Draft



Univ. "Federico II", Monte S. Angelo, 80125 Napoli, I,



1. Introduction



In the following I will illustrate a new interpretation of Popper's philosophy of science. It will result from a specific analysis of his texts which will quickly reconstruct its historical development. I will proceed by exploiting a result of my historical analysis on classical scientific theories, i.e. the great relevance of double negated statements which are not equivalent to the corresponding, positive statements (=DNS), in order to perform a penetrating analysis on the original texts pertaining to both a scientific theory and a philosophical theory.



In sect. 2 I will present the basic tenets of Popper's criticism to Vienna Circle (=VC). Then, in sect. 3 I will present his trust in classical logic although in his times this kind of logic was questioned by new logical attitudes, In sect. 4 I will illustrate the relevance of DNSs in order to interpret both a scientific theory and a philosophical theory. In sect. 5 I will scrutinise Popper's writings in order to collect his DNSs playing a crucial role in his arguments. In sect. 6 I will reconstruct by means of Popper's DNSs his discussion of the demarcation question, except for some meaningful differences which will be examined in sect. 7. Indeed, owing to Popper's distrust in non-classical logic, he was not faithful to a consistent use of non-classical logic; in particular, about the "symmetry" issue (sect. . To investigate more accurately Popper's position on logic, in sect. 9 his studies about "the new foundations of logic" are quickly examined; they confirm his distrust in any alternative to classical logic, although he gives room to non-classical logics. In sect. 10 Popper's actual uncertainty on the kind of logic is traced back to his unawareness of the incommensurability phenomenon underlying the foundations of various kinds of logic.



The conclusions propose a new collocation of Popper's philosophy inside the history of the philosophy of the 20th Century.



2. Popper's criticism to logical empiricism



K. Popper introduced a critical attitude on contemporary widespread certainties and wishful guesses. His analysis stressed a main point. According to the tradition of positivism, VC maintained that ingenuous induction is capable to achieve a physical law as a decisive conclusion of a process of cumulation of experimental data. To this basic tenet VC added mathematical logic and its rigorous deductive method. VC hoped to conciliate the two tenets by subsuming induction to mathematical logic and it devoted much energies for this aim.



Instead, in Popper's opinion, scientific research does not achieve a definitive truth by cumulating observational data, as naive positivism had claimed. Popper recalls that, according to classical Hume's criticism, a general law never can be supported by a finite number of experimental data. In addition to Hume's criticism he produced a new argument, pertaining to just that formal meta-theoretical level that neo-positivists had chosen, i.e. mathematical logic. According to him, mathematical logic does not offer a formalisation for induction process. In opposition to all previous justifications for induction, Popper concluded the question as if it allowed a yes-or-not solution: "… there is not such thing as induction"(1).



However, it is manifest that science is alive. Popper suggests that it proceeds through a long search by trial-and-error, whose results are eventually established in an inter-subjective way. Then, in order to investigate as formal steps the inner steps of scientific research Popper appealed to a new meta-theory of science, i.e. a probabilistic theory of a new kind, whose formalism had to generalise the formalism of mathematical logic.



But this rather vague characterisation of scientific initiative is then vindicated by offering a solution to the problem of the demarcation of a scientific sentence from metaphysics ("Kant problem", in Popper's words). In his autobiography he recalls that the failure of classical mechanics caused by the birth of special relativity suggested to him a new attitude about science as an essentially fallible construction(2). Only what is fallible by means of experimental data has to be accepted as science. According to him, "No more science is certainty". Hence, no decidability of a scientific sentence, if not in a negative sense; i.e. to discard it when confronted with negative results from experiments. His attitude may be called a fallibilistic rationalism.



A synthesis of the above criticisms is given by the following table.



Table 1: IDEAL STEPS BY POPPER TOWARDS A FALSIFICATIONIST THEORY



Warning: the table may not retain its integrity in the Forum and I may need to re-post the article after some more editing.



Received theses, denied by Popper

A blind certainty in science

A scientific sentence is obtained by induction

All scientific sentences can be decided in an empirical way

Science is certainty



Popper's conscious step

A physical theory may fail! (Newton/Einstein)

Induction cannot give conclusive empirical evidence

Demarcation problem: falsificationism rather than verification

Science is a historical process of testing previous results



Although in a philosophical way only, he had introduced an attitude favourable to undecidabilities in science. Instead in his times, Hilbert's dictum: "Non ignorebimus!" led most mathematicians to consider as impossible an undecidability. Hence, Popper had to overcome a hard barrier of incomprehension. It was Goedel's theorems that forced scientists to accept undecidable assertions in the very core of the most hard science (3). Afterwards, it became apparent that Popper had already represented a radical change in the basic attitudes on foundations of science, now forced to recognise impossibilities in logic, mathematics and science in general.



3. The role played by classical logic



It is reasonable to conclude that his philosophy represents a way out a common paradigm in philosophy of science. But which kind of new philosophical attitude?



According to a widespread pre-conception of these times, theoretical physics was the most advanced branch of science. According to logical neo-positivists too, physics was the first branch of science to be investigated in its very foundations. They focused the attention on classical mechanics, or little more; as if both electromagnetism and thermodynamics little had changed in the foundations of theoretical physics. Moreover, they devoted little attention to the crises on the foundations of theoretical physics occurred in the first years of 20th Century. Popper's philosophy was more advanced than logical neo-positivism also because it took in account the recent crisis in the foundations of Physics. In his autobiography, many times Popper remembers that his attitude originated from the failure of Newtonian paradigm. It was the replacement of classical mechanics by special relativity that suggested to him a new attitude about science, according to which science is an essentially fallible construction (4).



Let us consider in addition the status of the foundations of Logic in his times. Both logical neo-positivists and Popper did no see any problem in the foundations of logic, since the only one logic recognised in that times was classical logic. In the opinion of both logical neo-positivists and Popper it enjoyed an unquestioned status of a meta-theory assuring certitudes.



For instance, logical neo-positivists claimed for a full decidability of all scientific sentences. Popper concluded that induction does not exist. Both decisions presuppose the full validity of the law of excluded middle (LEM), i.e. classical logic(5).



In fact, several times Popper stated his trust in classical logic. His theory "… might be described as the theory of the deductive method of testing…";(p. 30) "… our world of experience… is to be distinguished by applying to it that deductive method which it is my aim to analyse, and to describe." (p. 39) "… the method of falsification presupposes no inductive inference, but only tautological transformations of deductive logic whose validity is not in dispute." (p. 42).



In Popper's main fight against induction, his main argument was to remark that in classical logic there is no room for obtaining an induction, except for a negative induction: "… note that there is only one type of argument which proceeds in an inductive direction: the deductive modus tollens, " (p. 314) which he always qualifies as belonging to "classical logic" (p. 41, 76). Hence, he did not even suspect a representation of induction in a different kind of logic(6).



These facts manifest a common trend in both VC and Popper for giving to classical logic an absolute value. Owing to this certainty, both neo-positivists and Popper devoted little attention to contemporary debate on the foundations of both Logic and Mathematics. Already around 1900, as a reaction to formalist attitude, Brouwer started a program for new foundations in both Mathematics and Logic. In Logic, by rejecting the formalist appeal to a full deductive method, he denounced the "unreliability of axioms"(7). In 1918 Lukasiewicz gave the celebrated farewell speech as a Rector of Warsaw University, on the "spiritual war" to be fought against the oppression by Aristotle's deductive method, based upon classical logic ( .



Brouwer's hint about a new Logic slowly produced in '30s (Heyting) a formal system which recognised some intuitionistic counterparts to classical logical laws. In the same years some authors (Glyvenko, Kolmogoroff, Goedel) showed that the two kinds of logic are mutually irreconcilable; then it was possible to recognise this "deviant" logic on the same foot of classical logic.



4. The relevance of double negated sentences in scientific literature



The words included in square brackets evidentiate that in each occurrence of a DNS the double negation law fails; i.e. dropping out the couple of negations results in a positive statement which is not equivalent to previous one, since the latter one is not a scientific statement for the lack of experimental evidence. For example, let us consider the following statement: "… a disagreement, as he was the first to stress, would show his theory to be untenable." The corresponding positive statement corresponds to VC thesis, which is contested by Popper.



Since 1989 I found out a list of similar DNSs in the original writings belonging to some scientific theories, i.e. classical chemistry, L. Carnot's calculus geometry and mechanics, S. Carnot's thermodynamics, Avogadro's atomic theory, Galois' theory, Einstein's special relativity (strategists' theories too, by Sun Tzu, L. Carnot, Clausewitz, and even S. Freud's psycho-analysis) (9). In some cases the author is so consistently arguing through DNS's and moreover the number of DNSs is so high to be enough for a re-construction of the essential content of his text; in order to explore this kind of reconstruction, one has to merely extract all DNS's and then grouping them in units of arguing, each one solving a specific problem, stated by means of a DNS, of the theory at issue.



The best instance of this kind of re-construction is represented by S. Carnot's book on thermodynamics, whose first part was entirely and deliberately written in a discursive way, except for a mathematical footnote (10). An analysis on this text gave 67 DNSs; the units of arguing are 7. This kind of analysis produced the first detailed interpretation of the theoretical part of his decisive book (11). Unfortunately, the subsequent formulation of the same theory by Clausius and Kelvin, who strangely enough added a first principle before the previously discovered second principle, obscured the investigative character of S. Carnot's thermodynamics; which, more than other physical theories, presents the characteristics features attributed by Popper to a scientific theory.



5. The origins of Popper's philosophy



Let us examine Popper's philosophy by means of this richer framework than he could recognise at his times. Was this philosophy influenced by this crisis in traditional logic? Where Popper saw an unique way, seventy years of investigations lead us to recognise divergent lines of development in Logic. Recent works on non-classical logic emphasised that the very borderline between intuitionistic logic and classical logic has to be put on the validity of the law of double negation rather than the law of excluded middle (12). That means that this borderline severs the two kinds of logic as being two entirely different world.



The change of emphasis from LEM to DNL in differentiating classical logic from non-classical logic is remarkable under one more aspect. Inside original scientific writings a failure of double negation law is easier than LEM to be recognised.



At a first glance, negated words (e.g., "fallibilism", "a never ended process", etc.) play a crucial role in Popper's writings; so that someone called Popper's philosophy a "negative epistemology" (13). This phenomenon is confirmed by an accurate analysis on Popper's writings, mainly the description of the origin of his main ideas.



In the autobiography Popper describes by means of the following negations his intellectual reaction to Einstein's criticisms which changed past framework of theoretical physics:



"No doubt, Einstein had all this, and especially his own theory, in mind when he wrote in another context: "There could be no fairer destiny for any physical theory than that it should point the way to a more comprehensive theory, in which it lives on as a limiting case." But what impressed me most was Einstein's own clear statement that he would regard his theory as untenable if it should fail in tests. Thus he wrote, for example: "If the redshift of spectral lines due to gravitational potential should not exist, then the general theory of relativity will be untenable."



Here was an attitude utterly different from the dogmatic [= not scientific] attitude of Marx, Freud, Adler, and even more so that of their followers. Einstein was looking for a crucial experiments whose agreement with his predictions would by no means establish his theory; while a disagreement, as he was the first to stress, would show his theory to be untenable.



This, I felt, was the true scientific attitude. It was utterly different from the dogmatic attitude [by VCs] which constantly claimed to find "verifications" for its favourite theories.



Thus I arrived, by the end of 1919, at the conclusion that the scientific attitude was the critical attitude, which did not look for verifications but for crucial tests; [negative] tests which could refute the theory tested, though they could never establish it." (14)



Let us remark that in the above texts the double negated statements constitute the most appropriate linguistic and logical phenomenon. Now the reader is invited to read again the above double negated statements ØØA about Einstein's revolution and the origin of Popper's main ideas, by addressing his attention to the contents of square brackets, each one of them suggests the sentence A; it results to be not equivalent to the double negated sentence ØØA:



"No doubt… [¹ it is true]

… no fairer destiny for any physical theory than [¹ a fair destiny for a physical theory is]

… he would regard his theory as untenable if it should fail in tests… [¹ as confirmed if it should proved by]

… not exist, then the general theory of relativity is untenable… [¹ If the redshift… should exist, then the general theory of relativity will be certain]

... different from the dogmatic… [¹ Here was an open attitude]

… a disagreement […] would show his theory to be untenable… [¹ an agreement… would show his theory to be certain]

.... different from the dogmatic attitude… [¹ it was an open attitude]

… [negative] tests which could refute the theory tested… [¹ positive tests establish]."



Apart the first one, the subsequent two double negations are very meaningful for Popper's philosophy. The whole first period makes apparent that he commonly argues in double negations. Then he reiterates double negations in a value judgement too, expressed two times. However, when illustrating his demarcation criterion, he always presents DNSs, except for the last statement, where the second negated word, i.e. "negative" is implicit and to be added for making more meaningful the sentence. By this minor integration, the whole text becomes perfectly consistent in itself and with Popper's basic ideas as represented by a list of double negated statements (=DNSs) (15).



In sum, for our purposes of interpreting Popper's philosophy the choice on the kind of logic is very relevant. In a first approximation, this choice may be considered as a dichotomic variable, being either classical logic or non-classical logic, in particular intuitionistic logic.



6. Interpretation of demarcation in science by means of a double negation sentence



Now I want to re-construct the intuitive path leading Popper to build his philosophy of science by relying upon mainly DNSs. Implicitly Popper appealed to non-classical logic even when he wanted to solve the main problem of his philosophy, i.e. to offer a demarcation criterion. We can resume his critical points by means of the following table illustrating his passage to the negation of VC theses and eventually to his own theses expressed by DNSs.



Table 2: COMPARISON BETWEEN VIENNA CIRCLE's THESES AND POPPER'S ONES



Warning: the table may not retain its integrity in the Forum and I may need to re-post the article after some more editing.



Vienna Circle: VERIFIABILITY

We can establish by means of observational data a scientific sentence, obtained through

induction

We cannot prove by means of observational data (and induction too) a metaphysical sentence





Popper: FALSIFIABILITY

Modus tollens only can represent an induction path; hence, in opposition to Vienna Circle's, an

all-or-none conclusion in experimental sciences: "Induction does not exist" (it is no more

useful)

Moreover, we cannot establish by means of observational data a not scientific sentence

But also we cannot establish a scientific sentence by means of a finite number of observational

data

Rather (Einstein's influence): We can disprove a scientific sentence by means of negative

observational data





We recognise the following steps from the positive sentence of VC to further Popper's DNSs upon science. VC denied any validity to metaphysical sentences and it wanted to assure full empirical evidence to all scientific statements. On the other hand, a sentence is metaphysical (i.e. non-scientific) in nature when it does not enjoys empirical evidence. The latter sentence is just the double negated of previous one. Inside classical logic the two statements are equivalent; in fact they circumscribe a same idea, one time from the inside of scientific realm, and the other time from the outside.



Rather, Popper considered VC positive statement as lacking of decisive support; then he suggested the double negated sentence of VC basic thesis; i.e. he suggested to qualify as metaphysical, i.e. non-scientific in nature, those sentences which cannot be rejected by negative empirical evidence. Let us remark that even in intuitionistic logic three negations are equivalent to a single negation; hence, previous statement qualifies non-scientific sentences. Instead, according Popper a sentence is a scientific one when it can be disproved by negative experimental results. This sentence is the double negated of VC sentence. The former VC thesis is obtained by dropping out the two negations in Popper's thesis. If Popper was in classical logic, this statement would be equivalent to VC sentence; only because Popper here implicitly assumed non-classical logic, his sentence is radically different from VC sentence and VC sentence is not implied by his sentence. Hence, it is just the radical difference in the kinds of logic that gives reason of the radical solution Popper gave to Kant's problem, in opposition to VC theses. In fact, he staunchly stressed the radical difference between his own thesis and VC one.



The great number of double negated sentences occurring in Popper's description of his enlightening ideas prove, according to the above recalled studies on foundations of Logic, that Popper was arguing in a non-classical logic. Let us consider for instance his characterisation of science:



"… three requirements which our empirical theoretical system will have to satisfy. First science must be synthetic, so that it may represent a non-contradictory, a possible world. Secondly, it must satisfy the criterion of demarcation…, i.e. it must not be metaphysical… Thirdly, it must be a system distinguished in some way from other such systems, as the one which represents our world of experience." (p. 39)



He comes till up to intend a deductive step according this attitude based upon double negated sentences: "A deduction is a sentence whose negation is not autocontradictory."16 This sentence includes four negations, being autocontradictory a double negated word.



One more proof of Popper's arguing in a non-classical logic is given by the following fact. In S. Carnot's book some units of arguing closes through an ad absurdum proof, which reaches as most as possible of the positive content of the initial DNS starting the problem at issue. The most important one is the celebrated S. Carnot's theorem on the efficiency of heat engines. Let us remark that, likely as in S. Carnot's theory, in Popper's exposition a reductio ad absurdum closes his list of double negated sentences in the above quotation of Popper's Autobiography, concerning his criterion of falsifiability; "otherwise we are obliged to be dogmatic". One more reductio ad absurdum occurs when he emphasises his distance from VC thesis about metaphysics; he stresses that his criterion is not a semantic criterion; otherwise "a not verified theory would be a senseless theory" (16).



7. Scientific principles and demarcation on quantifiers



Double negation sentences occur even when Popper is analysing the problem of falsifying a physical law or principle.



He first severs individual elementary sentences, concerning single experimental facts, from universal sentences, concerning a statement whose basis is experimental, but which plays a crucial role inside a systematic theory, as a law or even as a principle. Then he reformulates physical principles as sentences including quantifiers (17) and recalls the laws of classical logic about the negated quantifiers.



Indeed, let us observe that Popper's criterion of demarcation, when applied to physical principles, is not universal in nature (1 . It is enough to apply it to the principle of the impossibility of a perpetual motion (19). Not yet an instance of perpetual motion has been presented. Nevertheless, in theoretical physics there exist a counter-example of this principle, actually Newton's principle of inertia, on which was founded the three-secular theory of mechanics ("… perseveres in its state…"). It enjoys experimental evidence, at least by eliminating friction in vacuum tubes; hence, it offers a counter-example of previous principle. Nevertheless, none principle of the above two was discarded from science. In fact, Popper's criterion is valid no more than as a heuristic rule, in a no more different way than the verification rule.



Then, these quantifiers are restricted by him to be either negative existential sentences ("prohibitions or non-existence statements": "There cannot exist…") or universal sentences (" For all…").



The first source of the qui pro quo is his implicit assumption; i.e. any universal statement is either a positive statement or a negative statement; there is no place for double negated statements. This Popper's statement cannot refuted, owing to its inconclusive reference to empirical data. We have to conclude that science is more tolerant than Popper.



In fact, the introduction of the choice on the kind of logic - either the classical one or the non-classical one - opens new perspectives on the relationship between empirical corroboration and an hypothesis, expressed as an universal quantifier, since this hypothesis may be even a double negated hypothesis, i.e. a methodological principle (20). For instance, the assertion of the impossibility of a perpetual motion - an assertion often put by scientists as the very basis of their theories - is in itself a double negation (perpetual means in physical terms "without an end") (21). It is just in this case that induction concludes, and hence it works in science and it exists.



Let us remark that as a matter of fact, very few times Popper exhibits instances of the physical principles of the kind he is discussing. The only one time he gives two instances ("There is no perpetual motion machine… There is no electrical charge other than a multiple of the electrical elementary charge", p. 69) he was wrong; since these sentences are double negated statements; in fact, in the first principle the word "perpetual" does not pertain to the physical realm; Stevin, who first made use of this principle, wrote instead of "perpetual" some words restoring the second negation, "… without an end" (22); in the second principle he ignores the words other than which are the same as "if not", i.e. a second negation with respect to "There is no….". Hence, he makes use of a double negated statement by presenting it as a merely negated sentence.



An effort follows (p. 69) to show that "strictly or purely existential statements" and "strictly universal quantifiers" cannot be falsified are "non empirical" (p.6 inasmuch as require the impossible task to investigate the whole spatio-temporal region.



But his effort is futile since Popper ignores that each scientific theory refers to a specific spatio-temporal region, and even to no one (e.g., thermodynamics and chemistry). Actually he recognises that his "characterisation may seem dubious at first sight and not quite in accordance with the practice of empirical science."(p. 69) He even suggests a possible objection; "the existence of elements of certain atomic numbers". But he answers that actually much more is required than a prely existential statement. An unsatisfactory answer, since the empirical fact supporting a physical law are innumerable; it depends on our insistence on the side of the empirical support or the side of the final synthesis if the statement referring this theoretical fact includes a great number of facts or a one sentence.



Moreover Popper forgets a clear counter-example; a strict existential principle belonging to the most relevant part of theoretical physics, the action-reaction principle ("To any action there exists a reaction…"); but also in quantum mechanics the basic statement ("There exists a function y belonging to a Hilbert space which represents the state of the physical system…") to prove that science chose a different way than Popper's one.



Of course, after Popper discarded two out four of cases, in the remaining cases a negative test can correctly lead to discard a principle. But only in these cases, in the other cases no. Hence, Popper restricts the cases of possible quantifiers to those which fit his criterion of fallibilism, so that to obtain in physical theories predicate sentences of a decidable kind only. In other terms, by means of an ad hoc move, he wanted to present the application of his criterion as capable to be complete.



8. "Asymmetry" and non-classical logic



In fact, the main point of Popper's equivocation is constituted by the "asymmetry". As a matter of fact, the "asymmetry" is often emphasised by Popper himself as a phenomenon occurring inside his own philosophy (23). In a first time Popper he states that it "results from the logical form of universal statements." (p. 41) After having restricted the four cases of quantification to two only (p. 69) the phenomenon is referred to the relationship between singular statements and an universal statement. Popper sees the "asymmetry" in the two different roles an elementary sentence can play, either in confirming or in disclaiming an universal sentence; the former one cannot achieve a confirmation of the latter one, which yet can be disclaimed by it.



Such a trivial fact, in my opinion, does not deserves a special qualification. Rather, in the light of DNS's this asymmetry amounts to be a crucial fact, being an instance of the failure of double negation law in non-classical logic. According to previous interpretation this asymmetry amounts to be a mere instance of the failure of double negation law in non-classical logic. This constitutes a very radical phenomenon of asymmetry. No more greater difference than a difference in the kind of logic could mark the difference between Popper and VC.



9. Popper's studies on the foundations of logic



All that qualifies what Popper actually did not what he wrote.



It is well-known that Popper devoted much time to study the foundations of logic, not only for his meta-theory, probability theory, but also in general terms.



He surely represents one of the first scholars who suggested natural deduction. Indeed, he analysed all that is possible to develop by means of the deductive inference, intended as the basic connective of implication.



He devoted several papers (24) on this subject. He started with an analysis of dialectics, as a way of arguing according to a different logic (paper a) of previous footnote). Popper shows a complete negative attitude with respect to it: surely, it was the time of the extreme position of the "diamat"; but Popper did not accepted anything of dialectics. He rejected the idea of a logic of contradiction and excluded the possibility of a dialectic. He reiterated that logic is only the theory of classical deduction.(p. 322)



Ten years after he announces new foundations of logic; and then studies intensively his idea to base the whole logic in a more natural way upon the only notion of inference (papers b), c), d) e)). In the last paper he considers intuitionistic negation too in almost the same foot that the classical negation. But since the paper b) he rejects "the so-called "alternative systems of logic". From our point of view, these systems are not alternative systems of logic, but alternative ways of using certain labels such as, for example, then label of "negation"… All these concepts [of negation] can coexist, in one and the same model of language, as long as they are distinguished. Thus there is no quarrel between alternative systems. The rules of inference pertaining to the various concepts of negations are not identical, to be sure. But this is very satisfactorily explained by the simple fact that the various concepts have been given a meaning by rules which are not identical…. Ordinary language is not sufficiently precise for us to ask such questions about it….[and] we are still asking too much, without doubt, if we ask such a question in connection with it [artificial language of science]. But I suggest that there are good reasons why, for most purposes of science, the classical concept should be preferable to others - simply because it is stronger, more explicit. This does not prevent us from using, for certain purposes, especially in certain parts of mathematics, the interesting concept negationI [of intuitionism] side by side with the classical one." (p.219)



Here it is apparent that Popper incorrectly thinks that the coexistence of more kinds of negations does not weakens the whole system in which one is arguing to the lower level of negation (the same is reiterated in the last quoted sentence). Moreover, he closes the question with a common sense argument, i.e. classical logic is more capable to fit the needs of science since its negation is stronger, as if science required only a decidable logic (furthermore, in the last two pages he shape as the best analysis that whose meta-logic avoids negations). Clearly, this is an unsupported position.



In paper e) where Popper correctly studies various kinds of negations (classical, intuitionistic, minimal, etc.), he reiterates previous evaluation about coexistence of more kinds of negation in a different, more surprising way. "It is fairly clear, from this definition, that intuitionistic negation ai or its dual am, or perhaps both, may exist in a language in which classical negation does not exist: and we shall prove all this by an example (in Sect. V)." (p. 323) In fact, in sect. V (p. 324), he shows no more that on the same sentence the classical negation prevails on the other ones; but nothing is said about the coexistence of some sentences respectively negated by various kinds of negations. In the latter case the system as a whole is characterised by the weaker negation, contrarily to Popper's expectations.



This Popper's attitude can be recognised also in later papers h) and i). In the former one he stresses again that classical logic is necessary for his fallibilism; in the latter one the theses 15-19 emphasise that classical deduction is the organon of criticism; no other logic is considered.



This attitude is maintained although in the '50s Popper studies (papers f) and g)) logic for probabilities and he advances new insights about the possibility of non-classical logic24.



Unfortunately, Popper was unfriendly (beyond his declaration to be tributing a homage) with Brouwer's hints for a new attitude in both logic and mathematics and in particular for the relevance (stressed also by the studies of Kolmogoroff) of the failure of the double negated law (subjectivism is the main criticism) (25). No surprise if then he was not always consistent with the view previously illustrated.



10. Popper's slipping the double negation of his criterion



Tennant showed that fallibilism, contrarily to what Popper claimed, do not requires classical logic; minimal logic is enough (25b). Popper was surely overconfident about classical logic. It constitutes the fixed-point of Popper's innovative attitude; no fallibilism about it (26). Some facts support his view. Indeed, a singular sentence, inasmuch as it represents a mere fact; hence it belongs to classical logic, where ØØA=A.



But Popper's strong commitment is to his argument against induction, which he relies upon modus tollens:



p®q, Øq, Øq®Øp;



Its validity requires the use of the law of double negation; hence, it pertains to classical logic only, as the same Popper often underlines. No surprise if he concludes that "induction does not exist", because induction surely has to be represented in a different logic than the classical one (27).



He was able to present his framework inside classical logic provided that the residual double negated statement are presented, as more as possible as merely negative statements. Indeed, Popper often writes a negation only, by giving as implicit the second one (e.g., "it must be possible for an empirical scientific theory to be refused by [negative] experience" (p. 41); science fails "… owing to negative experimental data"). By apperceiving a single negation, he synthesised his thinking through the truncated sentence "Science may fail".



Moreover, whereas in a first time: "… a disagreement [with experiments] … would show his theory to be untenable." then, he often reiterates a shortened, less clear sentence: "Science is fallible", which cuts off the second negation (owing to negative experiments).



Yet one thing is to state to the actual failure owing to negative results from experiments and another thing is to state "fallible" in an absolute way, a word which leaves indeterminate the time of this failure, possible beyond any given term. The same Popper has to sever the two words, "falsicability" and "falsification" (2 . Indeed, "falsificability", when considered inside a formal context, represents a modality; it represents an indeterminate, future sentence (recall Aristotle's futuribles, e.g. a ship battle will be won by Athenians); hence, future contingent statements were the only sentences to which Aristotle allowed to follow a non-deductive pattern. As represented by they pertain to non-classical logic.



Let us remark that in discursive terms, this kind of sentence essentially referring to a future situation is translated in a modality: it may fail, it is fallible. the double negated sentences to Inside a discursive presentation of philosophy and methodology of science, it suggests the idea of science as a never ended process. It leads to a description of the scientific initiative according to both the subjectivist viewpoint of a single scientist and the social viewpoint of the institutionalised science. Hence, through the word "fallibilism" the analysis on science enters inside a socio-temporal framework; just what Popper wanted to do, by enlarging the scope of his philosophy to a political theory (29).



However, he had to pay a cost. His trust in classical logic while he introduces (covertly) DNS's, generates a fuzzy situation, which excludes certainty in psychological terms; "fallibilism" as a subjective feeling becomes the appropriate word for be representing a generic situation which he has no more the capability to define in a sharp way. His effort in order to open the door to a different logic aborts in an exploration of a probability framework for describing the fuzzy situation he created.



11. Conclusions



I resumes Popper's strategy by means of the following table, which compares both his moves and the solutions suggested by previous interpretation. Let us remark that this interpretation results from some achievements which have been obtained by the debate on foundations of science in mainly subsequent times to those of Popper (third column). Popper ignored them or at most was contemporary to these achievements; thus, no surprise if he was unable to anticipate them.



Table 4. POPPER'S STRATEGY AND ITS INTERPRETATION



Warning: This table may need to be edited after it first appears in the Forum.



POPPER'S STRATEGIC MOVES SUGGESTED INTERPRETATION DATE

Logic classical logic as meta-theory; no relevance of intuitionism several kinds of logic; intuitio-nism as an alternative logic 1990

modus tollens only; induction does not exist deduction and induction processes ~ 1960

halving the four cases of universal statements laws as both positive statements and double negated statement 1990

suppression of the second negation double negated statements (1931-) 1977

Scientific principles axioms only, negative existential quantifiers only axiom-principles and methodological principles 1928- 1977

Instances of physical principles There is no perpetual motion, No charges other than a multiple of the elementary one Actually, they are double negated statements. Different instances: inertia, action-reaction, entropy, c as a maximum, etc. 1990

Systematic organisation of a scientific theory deductive (axiomatic) organisation (AO) only Both apodictic organisation (AO) and problem-based organisation (PO) (1931-)1990

Instances of physical theories Schroedinger quantum mechanics in addition: chemistry, thermodynamics, Heisenberg quantum mechanics (1789-) 1990

The framework describing scientific theories probability theory (it allows several kinds of logic) philosophy of the several kinds of logic 1990

Science in society and in history problematic view (psychology of a researcher) two alternative views: idealistic view and problematic view 1990





Notice that in the Table the 1990 constitutes a rough indication of the time in which a change in these issues occurred.



In a time in which philosophy of science was either blocked in discussing primordial sentences of classical physics, I allude to positivism, or disconcerted by new shocking results generated by modern physics , i.e. space-time, quanta, wave-corpuscle. Wisely Popper pondered on the first process of abrupt theory-change, from classical mechanics to special relativity. .In the time in which alternative foundations were suggested for Logic, Popper's philosophy of science echoed the new logics. This fact allowed him to overturn VC thesis in the corresponding double negated statement belonging to non classical logic. Moreover, Popper attentively studied the novelty in Logic. But he maintained , likely as the great part of the scientists of his time (30) maintained that in natural science an alternative to classical logic cannot exist. This conviction supported the rejection of induction and hence led to search for a new suggestion from probability theory. On the other hand he resumed his philosophy of science by means of in the dictum "Science is fallible" where seemingly there is a single negative word only; instead, a second negative word is included in a necessary addition in order to make clear the meaning of the dictum ("owing to a negative comparison with experimental results").



Thanks to this dual role of his dictum, his trust in classical logic did not obstruct him to achieve an adequate representation of discursive kind, of scientific research in historical terms, i.e. as a process, and in heuristic terms. This fact resulted very innovative in the history of philosophy of science. But it does not cancel the incommensurability phenomenon in Popper's philosophy of science; it is generated by the coexistence of two different kinds of logic. Popper's inadequacy was to think that a conflict between two scientific theories has to result in the victory of one the rejection of the other; although in the same time mathematics and logic learnt to accept the coexistence of wave and corpuscle, or more in general, the coexistence of Quantum Mechanics and Special Relativity.



Notes



1. K. Popper: The Logic of Scientific Discovery, Harper, New York, 1959, p. 40. In the following text a reference to this book will be denoted by means of the no. of the page only.

2 In K. Popper: " Intellectual Autobiography", in P.A. Schilpp (ed.): The Philosophy of Karl Popper, The Library of Living Philosophers, Open Court, 1974, 3-181, p. 28-29, Popper himself stresses this point as having "a dominant influence on my thinking." (p. 2 .

3 K. Goedel: Collected Works, Oxford, 1986, 1931a.

4 I. Lakatos too recognises this element as decisive for the birth of Popper's attitude; in I. Lakatos and A. Musgrave (eds.): Criticism and the Growth of Knowledge, Cambridge, 1978, p. 92.

5 Let us remark that Hempel's paradox ("All unblack things are non-raven" implies "All ravens are black", which in its turn implies that black and unblack ravens dive evidence to the statement "All raven are black") shows that logical neo-positivists had not doubt in applying classical double negations; in non-classical logic the failure of this law interrupts the paradox chain since its first step. Hilbert too had no doubt upon the universal validity of classical logic, predicate calculus included; according to him it is necessary to a mathematician as the fits to a fighting boxer ("On infinity", in J. van Heijenoorth (ed.) From Frege to Goedel, Harvard, 19???, p. ????.

6 I suggested an interpretation of induction inside a non-classical logic: "The process of induction as a non-classical logic’s double negation: evidence from classical scientific theories", Mathware and soft computing, 3 (1996) 295-308.

7 L.E. Brouwer: "The unreliability of the logical axioms", in Collected Works, 1908C, North-Holland, Amsterdam, 1975.

8 J. Lukasiewicz: "Farewell Lecture" of 7/3/1918, in Selected Works, North- Holland, 84-86. Later, E.W. Beth several times insisted on the necessity of going away from the uniqueness of deductive method; he qualified his program by means of an historical analysis on past philosophy of science. See in particular, Foundations of Mathematics, Harper, New York, 1959, Ch. I, 2.

9 A. Drago: "Incommensurable scientific theories: The rejection of the double negation logical law", D. Costantini e M. G. Galavotti (eds.): Nuovi problemi della logica e della filosofia della scienza, CLUEB, Bologna, 1991, vol. I, 195-202. "The process of induction as a non-classical logic’s double negation: evidence from classical scientific theories", Mathware and soft computing, 3 (1996) 295-308. ???? Let us remark that Goedel's theorems too depends from double negated sentences. See my paper: "Is Goedel's incompleteness theorem a consequence of the two kinds of organization of a scientific theory?", in Z.W.K. Wolkowsky (ed.): First International Symposium on Goedel's Theorems, World Scientific, London, 1993, 107-135.

10 A. Drago e R. Pisano: "Interpretazione e ricostruzione delle Réflexions di Sadi Carnot mediante la logica non classica", Giornale di Fisica, 41 (2000) 195-215; "Interpretatioon and reconstruction of S. Carnot's Réflections through non-classical logic", submitted to J. History of Logic.

11 A. Drago and R. Pisano: "Interpretazione della nota matematica nelle Réflexions di S. Carnot mediante il metodo sintetico", submitted to Arch. Int. Histoire des Sci..

12 D. Prawitz and P.-E. Melmnaas: "A survey of some connections between classical, intuitionistic and minimal logic", in A. Schmidt and K. Schutte (eds.): Contributions to Mathematical Logic, North-Holland, 1968, 215-229. M. Dummett: Elements of Intuitionism, Oxford U. P., 1977. A firm point is the translation between the two kinds of logic, as suggested by Glyvenko, Kolmogoroff and Goedel; see S. Troelstra and D. van Dalen: Constructivism in Mathematics, 1, North-Holland, Amsterdam, p. 128.

13 W.O. Quine: "On Popper's negative ontology", in P.A. Schilpp (ed.): op. cit., 218-220.

14 K. Popper: "Intellectual Autobiography", op. cit., p. 29; Italic in the text. My underlying evidentiates those words which pertain to double negated sentences; between square brackets the corresponding, positive statement.

15 One may recall two more celebrated Popper's statements which are double negated statements: "To minimize [= reduce, annihilate] sufference"[¹ to enhance joy] and "To minimize the risks of bad rulers" [¹ to enhance good rulers].

16 K. Popper: The Logic…, op. cit., I, 1.

16 K. Popper: "Autobiography", op. cit., p. 63.

17 Some scholar objected to this suggestion. See G. Maxwell: "Corroboration without demarcation", in P. A. Schilpp (ed.): op. cit., 292-321, who discusses the classical sentence "All men are mortal" (p. 315-31 .

18 A similar remark was suggested by W.O. Quine: "On Popper's negative ontology", in P.A. Schilpp (ed.): op. cit., 218-220.

19

20 This is one aspect of the common objection Popper takes in accounts, according to which negative tests are not enough; since, according to Duhem thesis, the theoretical physicist always can change an hypothesis for changing a negative results in a confirming result. Actually, the change in logic gives a change in the whole organisation of hypotheses.

21 See Einstein who considered this principle as the best instance of a principle in "constructive theories", vs. "principle theories".

22 R. Dugas: Histoire de la Mécanique, Griffon, Neuchatel, 1955, p. 121. In Einstein's opinion, physical theories can be severed in two sets of different kind, the set of "constructive theories", e.g. thermodynamics, and the set of "principle theories", e.g. statistical mechanics; the above principle is the best instance pertaining to constructive theories.

23 K. Popper: The Logic…, op. cit. I, 6; "Postscript", § 22. V. Kraft ("Popper and Vienna Circle", in A. Schilpp (ed.): op. cit., p. 185-203 , p. 190) states that asymmetry was a "revolutionary discovery" by Popper.

24 K. Popper: a) "What is dialectics?", (1937) re-printed in Conjectures and Refutations. The Growth of Scientific Research, Routledge and Keagan, London, 1965, as Ch. 15; b) "New foundations of Logic", Mind, 56 (1947) 193-235; c) "Functional Logic without Assumptions or Primitive Rules of Inference", Aristot. Soc. Proc., 47 (1947) 251-292; d) "On the Theory of Deduction. Part I: Derivation and its Generalization", Comp. Math., 10 (194 173-183; e) "Part II: The Definition of Classical and Intuitionistic Negation", ibidem, 322-331; f) "Two Autonomous Systems of the Calculus of Probabilities", (1951); g) "Appendices on Logical Calculus for Probabilities" to The Logic of …., op. cit.; h) "A Realistic View of Logic, Physics and History", in A.D. Breck and W. Yourgrau (eds.): Physics, Logic and History, Plenum Press, 1970, 1-30; i) "The logic of social sciences", in T. Adorno et al. (eds.): The Positivistic Dispute in German Philosophy, Heineman, 1976.

24 See the several references to Popper's work on this subject in the author's index of D. Gabbey and F. Guenther (eds.): Handbook of Philosophical Logic, Reidel, 1983.

25 See K. Popper: Objective Knowledge. An Evolutionary Approach, Oxford, 1972; § 3.6.

25 N. Tennant: "Minimal Logic is adequate for Popperian Fallibilism", Brit. J. Phil. Sci,, 36 (1985) 325-329.

26 Also S. Haak: Deviant Logic, Oxford, 1974, p. 38, remarks that Popper's "proposal would leave logic totally immune from criticism."

27 I tried to explain induction just by means of double negated statements: "The process of induction as a non-classical logic’s double negation: evidence from classical scientific theories", Mathware and soft computing, 3 (1996) 295-308.

28 K. Popper: The Logic…, p.????

29 However, his misunderstanding of Brouwer's relevance in the development of logic obstructs him to understand dialectic too; by him it is compressed in deductive schemes of classical logic only. K. Popper: Conjectures… and Refutations. The Growth of Scientific Research, Routledge and Keagan, London, 1965, ch. 15.

30 For instance, the authoritative couple of scientists Birkhoff G., von Neumann J. in the last sect. of their paper "The logic of Quantum Mechanics", Annals of Mathematics, 37 (1936), pp. 823-843 which discovered quantum logic, confess to have deliberately avoided intuitionistic law.