this sentence'), and as the lawfulness of asserting a (true) sentence ('this sentence is false') as an accepted one. And in both cases it has to be used together with the negation, and the negation is employed as both subjective and objective (see the first report for this distinction in pre-Platonic Greek). If these two aspects are the same we underlined recalling the analysis of the ''negative judgment paradox'', why did the ''liar'' paradox survive to the Aristotelian 'solution'? Regarding the functional aspect of truth, we have to remind that the solution was linked to the architecture of the syntactic paradigm. There we defined recursively the truth of complex sentences, until the Forms' blending or unblending for the simplest sentences. Now, this solution cannot work now, for it had to be applied to 'this sentence' in verifying its correspondence to an unblending of Forms. Simply 'this sentence' cannot be analyzed as a subject/verb pair, and hence is not a 'sentence' at all! Regarding the assertive aspect, the solution was based on the mind/reality dichotomy. In our paradox it should mean that we have to deal with the subjective denial of an existing fact. However, 'this sentence' is not an 'existing fact', and in our language the ancient distinction subjective/objective negative is lost.
The reducibility of these paradoxes to an unified usage of different aspects is sketched also in (WITTGENSTEIN 1956), referred to Russell's and Grelling's paradoxes. In these paradoxes from a definition as : heterological (p) iff not p(p), setting p = heterological, we get the contradiction: heterological (heterological) iff not heterological (heterological).
Wittgenstein writes, apparently about the different roles played by the substitution '' The one time is the unabbreviated assertion, the other time the assertion abbreviated according to the definition.''(III.79) Even though the point is not thoroughly clear, it can become more precise, comparing it with the ''liar''. Here, we consider the truth as the lawfulness of asserting: true(p) means to assert 'p', and as the syntactic recursive definition: 'true(not(p))' means 'false(p)'. If p = 'false(p)', we get by substitution true(p) = 'true(not(p))', accomplishing two different kinds of substitution, the first as a simple assertion and the second as a recursive definition, as asserted by Wittgenstein. In addition, it is quite straightforward to recognize the correspondence of these aspects with the assertive and functional aspects of the truth embedded in the Aristotelian foundation, and to observe that the contradiction arises only if we let the two different truths coincide. The ''liar'' and the analogous paradoxes seem then to be the ruins of the ''negative judgment paradox'', which have been forgotten, but as riddles, for centuries, until the progressive vanishing of the 'mental' world in the ''syntactic paradigm'' has given them again a crucial role.