Abstract
Peirce’s claims that methodeutic “concerns abduction alone” and that “pragmatism contributes to the security of reasoning but hardly to its uberty” are explained. They match as soon as a third claim is taken into account, namely that “pragmatism is the logic of abduction,” not of deduction or induction. Since methodeutic concerns abduction and not deduction or induction, it follows that pragmatism is a maxim of methodeutic. Then, since pragmatism contributes to the security of reasoning but not to its uberty, it follows that methodeutic contributes to the security of the only reasoning it is concerned with, namely abduction. We then explain two related issues of methodeutic of abduction. First, in addition to the maxim of pragmatism, which suggests how to choose among experimental hypotheses contributing to the security of reasoning, there is the maxim of simplicity, which suggests hypotheses that are preferable for investment and which contributes to uberty of reasoning. Second, a third maxim of abduction is economy, which suggests adopting hypotheses that contribute to the advantageousness of reasoning even when pragmatism and simplicity cease to apply. These three maxims—experientiality for security, simplicity for uberty, and economy for advantageousness—are the bedrocks of Peirce’s methodeutic of abduction.
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Notes
- 1.
The preserved chapters and their sections are as follows. Chapter 1 (R 425, 176 ms pages): Intended Characters of this Treatise. § 1. Logic’s Promises. § 2. Of Minute Accuracy. § 3. Different Methods in Logic. § 4. Synopsis of Contents of this Book; Chapter 2: Prelogical Notions. § 1. Classification of the Sciences (R 426–427, 291 ms pages), § 2. Why Study Logic? (R 428, 128 ms pages); Chapter 3. § 1. The Essence of Mathematics (R 431, 200 ms pages), § 2. Division of Pure Mathematics (R 430, 108 ms pages)/Specimens of Mathematical Reasoning (R 430, 68 ms pages); Chapter 4: Ethics (R 432–434, 234 ms pages). According to the plan and its revisions (“List of Proposed Memoirs on Minute Logic,” R 1578), at least 33 chapters were to be accomplished in total, together with a lucrative pay-check from Peirce’s friend Francis Lathrop, who had promised to pay $150 for each chapter. According to this plan, the memoirs on methodeutics were by title “27: Methodeutic. Discussion of the Method which it should possess”; “28: Economics of Research”; “29: Clearness of Ideas. Discussion of Pragmatism”; “30: Methodeutic of logical division and natural classification”; “31: On the Conduct of Inquiries and the Economics of Research”; “32: Criticism of Methods actually in use”; and “33: Objective Logic and Metaphysics.”
- 2.
Final version and drafts in RL 75; see [1] for an introduction to the reconstruction of this set of documents, which amounts to over 500 manuscript pages in the Harvard Peirce Papers (see also NEM IV, pp. 13–73).
- 3.
See [2], pp. 278–289.
- 4.
All mathematical reasoning, he states a few years later, “relates to some schema of the nature of a diagram, that is, a sign having parts related similarly to the objects denoted, and having letters or other indices to distinguish those parts” (R 87, c.1905). Soon after, he needs to revisit these and other earlier statements of his, rethink, and widen the scope of methodeutic.
- 5.
See [4]. In R 70(s) (9 August 1906) Peirce defines properties of positive integers in the language of Beta graphs by (i) the two-place successor function (namely that each positive integer N has a successor S(N)), (ii) that successor to be unique, and that zero is not a successor of any positive integer, and (iii) the principle of finite induction. He then derives further axioms such as that being a successor of two positive integers implies the identity of those integers, and the property that 0 is the only integer (natural number) not a successor of any integer, among other properties.
- 6.
- 7.
- 8.
Namely R 752 (15 March 1914). In it, he talks about “orders” of reasoning, which signals their double nature of being both “stages” and “kinds” (compare these with the terms “grades” in R 843, 1908, and “types” in R 905, 1908).
- 9.
Reasonably assuming that P ≠ NP.
- 10.
R 753 appears to be an earlier, c.1906, segment of a draft lecture on retroduction, with manuscript pages 3–8 preserved. Its paragraphs that concern retroduction read as follows: “Now without taking the audience more thoroughly into this matter, I, who have gone through the whole matter far more systematically, can point out some characteristic features of these three modes of reasoning which will show their entire unlikeness to one another and what each depends upon. ¶ First, Retroductive reasoning is the only one of the three which produces any new idea. It originates a theory. Now if the multitude of possible theories to account for any collection of facts be not strictly infinite, it may be very moderately estimated as a million. Of these only one can be the true one. Therefore, considering that the testing of one theory is a matter that may cost a million dollars, several lives, and a full generation, it is easy to see that history is not long enough to account for our having reached any tolerably approximate theories, without we suppose that the human mind has a natural talent for guessing at the truth of nature – meaning by nature, not only what is artificial, but all that is not downright fictitious[suffix or word(s) torn out] a product of the theory. Of course, I do not mean [word(s) possibly torn out] that a guess is probably right, but merely that it [word(s) possibly torn out] be true, to be evident. Being evident, it is not doubted. Not being doubted, it cannot really be criticized. For a doubt is a real state of dissatisfaction; and the common practice of making believe to doubt, and then offering considerations to appease that make-believe doubt, is a foolish waste of time, since the man who does not doubt can realize the state of mind of the man who does, only to a very limited extent, and he thus quite fails in all difficult cases to appease any real doubt that may exist, or appeases it quite otherwise than by his attempted reasonings. Another man’s doubt will usually set up a doubt in my mind and that I can handle intelligently; but when his doubt excites no doubt in my mind, it is either that I know by my own experience where his mind has probably stumbled (and if I do he can be set right in a moment) or it is that he has thought that I have never had, and probably understands the matter better than I. But what appears to be evident is past all possible genuine criticism, until it ceases to appear evident. ¶ It will be observed that this justification of retroduction supposes that the hypothesis or conjecture which is justified has been the result of some normal way of forming such a conjecture, though the justification does not depend upon what that process is” (R 753, pp. 3–5).
- 11.
Of the three types of reasoning, abduction is “persuasive, seductive” (R 754, 1907). We are “compelled to begin with [it] if we are ever to discover a law or the rationale of any phenomenon” (R 843, p. 41, “Neglected Argument,” (copy-text?) draft, crossed out, c.1907). Peirce talks about “irresistible persuasiveness” that abductive conclusions have in scientific discovery: “You will find that our rational instinct often prompts us to reasoning such that no conceivable mass of similar data would render its conclusion either certain or, in the strict sense above defined, probable, and it appears to be evident on examination that it is impossible absolutely to prove that these arguments have any value whatever. Nevertheless, it seems that many of them have an almost, if not quite, irresistible persuasiveness, that many of them have caused great discoveries and apparent great advances in science; and finally the most decisive circumstance of all in their favour is that unless these arguments have do tend [sic] to carry us toward new truth in the whole, we must abandon all hope of penetrating further into the secrets of the universe than we have done already” (R 652, pp. 12–13, July 12, 1910). Peirce draws an analogy from the game of Whist, in which players may be led to situations which “full warrant a player for acting on the hopeful hypothesis” (ibid., p. 14). This is abduction, namely “reasoning from consequent to antecedence”; reasoning “which from a consequent and a consequence infers an antecedent” (ibid., p. 15). It is “infers” that Peirce accentuates: abduction is inverse inference and has a certain logical form.
- 12.
Letter preserved in the Harvard University Archives.
- 13.
It is only later in the process that “we correct the errors of our Retroductions [abductions] by processes of Adduction [induction]” (R 764).
- 14.
Here we may offer also another, and related, explanation of why uberty is not mere “fruitfulness.” Peirce explains that non-uberous hypotheses can be taken to be as fruitful as we like and yet be conclusively overturned as soon as compelling evidence emerges that indicates refutation. True and false hypotheses may be both fruitful but only the former can be truly uberous. In contemporary terms, Peirce may be seen to emphasise the importance of non-epistemic values in science, which are not in fact distinct from epistemic values [11]. The uberty of retroductive conclusions suggests that non-epistemic values can be logically analysed, as he notices in the letter to F. A. Woods quoted above: “I think logicians should have two principle aims: First, to bring out the amount and kind of security (approach to certainty) of each kind of reasoning, and second, to bring out the possible and esperable uberty, or value in productiveness, of each kind” (RL 477). It needs to be conceivable that those values can be logically analysed, at least in principle. One way may be to apply dynamic modal logics of conjecture-making, which is a theory of abductive reasoning at the level of pre-beliefs, [12]. There is thus a connection between uberty and human hope that things will turn out the way anticipated: it springs from uberty that the searching questions of science upon which our rational hopes are built are amenable to final decisions. These two explanations are related, since uberty of reasoning has a link with truth: all good reasoning rests on logic. The defence of the latter is not to be taken up here.
- 15.
In one of his last essays, “Essay toward Improving Our Reasoning in Security and in Uberty,” Peirce takes abductive conjectures to be distinguishable from normal formulations of interrogative moods in the sense that the former are “actually gravid with living and prolific truth” (R 683, p. 8). Not truth per se but “living truth”: truth that might arise to our view in the case our inquiries were pushed to their utmost limits. Conjectures that are prolific in truth are also of “value in increasing knowledge” (ibid.). Peirce does not base his revisions of the logical schema of abduction directly on the shortcomings of his earlier, 1903 Harvard schema. For one, he is not led to acclaim that conjectures increase our knowledge. His later revisions concern how Modus Tollens, an inverse reasoning from effects to causes, can result in conclusions in the special “investigand” mood (RL 463; [14, 15]). The increase in knowledge follows from the imperative part of the investigand, because epistemic import cannot solely derive from the interrogative mood of questions in science. Questions do not cash out the added value in terms of knowledge. Methodological advances that Peirce refers to draw not from abduction as such but from the scientific values that imperative moods have towards resolution of pertinent research questions.
- 16.
Many measures of simplicity abound, from model selection tasks by least-effort path principles and minimal description lengths to levels of symmetry and degrees of conservativity in guessing at new equations. What is common are the considerations of structural relationships in the inferential framework when moving from premises to conclusions.
- 17.
This assumes carrying out the entire tri-partite, repetitive cycles of reasoning: abduction, deduction and the experimental testing of the outcomes of the previous two stages of reasoning by induction. Indeed the action of induction is to conclude “from the results of [abduction and deduction] to what extent it will be safe to rely upon the hypothesis” (R 478, p. 102).
- 18.
Peirce’s pertinent observation was that “now economy, in general, depends upon three kinds of factors; cost; the value of the thing proposed, in itself; and its effect upon other projects. Under the head of cost, if a hypothesis can be put to the test of experiment with very little expense of any kind, that should be regarded as a recommendation for giving it precedence in the inductive procedure” (CP 7.220, 1901). On Peirce’s economy of research, see e.g. [17,18,19,20]. [21] has introduced related concepts of eco-cognitive openness, optimization of situatedness and eco-cognitive model of abduction and can be used in explaining further the phenomenological aspects of creativity in scientific reasoning.
- 19.
The Nobel Prize in economics went in 2019 to poverty research that did not ask the unanswerable “How to get rid of poverty?” but, for instance, “Could the rate of access to clean water be expedited?”, etc.
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Acknowledgments
Supported by the Estonian Research Council personal research grant PUT 1305, Abduction in the Age of Fundamental Uncertainty (A.-V. Pietarinen, 2016-2018), and the HSE University Basic Research Program funded by the Russian Academic Excellence Project ‘5-100’ (A.-V. Pietarinen, 2018-2020).
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Appendix
Appendix
R 754 [p. 1]
1.1 Second Talk to the Philosophical Club
Deduction. Define it. Several years’ reconsideration leads to this new definition.
Not necessary but compulsive reasoning.
Retroduction persuasive, seductive.
Induction appeals to you as a reasonable being.
Deduction points to the premises and to their relation and then shakes its fist in your face and tells you: “By the eternal powers, you have got to admit the conclusion.” Effort required distinguishes deduction. That [it is] not necessary [is] shown by definition of the necessary and argument about gold. [See the last paragraph of this R 754 for the argument about gold.]
Compulsive means that you are logically forced to admit the conclusion.
Now, resume the division of reasoning. You begin with perception which brings surprise. To make that reasonable you resort to Retroduction. That finally finished This resembles perception in bringing the new. And after this no new idea enters into reasoning. Now comes Deduction by which this new forces you to join with it a transformation,—no new matter but only a new form. And Deduction is defined as including all that. Finally Induction, or the Experimental Method tests its truth.
Observe Regular Gradation from the incoming of the new, first in Perception, then in Retroduction; then, formally only in Deduction, while the reasonableness, quite absent in perception begins to appear in Retroduction, [and] still more forcibly in Deduction and in its highest development in Induction.
[p. 2, discontinuous] […] the depths of consciousness. Question of number of premises. Effort required. Theôric element never or hardly ever absent from deduction.
I must confess I have nothing but such mere Generalities to offer, when what is wanted is a Method for the discovery of Methods. Utter weakness of all the Books. One obvious and old remark is that a good way to prove that a thing is possible is to show how it can be brought about. Yet that that is not indispensable is shown by a celebrated proposition in the elements of higher arithmetic.
The Mathematicians [are] the only skilled and correct reasoners because all reasoning depends on deduction and I will not consent to my admired friend Bôchler’s limitation of mathematics to any particular class of Deductions. Mathematics is the practice of deduction.
It seems to me rather remarkable that though it will not be disputed that I have shown high powers in regard to the theory of deduction, I have but a very moderate mathematical talent. Perhaps what I mostly [blame] imperfection is my
Proof of abnumeral multitudes.
Next the proof which I was the first to give [was] that Linear Associative Algebra is the same things as the Theory of Matrices.
Obvious enough, perhaps, yet disputed by Sylvester. Then my proof that only three algebras have determinate division [quaternions, octonions, nonions].
Then various trifles [are] relative to computations in which my lack of generalizing power is most prominent. (Duplication of the cube. Entire circulating decimals.) [p. 3]
The more I consider the matter, the more I am convinced that corollarial reasonings are the highest.
Among these is my proof about multitude.
Also De Morgan’s Syllogisms of Transposed Quantity.
Also the Fermatian Inference commonly but most absurdly called Mathematical Induction.
But those stupendous proofs of Gauss appear to be Theôric. That is why he can prove a proposition in so many different ways. There can be but one sound Corollarial proof of any thing from given premises, though the order of procedure may be somewhat varied. [p. 4]
Second talk. On Deduction.
It was Monday. I was overcome by this Boston April: how ill it always made me and nearly drove me frantic. Was so ill at last lecture and so gave to sundry points colors I had never intended.
I began by speaking of myself,—a very hazardous and rash thing to do;—one runs away with oneself so easily and finds oneself brought up in the court of self to answer the charge of driving beyond the limit of speed.
I mentioned my unusual turn for the subtleties of logic, but entirely forgot to mention, what I had fully intended to emphasize, the quite adventitious character of such gifts. A talent for logic, or any other, is no more a thing to take credit to oneself for than is the inheritance of a fortune. The parable of the talents embodies deep truths. The talents are represented as capable of growth if properly put to use. This is very true, indeed. But the parable leaves us to think that the talents are more entirely extraneous than they really are, that they are as passive as so many pieces of gold, and that the issue of the matter depends entirely upon the personal character of the man, regardless of the talents. It is true that there is an attempt, in the parable to [p.6] correct this error by representing that it is only the man who has one isolated talent who fails to put it to use, being fearful lest he should lose his all. Well, I am a Christian, as we all are, did we but recognize it. But the gospel of Christ is a gospel at once of liberty and of duty to think out things for ourselves, and we have no warrant whatever for regarding the parables as anything else than appeals to our reason. Now this talent that we receive, even such a limited endowment as mine, is more than a fixed sum. It is capable of development so as to dominate and elevate the whole character; and therefore I prefer that other parable or metaphor of the Nancy hypnotists of multiple personality;—an ancient and Christian [illeg.] it is too, of a spirit that grows within a man and supplants the “old man.” [Added in margin:] It is a singular result of well-known circumstances, that a man is not permitted to express his belief in a personal God in any impassioned way, no matter how overwhelming that belief may be.
Dr. Huntington at the end of the other evening’s talk asked how I would regard the great hypotheses of pure mathematics; and whether it was that I was fatigued or what, I replied quite stupidly. I certainly regard from considering what for want of a better word I may call the facts of mathematics. When a coefficient of an equation gradually changes so that two adjoining roots after coalescing [p. 5] disappear, our explanatory hypothesis to account for the phenomena if you will permit me to so to call it, is that the roots, in consequence of the collision have struck off into another dimension. It is a hypothesis, because it is a way of rendering the phenomena comprehensible. But it is only a formal hypothesis, not a material hypothesis[.] Definition or Division are also retroductive. Deal with them separately.
Deduction. I have hitherto defined this as necessary reasoning; and no doubt much, perhaps most, possibly all deduction is necessary. But on reviewing the subject for this talk, it seems to me more correct to define Deduction as compulsive reasoning. Retroduction seduces you. Induction appeals to you as a reasonable being. But Deduction first points to the premises and their relation, and then shakes its fist in your face and tells you “Now by God, you’ve got to admit the conclusion.” I beg your pardon, with all my heart, I meant to say, “Now by the eternal world forces spiritual and personal [illeg.]” Necessary reasoning is reasoning from the truth of whose premises it not only follows that the conclusion is true, but that it would be so under all circumstances.
Gold. Somebody submits a proposal for a new coinage, with a specimen the Master of the mind says, This is pure gold and therefore it is too soft for the purpose. That is not necessary reasoning for it is quite conceivable that gold should be hardened as pure copper and pure iron can be. But in existing circumstances, no such way of hardening gold being known, it is now quite compulsive. [End of R 754]
1.2 Third Lecture on Methodeutic [R 774]
Induction
Apology. Had been really very ill for 24 h previously and had had no sleep. I am still far from well, but hope to do better than last night.
That proof which I failed to give is as follows. No matter what objects the Ns may be, so long as they have independent identities, I ask whether there can be any relation in which every conceivable collection of Ns stands to some N to which no other describable collection of Ns stands in the same relation.
Take any relation you please, and call it R. Then I divide the Ns into 4 classes:
First class consists of Ns to each of which no collection of Ns is R.
Second class consists of Ns to each of which 2 or more collections of Ns are R.
Third and fourth classes consist of Ns to which of which a single collection of Ns is R.
The third class consists of those of these Ns each of which is contained in their collection of Ns that is R to it. The fourth class of those each of which is uncontained in the collection of Ns that R to it.
Now I will describe a collection of [p. 2] Ns which is not R to any N, unless there be another collection of Ns that is R to it. It shall contain all the Ns of class 4 none of class 3 and [2]. This is the collection I choose as a test. Is this the sole R of any N. Not of any N of the first class, since no such has any R at all. N of any N of the second class since all such have two Rs so that [illeg.] is sole R to any of these. Nor is the collection I have described R to any N of the third class, since no N of the third class is contained in it, while every N of the third class is contained in the sole collection that is R to it. Nor is this collection R to any N of the fourth class, since every such N of [sic.] is contained in it, while no N of the fourth class is contained in the sole collection that is R to it. So here is a collection described as definitely as you please which is not the sole R of any N.
I spoke of Deduction as the compulsive kind of reasonings. Almost all the theoric inferences are positively creative. That is, they create, not existent things, but entia rationis which are quite as real. This blackboard is black. Theoric deduction concludes that the board possesses the quality of blackness and that blackness is a simple object, called [p. 3] an ens rationis because that theoric thought creates it.
Object and Interpretant of a Sign.
The interpretant of Deductive Reasoning is Energetic, or as I call its generalization Existential. The Object of it, however, is purely Hypothetic. If that situation which Deduction reasons about happens to exist, that has no bearing at all on its reasoning. It is not so with Retroduction which starts from the actually perceived. Still less is it true of Induction which as we shall see, relates entirely to the actual course of Experience as its Object.
Induction, as I use the word, and I confess I cannot defend my case of the word,—is simply the acceptance of a hypothesis in so far as it has supported tests. Every test of a hypothesis consists in making it the basis for a conditional prediction, and if the prediction turns out to be verified, we reasonably accept the hypothesis provisionally. That is the highest certainty attainable by positive science.
The justification of it is that if it be false[,] perseverance in the method will correct its own result. Induction has three grades. The first is the simple assumption that the future will be like the past. Hence we deny fairies, ghosts, telepathy, etc.
The second is reasoning from a random sample. [p. 4]
The third is where a random sample is impossible because no counting or measurement is possible, but where we know that such a method must bring indefinite approximation to truth in the long run.
Organic chemistry. Aristotle. Plato and his Xth letter. Basil Valentine. [End of R 774]
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Bellucci, F., Pietarinen, AV. (2021). Methodeutic of Abduction. In: Shook, J.R., Paavola, S. (eds) Abduction in Cognition and Action. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 59. Springer, Cham. https://doi.org/10.1007/978-3-030-61773-8_5
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