Abstract
In the present paper Peirce’s inferential processes are accurately defined against the background of the four ways of reasoning in Computability theory, i.e. general recursion, unbounded minimalization, oracle and undecidabilities. It is shown that Peirce anticipated almost all them.
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Notes
- 1.
However, the author adds on the same page: “The reader may be sure that any time he spends with Peirce…. [he] will splendidly widen his intellectual horizons.”.
- 2.
I recall that two numbers in round brackets refer to (Peirce 1931), the first number means the volume and the second number the section.
- 3.
For an introduction to Peirce’s thinking on inference processes, see Fann (1970).
- 4.
In particular, two different classifications of the several meanings that the notion abduction may assume have been suggested by two authors (Schurz 2008; Hoffmann 2010). The latter author obtains fifteen forms of abductions, yet he classifies them according to the ‘outside’, i.e. by means of external issues and the resulting effects; in my opinion these are superficial features. The classification of the former author will be analysed later.
- 5.
- 6.
This thesis is the same as Hintikka’s (1998, p. 507).
- 7.
Peirce implicitly referred his reflections to his wide knowledge of scientific theories, mainly chemistry; (Drago 2015) but his scholars rarely referred his reasoning to some inductive processes in the history of classical chemistry. Among the philosophers of chemistry, only van Brakel (2000, pp. 21–22) devoted a page to Peirce. Rather two other instances of inductive processes have been studied: (1) Kepler’s theory of the orbit of Mars (1.73); which was further illustrated by (Hanson, ch. IV). Nickles made valid criticisms of this analysis (1980, pp. 22–23). In any case, Kepler’s theory belongs to a too informal context for suggesting certain conclusions on his underlying logic. (2) The kinetic theory of gas (2.639) which however, is too loosely treated by Peirce; moreover, this instance was contested as inappropriate to an investigation of his suggestions (Achinstein 1987).
- 8.
In the following, the negated words belonging to doubly negated proposition will be underlined for an easy inspection by the reader. Notice that even a single word may be equivalent to such a kind of proposition, in particular a modal word; e.g. possible = it is not the case that is not (more in general, it is well-known that the modal logic is translated into intuitionist logic by means of S4 model) such a kind of word will be bold.
- 9.
This point deserves particular attention, because the current usage of the English language exorcises DNPs as representing a characteristic feature of a primitive language, English linguists are dominated by a long tradition which L. Horn called a “dogma” (Horn 2001, pp. 79ff.; Horn 2008). This linguistic dogma asserts the absolute validity of the double negation law: whenever a DNP is found in a text, it has to be changed into the corresponding affirmative statement, because those who speak by means of DNPs want to be, for instance, unclear. Evidence for this “dogma” is the small number of studies on double negations in comparison with the innumerable studies on a single negation. The following three well-known DNPs belonging to mathematics, physics and classical chemistry show that this logical-linguistic feature pertains to scientific research since its origin. In Mathematics it is usual to assure that a theory is “without contradictions”; to state the corresponding affirmative statement, i.e. the consistency of the theory, is impossible, owing to Goedel’s theorems. In theoretical physics it is usual to study the in-variant magnitudes. The proposition does not mean that the magnitudes stay fixed. In order to solve the problem of what the elements of matter are, Lavoisier suggested defining these unknown entities by means of a DNS: “If… we link to the name of elements or principles of corps the idea of last term to which [through chemical reactions] arrive at the analysis, all the substances which we were not capable to decompose through any tool are for us, elements”, where the word ‘decomposable’ naturally carries a negative meaning and stands for ‘non-ultimate’ or ‘non-simple’ (Lavoisier 1862–92, p. 7). As a matter of fact, Grzegorczyk (1964) independently proved that scientific research may be formalized through propositions belonging to intuitionist logic, hence through DNPs.
- 10.
Notice that it is commonly maintained that ad absurdum proofs can be all translated into direct proofs (Gardiès 1991). Unfortunately, it is rarely remarked that this translation is possible only by applying classical logic to its conclusion, i.e. the absurdity of the negated thesis ¬Ts, that one wants to prove, ¬¬Ts. But this last step, the translation of ¬¬Ts in Ts, is interdicted by intuitionist logic (otherwise a direct proof would be more appropriate). Hence, an essentially ad absurdum proof is a characteristic argument of intuitionist logic.
- 11.
As it will be illustrated in Sect. 6 this principle (itself a DNP: “Nothing is without reason”) allows us to change a DNP into its corresponding affirmative proposition (in our case in the last quotation).
- 12.
More versions of the same theses occur on pp. 223, 228, 240, 241. Two more incapacities are presented by Peirce in 5.200 “…because all our ideas being more or less vague and approximate, what we mean by saying that a theory is true can only be that a theory is very near true. But they [some logicians] do not allow to say that anything they put forth as an anticipation of experience should assert exactitude, because exactitude in experience would imply experience in endless series, which is impossible…. We therefore have a right, they [the logicians] will say, to infer that something never will happen, provided it be of such a nature that it would not occur without being detected.” (Peirce 1965, p. 124, 5.199 and 5.200). These propositions are also DNPs. By incidence, let us remark that these propositions anticipated the great debate among mathematicians that took place in the middle of the last century. Almost to defy such statements by Peirce, Hilbert had solemnly affirmed: “In mathematics there is no Ignorabimus” (Hilbert 1902, p. 445). But Goedel’s theorems subsequently disproved Hilbert’s thesis, leading mathematicians to accept innumerable undecidabilities.
- 13.
Let us recall that thermodynamics’ principle of the impossibility of perpetual motion suggested to Einstein the trigger idea for building special relativity; he was led by it to think that a body’s velocity greater than or equal to the speed of light is impossible.
- 14.
Why did he not regard this reasoning about incapacities as a kind of reasoning that was independent of the others? One may suggest two reasons. First, his above-mentioned three writings were a polemic against Descartes, rather than an investigation of kinds of reasoning. Second, the strategy of the kind of reasoning on incapacities is the opposite of his usual strategy; he was interested in the ampliative, rather than the restrictive, kinds of reasoning -, which is what incapabilities are. (About the notion of strategy in Peirce, see Hintikka (1998, pp. 51ff). As such this kind of reasoning is quite different from the usual ones. In particular, since they refer to totalities undecidabilities are essentially different from deductions and inductions.
- 15.
Peirce (1881), “On the Logic of Numbers” (3.252–288). Notice that the Dedekind’s most celebrated work on the same subject was edited seven years later (Dedekind 1888).
- 16.
Notice that recursion, although apparently a very simple technique, may make use of the actual infinity. Indeed, the general recursive functions are defined e.g. by a diagonalization process on all the elementary ones (see Davis et al. 1994, p. 105ff.). Also unbounded minimalization introduces non constructive elements (see Davis et al. 1994, pp. 57–58). The oracle manifestly constitutes a non-constructive move. It is roughly defined as a black box which is able to decide certain decision-making problems, otherwise unsolvable, through a single operation. It is more precisely defined as follows: “A number m that is replaced by G(m) in the course of a G-computation… is called an oracle to query to the G-Computation” (where a G-computation is a computation of a partial recursive function G under specific conditions; Davis et al. 1994, p. 197).
- 17.
Notice that in the history of science an abduction was an elusive notion. In the case of the Kinetic Theory of Gases, the atomic hypothesis started as a mere guess in Galilei and Boyle, became an abduction-oracle in the first attempts at building a theory through its consequences (Newton and D. Bernoulli attributed reality to atoms without any experimental evidence, but only to obtain new results in a deductive development from this hypothesis) and subsequently became an extremant when Avogadro calculated his celebrated number; eventually, in the 20th century it received experimental evidence. It is not possible to study it with the expedient of the historical case of D. Bernoulli and Newton, because these authors communicated their logical inferences in the language of mathematics, without references to their logical processes. In other words, in the history of science abduction pertains to a genetic stage of the historical development of a theory, without certain documents that need to be analyzed.
- 18.
Also the conclusions of the forms 2.702 and 2.706 are non-affirmative statements: “… probably and approximately….”.
- 19.
I took advantage of Schurz’s comprehensive appraisal of the several meanings of an abduction processes: “I will classify patterns of abduction along three dimensions: (1) along the kind of hypothesis which is abduced, i.e. which is produced as a conjecture. (2) along the kind of evidence which the abduction intends to explain, and (3) according to the beliefs or cognitive mechanisms which drive the abduction.” (Schurz 2008, p. 205). His decisive “Result 3” seems to summarize the previous dimensions: “In all cases the crucial function of a pattern of abduction… consists in its function as a research strategy which leads us, for a given kind of scenario,… to a most promising explanatory conjecture which is then subject to further test.” (Schurz 2008, p. 205). I agree with both dimensions and Result 3 obtained by Schurz’ since in my opinion he intuitively referred to the two above illustrated dichotomies on the kind of organization of a theory (“scenario”) and the kind of infinity (“cognitive mechanics which drive the abduction”). In particular, the word “belief” in Schurz’s third dimension (Schurz 2008, p. 205) corresponds to the logical step which plays the role of the conclusion of a PO theory.
- 20.
For instance, Kleene states Church’s thesis as follows: “… it cannot conflict…” (Kleene 1952, pp. 318–319). Although ignoring the exact linguistic expression of a DNP, several authors presented Turing-Church’s thesis through similar words: Goedel called it a “heuristic principle”, (Davis, p. 44). Post a “working hypothesis” (Davis 1965, p. 291). The same Church presented it through the words: “… it is thought to correspond satisfactorily…” (Davis 1965, p. 90) Remarkable is the masked proposition in Davis et al. (1994, pp. 68–69): “we have reason to believe that…”, where the word “believe” is a subjective word which actually is the DNP: “it is not false that it is …”.
- 21.
This definition is a modification of Schurz’ definition (Schurz 2008, p. 202).
- 22.
For instance in the history of theoretical physics the approximating series mattered. The case of an ideal body for building the theory of impacts was very controversial. Wallis and Newton have suggested a perfectly hard body; it does not change its shape however violent the impact; of course the conservation of energy is no longer allowed. This ideal body cannot be approximated by a series of ever harder bodies because its definition refers only to the behaviour of the final body of any possible approximation series. Instead Leibniz has suggested as an ideal body the perfectly elastic body. The approximating instances of elasticity, by pertaining to the observable domain, may be represented by a series of values of a parameter characterizing this specific feature; in such a way one obtains a mathematical converging series of the values of the elasticity index of the material objects. This process may be defined also—in physical terms—as a process that obtains a final element by operative means. It was not until 1850 that the hard body model was dismissed and the conservation of energy was considered a general law. In conclusion, since lacking any approximating series, the abstraction of a perfectly hard body was a false induction, a mere guess. In theoretical physics the inertia principle, the door to modern physics, is not the a result of an approximation process; it holds true only when one considers a body without friction and gravity (a situation which cannot be approximated—as Galileo put it—by a variable incline). For this reason it is a theoretical principle, not derived from other principles.
- 23.
Being a merely probable proposition, it has to be correctly stated as “It is not the case that it is not…”, i.e. a DNP. The lack of this accurate logical version of his notion surely made Peirce’s logical elaboration of these inference processes difficult.
- 24.
Why was it possible to compare Peirce’s reflection on inferential processes with those of CT rather than with those of physical theories? Because each physical theory at most exploited one inferential process only. Only Mendeleev may have exploited all these processes together in his obscure way in order to build the table of chemical elements (Drago 2015).
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I acknowledge Prof. David Braithwaite who corrected my poor English.
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Drago, A. (2016). Defining Peirce’s Reasoning Processes Against the Background of the Mathematical Reasoning of Computability Theory. In: Magnani, L., Casadio, C. (eds) Model-Based Reasoning in Science and Technology. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-38983-7_21
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