View source for Abductive reasoning

{{Short description|Inference seeking the simplest and most likely explanation}}
[[File:Mastermind beispiel png.png|thumb|150px|A [[Software:Mastermind (board game)|Mastermind]] player uses abduction to infer the secret colors ''(top)'' from summaries ''(bottom left)'' of discrepancies in their guesses ''(bottom right)''.]] 
'''Abductive reasoning''' (also called '''abduction''',<ref name="Josephson">For example: {{cite book |editor1-last=Josephson |editor1-first=John R. |editor2-last=Josephson |editor2-first=Susan G. |date=1994 |title=Abductive Inference: Computation, Philosophy, Technology |location=Cambridge, UK; New York |publisher=Cambridge University Press |isbn=978-0521434614 |oclc=28149683 |doi=10.1017/CBO9780511530128|url=https://books.google.com/books?id=uu6zXrogwWAC }}</ref> '''abductive inference''',<ref name="Josephson"/> or '''retroduction'''<ref>{{Cite web|url = http://www.commens.org/dictionary/term/retroduction|title = Retroduction |access-date = 2014-08-24|website = Commens – Digital Companion to C. S. Peirce|publisher = Mats Bergman, Sami Paavola & João Queiroz|archive-url = https://web.archive.org/web/20140826115846/http://www.commens.org/dictionary/term/retroduction|archive-date = 2014-08-26|url-status = dead}}</ref>) is a form of logical inference that seeks the simplest and most likely conclusion from a set of observations. It was formulated and advanced by American [[Biography:Philosopher|philosopher]] [[Biography:Charles Sanders Peirce|Charles Sanders Peirce]] beginning in the latter half of the 19th century.

Abductive reasoning, unlike [[Deductive reasoning|deductive reasoning]], yields a plausible conclusion but does not definitively verify it. Abductive conclusions do not eliminate uncertainty or doubt, which is expressed in retreat terms such as "best available" or "most likely". One can understand abductive reasoning as inference to the best explanation,<ref>{{cite book |last=Sober |first=Elliott |date=2013 |title=Core Questions in Philosophy: A Text with Readings |edition=6th |location=Boston |publisher=Pearson Education |isbn=9780205206698 |oclc=799024771 |page=28 |quote=I now move to abduction—inference to the best explanation.}}</ref> although not all usages of the terms ''abduction'' and ''inference to the best explanation'' are equivalent.<ref>{{cite journal |last=Campos |first=Daniel G. |date=June 2011 |title=On the distinction between Peirce's abduction and Lipton's inference to the best explanation |journal=Synthese |volume=180 |issue=3 |pages=419–442 |doi=10.1007/s11229-009-9709-3 |s2cid=791688 |quote=I argue against the tendency in the philosophy of science literature to link abduction to the inference to the best explanation (IBE), and in particular, to claim that Peircean abduction is a conceptual predecessor to IBE. [...] In particular, I claim that Peircean abduction is an in-depth account of the process of generating explanatory hypotheses, while IBE, at least in [[Biography:Peter Lipton|Peter Lipton]]'s thorough treatment, is a more encompassing account of the processes both of generating and of evaluating scientific hypotheses. There is then a two-fold problem with the claim that abduction is IBE. On the one hand, it conflates abduction and induction, which are two distinct forms of logical inference, with two distinct aims, as shown by Charles S. Peirce; on the other hand it lacks a clear sense of the full scope of IBE as an account of scientific inference.}}</ref><ref>{{cite journal |last=Walton |first=Douglas |author-link=Douglas N. Walton |date=2001 |title=Abductive, presumptive and plausible arguments |journal=Informal Logic |volume=21 |issue=2 |pages=141–169 |doi=10.22329/il.v21i2.2241 |quote=Abductive inference has often been equated with inference to the best explanation. [...] The account of abductive inference and inference to the best explanation presented above has emphasized the common elements found in the analyses given by Peirce, Harman and the Josephsons. It is necessary to add that this brief account may be misleading in some respects, and that a closer and more detailed explication of the finer points of the three analyses could reveal important underlying philosophical differences. Inferences to the best explanation, as expounded by Harman and the Josephsons, can involve deductive and inductive processes of a kind that would be apparently be excluded by Peirce's account of abduction.|citeseerx=10.1.1.127.1593 }}</ref>

In the 1990s, as computing power grew, the fields of law,<ref>See, e.g. ''Analysis of Evidence, 2d ed.'' by Terence Anderson (Cambridge University Press, 2005)</ref> [[Computer science|computer science]], and [[Artificial intelligence|artificial intelligence]] research<ref name="Josephson, Magnani">For examples, see "[https://web.archive.org/web/20110720020440/http://www.cse.ohio-state.edu/lair/Projects/Abduction/abduction.html Abductive Inference in Reasoning and Perception]", John R. Josephson, Laboratory for Artificial Intelligence Research, Ohio State University, and ''Abduction, Reason, and Science. Processes of Discovery and Explanation'' by Lorenzo Magnani (Kluwer Academic/Plenum Publishers, New York, 2001).</ref> spurred renewed interest in the subject of abduction.<ref>
{{cite book | editor1-last=Flach | editor1-first=P. A. | editor1-link= Peter Flach | editor2-first= A. C. | editor2-last= Kakas |date=2000 |title=Abduction and Induction: Essays on their Relation and Integration |url={{google books |plainurl=y |id=E7fnCAAAQBAJ|page=13}} |publisher=Springer |page=xiii |access-date=31 October 2016|quote=This book grew out of a series of workshops on this topic. [Budapest 1996; Nagoya 1997; Brighton 1998] 
}}</ref>
Diagnostic [[Expert system|expert system]]s frequently employ abduction.<ref>Reggia, James A., et al. "[http://www.cs.umd.edu/~nau/papers/reggia1985answer1.pdf Answer justification in diagnostic expert systems-Part I: Abductive inference and its justification]." IEEE transactions on biomedical engineering 4 (1985): 263-267.</ref>

== Deduction, induction, and abduction==
{{Main|Philosophy:Logical reasoning}}

=== Deduction ===
{{Main|Deductive reasoning}}
Deductive reasoning allows deriving <math>b</math> from <math>a</math> only where <math>b</math> is a formal [[Logical consequence|logical consequence]] of <math>a</math>. In other words, deduction derives the consequences of the assumed. Given the truth of the assumptions, a valid deduction guarantees the truth of the conclusion. For example, given that "Wikis can be edited by anyone" (<math>a_1</math>) and "Wikipedia is a wiki" (<math>a_2</math>), it follows that "Wikipedia can be edited by anyone" (<math>b</math>).

=== Induction ===
{{Main|Philosophy:Inductive reasoning}}
Inductive reasoning is the process of inferring some ''general'' principle <math>b</math> from a body of knowledge <math>a</math>, where <math>b</math> does not necessarily follow from <math>a</math>. <math>a</math> might give us very good reason to accept <math>b</math>, but does not ensure <math>b</math>. For example, if all swans that we have observed so far are white, we may induce that the possibility that all swans are white is reasonable. We have good reason to believe the conclusion from the premise, but the truth of the conclusion is not guaranteed. (Indeed, it turns out that some swans are black.)

=== Abduction ===
Abductive reasoning allows inferring <math>a</math> as an explanation of <math>b</math>. As a result of this inference, abduction allows the precondition <math>a</math> to be abducted from the consequence <math>b</math>. [[Deductive reasoning]] and abductive reasoning thus differ in which end, left or right, of the proposition "<math>a</math> entails <math>b</math>" serves as conclusion. For example, in a billiard game, after glancing and seeing the ''eight'' ball moving towards us, we may abduce that the cue ball struck the eight ball. The strike of the cue ball would account for the movement of the eight ball. It serves as a hypothesis that explains our observation. Given the many possible explanations for the movement of the eight ball, our abduction does not leave us certain that the cue ball in fact struck the eight ball, but our abduction, still useful, can serve to orient us in our surroundings. Despite many possible explanations for any physical process that we observe, we tend to abduce a single explanation (or a few explanations) for this process in the expectation that we can better orient ourselves in our surroundings and disregard some possibilities. Properly used, abductive reasoning can be a useful source of [[Prior probability|priors]] in [[Bayesian statistics]].

==Formalizations of abduction==

===Logic-based abduction===
In [[Logic|logic]], [[Philosophy:Explanation|explanation]] is accomplished through the use of a logical theory <math>T</math> representing a [[Philosophy:Domain of discourse|domain]] and a set of observations <math>O</math>. Abduction is the process of deriving a set of explanations of <math>O</math> according to <math>T</math> and picking out one of those explanations. For <math>E</math> to be an explanation of <math>O</math> according to <math>T</math>, it should satisfy two conditions:

* <math>O</math> follows from <math>E</math> and <math>T</math>;
* <math>E</math> is consistent with <math>T</math>.

In formal logic, <math>O</math> and <math>E</math> are assumed to be sets of [[Literal (mathematical logic)|literal]]s. The two conditions for <math>E</math> being an explanation of <math>O</math> according to theory <math>T</math> are formalized as:

:<math>T \cup E \models O;</math>
:<math>T \cup E</math> is consistent.

Among the possible explanations <math>E</math> satisfying these two conditions, some other condition of minimality is usually imposed to avoid irrelevant facts (not contributing to the entailment of <math>O</math>) being included in the explanations. Abduction is then the process that picks out some member of <math>E</math>. Criteria for picking out a member representing "the best" explanation include the [[Philosophy:Simplicity|simplicity]], the [[Prior probability|prior probability]], or the explanatory power of the explanation.

A [[Proof theory|proof-theoretical]] abduction method for [[First-order logic|first-order]] classical logic based on the [[Sequent calculus|sequent calculus]] and a dual one, based on semantic tableaux (analytic tableaux) have been proposed.<ref>Cialdea Mayer, Marta and Pirri, Fiora (1993) "First order abduction via tableau and sequent calculi" Logic Jnl IGPL 1993 1: 99–117; {{doi|10.1093/jigpal/1.1.99}}. Oxford Journals</ref> The methods are sound and complete and work for full first-order logic, without requiring any preliminary reduction of formulae into normal forms. These methods have also been extended to [[Modal logic|modal logic]].<ref>Cialdea Mayer, Marta and Pirri, Fiora (1993) "Propositional abduction in modal logic" Logic Jnl IGPL 1995 3(6) 907–919; {{doi|10.1093/jigpal/3.6.907}}. Oxford Journals</ref>

[[Abductive logic programming]] is a computational framework that extends normal [[Logic programming|logic programming]] with abduction. It separates the theory <math>T</math> into two components, one of which is a normal logic program, used to generate <math>E</math> by means of backward reasoning, the other of which is a set of integrity constraints, used to filter the set of candidate explanations.

===Set-cover abduction===
A different formalization of abduction is based on inverting the function that calculates the visible effects of the hypotheses. Formally, we are given a set of hypotheses <math>H</math> and a set of manifestations <math>M</math>; they are related by the domain knowledge, represented by a function <math>e</math> that takes as an argument a set of hypotheses and gives as a result the corresponding set of manifestations. In other words, for every subset of the hypotheses <math>H' \subseteq H</math>, their effects are known to be <math>e(H')</math>.

Abduction is performed by finding a set <math>H' \subseteq H</math> such that <math>M \subseteq e(H')</math>. In other words, abduction is performed by finding a set of hypotheses <math>H'</math> such that their effects <math>e(H')</math> include all observations <math>M</math>.

A common assumption is that the effects of the hypotheses are independent, that is, for every <math>H' \subseteq H</math>, it holds that <math>e(H') = \bigcup_{h \in H'} e(\{h\})</math>. If this condition is met, abduction can be seen as a form of set covering.

===Abductive validation===
Abductive validation is the process of validating a given hypothesis through abductive reasoning. This can also be called reasoning through successive approximation.{{Citation needed|date=July 2020}} Under this principle, an explanation is valid if it is the best possible explanation of a set of known data. The best possible explanation is often defined in terms of simplicity and elegance (see [[Occam's razor]]). Abductive validation is common practice in hypothesis formation in [[Science|science]]; moreover, Peirce claims that it is a ubiquitous aspect of thought:

{{blockquote|
Looking out my window this lovely spring morning, I see an azalea in full bloom. No, no! I don't see that; though that is the only way I can describe what I see. That is a proposition, a sentence, a fact; but what I perceive is not proposition, sentence, fact, but only an image, which I make intelligible in part by means of a statement of fact. This statement is abstract; but what I see is concrete. I perform an abduction when I so much as express in a sentence anything I see. The truth is that the whole fabric of our knowledge is one matted felt of pure hypothesis confirmed and refined by induction. Not the smallest advance can be made in knowledge beyond the stage of vacant staring, without making an abduction at every step.<ref>Peirce MS. 692, quoted in Sebeok, T. (1981) "[http://www.visual-memory.co.uk/b_resources/abduction.html You Know My Method]" in Sebeok, T., ''The Play of Musement'', Bloomington, IA: Indiana, page 24.</ref>
}}

It was Peirce's own maxim that "Facts cannot be explained by a hypothesis more extraordinary than these facts themselves; and of various hypotheses the least extraordinary must be adopted."<ref>Peirce MS. 696, quoted in Sebeok, T. (1981) "[http://www.visual-memory.co.uk/b_resources/abduction.html You Know My Method]" in Sebeok, T., ''The Play of Musement'', Bloomington, IA: Indiana, page 31.</ref> After obtaining possible hypotheses that may explain the facts, abductive validation is a method for identifying the most likely hypothesis that should be adopted.

===Subjective logic abduction===
[[Subjective logic]] generalises [[Probabilistic logic|probabilistic logic]] by including degrees of epistemic [[Uncertainty quantification|uncertainty]] in the input arguments, i.e. instead of probabilities, the analyst can express arguments as [[Subjective logic|subjective opinions]]. Abduction in subjective logic is thus a generalization of probabilistic abduction described above.<ref name="Josang2016-SL" /> The input arguments in subjective logic are subjective opinions which can be binomial when the opinion applies to a binary variable or multinomial when it applies to an ''n''-ary variable. A subjective opinion thus applies to a state variable <math>X</math> which takes its values from a domain <math>\mathbf{X}</math> (i.e. a state space of exhaustive and mutually disjoint state values <math>x</math>), and is denoted by the tuple <math>\omega_{X}=(b_{X}, u_{X}, a_{X})\,\!</math>, where <math>b_{X}\,\!</math> is the belief mass distribution over <math>\mathbf{X}</math>, <math>u_{X}\,\!</math> is the epistemic uncertainty mass, and <math>a_{X}\,\!</math> is the [[Base rate|base rate]] distribution over <math>\mathbf{X}</math>. These parameters satisfy <math>u_{X}+\sum b_{X}(x) = 1\,\!</math> and <math>\sum a_{X}(x) = 1\,\!</math> as well as <math>b_{X}(x),u_{X},a_{X}(x) \in [0,1]\,\!</math>.

Assume the domains <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> with respective variables <math>X</math> and <math>Y</math>, the set of conditional opinions <math>\omega_{X\mid Y}</math> (i.e. one conditional opinion for each value <math>y</math>), and the base rate distribution <math>a_{Y}</math>. Based on these parameters, the subjective [[Bayes' theorem]] denoted with the operator <math>\;\widetilde{\phi}</math> produces the set of inverted conditionals <math>\omega_{Y\tilde{\mid} X}</math> (i.e. one inverted conditional for each value <math>x</math>) expressed by:

:<math>\omega_{Y\tilde{|}X}=\omega_{X|Y}\;\widetilde{\phi\,}\;a_{Y}</math>.

Using these inverted conditionals together with the opinion <math>\omega_{X}</math> subjective [[Deductive reasoning|deduction]] denoted by the operator <math>\circledcirc</math> can be used to abduce the marginal opinion <math>\omega_{Y\,\overline{\|}\,X}</math>. The equality between the different expressions for subjective abduction is given below:

:<math>\begin{align}
\omega_{Y\,\widetilde{\|}\,X} 
&=  \omega_{X\mid Y} \;\widetilde{\circledcirc}\; \omega_{X}\\
&= (\omega_{X\mid Y} \;\widetilde{\phi\,}\; a_{Y}) \;\circledcirc\;\omega_{X}\\
&= \omega_{Y\widetilde{|}X} \;\circledcirc\;\omega_{X}\;.
\end{align}</math>

The symbolic notation for subjective abduction is "<math>\widetilde{\|}</math>", and the operator itself is denoted as "<math>\widetilde{\circledcirc}</math>". The operator for the subjective Bayes' theorem is denoted "<math>\widetilde{\phi\,}</math>", and subjective deduction is denoted "<math>\circledcirc</math>".<ref name="Josang2016-SL">A. Jøsang. ''Subjective Logic: A Formalism for Reasoning Under Uncertainty'', Springer 2016, {{ISBN|978-3-319-42337-1}}</ref>

The advantage of using subjective logic abduction compared to probabilistic abduction is that both aleatoric and epistemic uncertainty about the input argument probabilities can be explicitly expressed and taken into account during the analysis. It is thus possible to perform abductive analysis in the presence of uncertain arguments, which naturally results in degrees of uncertainty in the output conclusions.

==History==

The idea that the simplest, most easily verifiable solution should be preferred over its more complicated counterparts is a very old one. To this point, [[Biography:George Pólya|George Pólya]], in his treatise on problem-solving, makes reference to the following Latin truism: ''simplex sigillum veri'' (simplicity is the seal of truth).<ref>{{cite book |last1=Pólya |first1=George |title=How to solve it: a new aspect of mathematical method |date=1945 |publisher=Princeton University Press |location=Princeton [N.J.] |isbn=0-691-11966-X |page=45 |edition=Expanded Princeton Science Library (2004)}}</ref>


===Introduction and development by Peirce===
====Overview====
The American philosopher Charles Sanders Peirce introduced abduction into modern logic. Over the years he called such inference ''hypothesis'', ''abduction'', ''presumption'', and ''retroduction''. He considered it a topic in logic as a normative field in philosophy, not in purely formal or mathematical logic, and eventually as a topic also in economics of research.

As two stages of the development, extension, etc., of a hypothesis in scientific [[Inquiry|inquiry]], abduction and also [[Philosophy:Inductive reasoning|induction]] are often collapsed into one overarching concept—the hypothesis. That is why, in the [[Scientific method|scientific method]] known from [[Biography:Galileo Galilei|Galileo]] and [[Biography:Francis Bacon|Bacon]], the abductive stage of hypothesis formation is conceptualized simply as induction. Thus, in the twentieth century this collapse was reinforced by [[Biography:Karl Popper|Karl Popper]]'s explication of the [[Hypothetico-deductive model|hypothetico-deductive model]], where the hypothesis is considered to be just "a guess"<ref>{{cite book|last=Popper|first=Karl|title=Conjectures and Refutations: The Growth of Scientific Knowledge|location=London|publisher=Routledge|edition=2|date=2002|page=536}}</ref> (in the spirit of Peirce). However, when the formation of a hypothesis is considered the result of a process it becomes clear that this "guess" has already been tried and made more robust in thought as a necessary stage of its acquiring the status of hypothesis. Indeed, many abductions are rejected or heavily modified by subsequent abductions before they ever reach this stage.

Before 1900, Peirce treated abduction as the use of a known rule to explain an observation. For instance: it is a known rule that, if it rains, grass gets wet; so, to explain the fact that the grass on this lawn is wet, one ''abduces'' that it has rained. Abduction can lead to false conclusions if other rules that might explain the observation are not taken into account{{mdash}}e.g. the grass could be wet from dew. This remains the common use of the term "abduction" in the [[Social:Social science|social science]]s and in [[Artificial intelligence|artificial intelligence]].

Peirce consistently characterized it as the kind of inference that originates a hypothesis by concluding in an explanation, though an unassured one, for some very curious or surprising (anomalous) observation stated in a premise. As early as 1865 he wrote that all conceptions of cause and force are reached through hypothetical inference; in the 1900s he wrote that all explanatory content of theories is reached through abduction. In other respects Peirce revised his view of abduction over the years.<ref>See Santaella, Lucia (1997) "The Development of Peirce's Three Types of Reasoning: Abduction, Deduction, and Induction", 6th Congress of the IASS. [http://www.pucsp.br/~lbraga/epap_peir1.htm Eprint].</ref>

In later years his view came to be:
* Abduction is guessing.<ref name="guess">Peirce, C. S. 
* "On the Logic of drawing History from Ancient Documents especially from Testimonies" (1901), ''Collected Papers'' v. 7, paragraph 219.
* "PAP" ["Prolegomena to an Apology for Pragmatism"], MS 293 c. 1906, ''New Elements of Mathematics'' v. 4, pp. 319–320.
* A Letter to F. A. Woods (1913), ''Collected Papers'' v. 8, paragraphs 385–388.
(See under "[http://www.helsinki.fi/science/commens/terms/abduction.html Abduction]" and "[http://www.helsinki.fi/science/commens/terms/retroduction.html Retroduction]" at ''Commens Dictionary of Peirce's Terms''.)</ref> It is "very little hampered" by rules of logic.<ref name="HL">Peirce, C. S. (1903), Harvard lectures on pragmatism, ''Collected Papers'' v. 5, [http://www.textlog.de/7664-2.html paragraphs 188–189].</ref> Even a well-prepared mind's individual guesses are more frequently wrong than right.<ref>Peirce, C. S. (1908), "A Neglected Argument for the Reality of God", ''Hibbert Journal'' v. 7, pp. 90–112, see §4. In ''Collected Papers'' v. 6, see paragraph 476. In ''The Essential Peirce'' v. 2, see p. 444.</ref> But the success of our guesses far exceeds that of random luck and seems born of attunement to nature by instinct<ref name="NA">Peirce, C. S. (1908), "A Neglected Argument for the Reality of God", ''Hibbert Journal'' v. 7, pp. 90–112. See both part III and part IV. Reprinted, including originally unpublished portion, in ''Collected Papers'' v. 6, paragraphs 452–85, ''Essential Peirce'' v. 2, pp. 434–50, and elsewhere.</ref> (some speak of [[Philosophy:Logical intuition|intuition]] in such contexts<ref>Peirce used the term "intuition" not in the sense of an instinctive or anyway half-conscious inference as people often do currently. Instead he used "intuition" usually in the sense of a cognition devoid of logical determination by [[Philosophy:A priori and a posteriori|previous cognitions]]. He said, "We have no power of Intuition" in that sense. See his "Some Consequences of Four Incapacities" (1868), [http://www.peirce.org/writings/p27.html Eprint] {{Webarchive|url=https://web.archive.org/web/20110514094121/http://www.peirce.org/writings/p27.html |date=2011-05-14 }}.</ref>).
* Abduction guesses a new or outside idea so as to account in a plausible, instinctive, economical way for a surprising or very complicated phenomenon. That is its proximate aim.<ref name="NA" />
* Its longer aim is to economize [[Inquiry|inquiry]] itself. Its rationale is inductive: it works often enough, is the only source of new ideas, and has no substitute in expediting the discovery of new truths.<ref>For a relevant discussion of Peirce and the aims of abductive inference, see McKaughan, Daniel J. (2008), "[https://muse.jhu.edu/article/252833/summary From Ugly Duckling to Swan: C. S. Peirce, Abduction, and the Pursuit of Scientific Theories]", ''Transactions of the Charles S. Peirce Society'', v. 44, no. 3 (summer), 446–468.</ref> Its rationale especially involves its role in coordination with other modes of inference in inquiry. It is inference to explanatory hypotheses for selection of those best worth trying.
* [[Philosophy:Pragmatism|Pragmatism]] is the logic of abduction. Upon the generation of an explanation (which he came to regard as instinctively guided), the [[Philosophy:Pragmatic maxim|pragmatic maxim]] gives the necessary and sufficient logical rule to abduction in general. The hypothesis, being insecure, needs to have conceivable<ref>Peirce means "conceivable" very broadly. See ''Collected Papers'' v. 5, paragraph 196, or ''Essential Peirce'' v. 2, p. 235, "Pragmatism as the Logic of Abduction" (Lecture VII of the 1903 Harvard lectures on pragmatism): {{blockquote|It allows any flight of imagination, provided this imagination ultimately alights upon a possible practical effect; and thus many hypotheses may seem at first glance to be excluded by the pragmatical maxim that are not really so excluded.}}</ref> implications for informed practice, so as to be testable<ref name="L75">Peirce, C. S., Carnegie Application (L75, 1902, ''New Elements of Mathematics'' v. 4, pp. 37–38. See under "[http://www.helsinki.fi/science/commens/terms/abduction.html Abduction]" at the ''Commens Dictionary of Peirce's Terms'': {{blockquote|Methodeutic has a special interest in Abduction, or the inference which starts a scientific hypothesis. For it is not sufficient that a hypothesis should be a justifiable one. Any hypothesis which explains the facts is justified critically. But among justifiable hypotheses we have to select that one which is suitable for being tested by experiment.}}</ref><ref name="prag">Peirce, "Pragmatism as the Logic of Abduction" (Lecture VII of the 1903 Harvard lectures on pragmatism), see parts III and IV. Published in part in ''Collected Papers'' v. 5, paragraphs 180–212 (see 196–200, [http://www.textlog.de/7663.html Eprint] and in full in ''Essential Peirce'' v. 2, pp. 226–241 (see sections III and IV). {{blockquote|.... What is good abduction? What should an explanatory hypothesis be to be worthy to rank as a hypothesis? Of course, it must explain the facts. But what other conditions ought it to fulfill to be good? .... Any hypothesis, therefore, may be admissible, in the absence of any special reasons to the contrary, provided it be capable of experimental verification, and only insofar as it is capable of such verification. This is approximately the doctrine of pragmatism.}}</ref> and, through its trials, to expedite and economize inquiry. The economy of research is what calls for abduction and governs its art.<ref name="econ">Peirce, C.S. (1902), application to the Carnegie Institution, see MS L75.329-330, from [http://www.cspeirce.com/menu/library/bycsp/l75/ver1/l75v1-08.htm#m27 Draft D] {{Webarchive|url=https://web.archive.org/web/20110524021101/http://www.cspeirce.com/menu/library/bycsp/l75/ver1/l75v1-08.htm#m27 |date=2011-05-24 }} of Memoir 27:
{{blockquote|Consequently, to discover is simply to expedite an event that would occur sooner or later, if we had not troubled ourselves to make the discovery. Consequently, the art of discovery is purely a question of economics. The economics of research is, so far as logic is concerned, the leading doctrine with reference to the art of discovery. Consequently, the conduct of abduction, which is chiefly a question of [[Heuristic|heuristic]] and is the first question of heuristic, is to be governed by economical considerations.}}</ref>

Writing in 1910, Peirce admits that "in almost everything I printed before the beginning of this century I more or less mixed up hypothesis and induction" and he traces the confusion of these two types of reasoning to logicians' too "narrow and formalistic a conception of inference, as necessarily having formulated judgments from its premises."<ref>Peirce, A Letter to Paul Carus circa 1910, ''Collected Papers'' v. 8, paragraphs 227–228. See under "[http://www.helsinki.fi/science/commens/terms/hypothesis.html Hypothesis]" at the ''Commens Dictionary of Peirce's Terms''.</ref>

He started out in the 1860s treating hypothetical inference in a number of ways which he eventually peeled away as inessential or, in some cases, mistaken:
* as inferring the occurrence of a character (a characteristic) from the observed combined occurrence of multiple characters which its occurrence would necessarily involve;<ref name="NCA">(1867), "On the Natural Classification of Arguments", ''Proceedings of the American Academy of Arts and Sciences'' v. 7, pp. 261–287. Presented April 9, 1867. See especially starting at [{{google books |plainurl=y |id=nG8UAAAAYAAJ|page=284}} p. 284] in Part III §1. Reprinted in ''Collected Papers v. 2, paragraphs 461–516 and ''Writings'' v. 2, pp. 23–49.''</ref> for example, if any occurrence of ''A'' is known to necessitate occurrence of ''B, C, D, E'', then the observation of ''B, C, D, E'' suggests by way of explanation the occurrence of ''A''. (But by 1878 he no longer regarded such multiplicity as common to all hypothetical inference.<ref name="DIH">Peirce, C. S. (1878), "Deduction, Induction, and Hypothesis", ''Popular Science Monthly'', v. 13, pp. 470–82, see [{{google books |plainurl=y |id=u8sWAQAAIAAJ|page=472}} 472]. ''Collected Papers'' 2.619–44, see 623.</ref>[https://en.wikisource.org/wiki/Popular_Science_Monthly/Volume_13/August_1878/Illustrations_of_the_Logic_of_Science_VI Wikisource])
* as aiming for a more or less probable hypothesis (in 1867 and 1883 but not in 1878; anyway by 1900 the justification is not probability but the lack of alternatives to guessing and the fact that guessing is fruitful;<ref name="L2L">A letter to Langley, 1900, published in ''Historical Perspectives on Peirce's Logic of Science''. See excerpts under "[http://www.helsinki.fi/science/commens/terms/abduction.html Abduction]" at the ''Commens Dictionary of Peirce's Terms''.</ref> by 1903 he speaks of the "likely" in the sense of nearing the truth in an "indefinite sense";<ref>"A Syllabus of Certain Topics of Logic'" (1903 manuscript), ''Essential Peirce'' v. 2, see p. 287. See under "[http://www.helsinki.fi/science/commens/terms/abduction.html Abduction]" at the ''Commens Dictionary of Peirce's Terms''.</ref> by 1908 he discusses ''plausibility'' as instinctive appeal.<ref name="NA" />) In a paper dated by editors as ''circa'' 1901, he discusses "instinct" and "naturalness", along with the kind of considerations (low cost of testing, logical caution, breadth, and incomplexity) that he later calls methodeutical.<ref>Peirce, C. S., "On the Logic of Drawing History from Ancient Documents", dated as ''circa'' 1901 both by the editors of ''Collected Papers'' (see CP v. 7, bk 2, ch. 3, footnote 1) and by those of the ''Essential Peirce'' (EP) ([http://www.iupui.edu/~peirce/ep/ep2/headers/ep2headx.htm#8 Eprint] {{Webarchive|url=https://web.archive.org/web/20120905022758/http://www.iupui.edu/~peirce/ep/ep2/headers/ep2headx.htm#8 |date=2012-09-05 }}. The article's discussion of abduction is in CP v. 7, paragraphs 218–31 and in EP v. 2, pp. 107–14.</ref>
* as induction from characters (but as early as 1900 he characterized abduction as guessing<ref name="L2L" />)
* as citing a known rule in a premise rather than hypothesizing a rule in the conclusion (but by 1903 he allowed either approach<ref name="HL" /><ref name="newidea">Peirce, C. S., "A Syllabus of Certain Topics of Logic" (1903), ''Essential Peirce'' v. 2, p. 287:
{{blockquote| The mind seeks to bring the facts, as modified by the new discovery, into order; that is, to form a general conception embracing them. In some cases, it does this by an act of ''generalization''. In other cases, no new law is suggested, but only a peculiar state of facts that will "explain" the surprising phenomenon; and a law already known is recognized as applicable to the suggested hypothesis, so that the phenomenon, under that assumption, would not be surprising, but quite likely, or even would be a necessary result. This synthesis suggesting a new conception or hypothesis, is the Abduction.}}</ref>)
* as basically a transformation of a deductive categorical syllogism<ref name="DIH" /> (but in 1903 he offered a variation on ''modus ponens'' instead,<ref name="HL" /> and by 1911 he was unconvinced that any one form covers all hypothetical inference<ref name="kehler">A Letter to J. H. Kehler (1911), ''New Elements of Mathematics'' v. 3, pp. 203–4, see under "[http://www.helsinki.fi/science/commens/terms/retroduction.html Retroduction]" at ''Commens Dictionary of Peirce's Terms''.</ref>).

====''The Natural Classification of Arguments'' (1867)====
In 1867, Peirce's "On the Natural Classification of Arguments",<ref name="NCA" /> hypothetical inference always deals with a cluster of characters (call them ''P′, P′′, P′′′,'' etc.) known to occur at least whenever a certain character (''M'') occurs. Note that categorical syllogisms have elements traditionally called middles, predicates, and subjects. For example: All ''men'' [middle] are ''mortal'' [predicate]; ''Socrates'' [subject] is a ''man'' [middle]; ergo ''Socrates'' [subject] is ''mortal'' [predicate]". Below, 'M' stands for a middle; 'P' for a predicate; 'S' for a subject. Peirce held that all deduction can be put into the form of the categorical [[Philosophy:Syllogism|syllogism]] [[Philosophy:Syllogism#Barbara (AAA-1)|Barbara (AAA-1)]].

<blockquote>
{| cellspacing=1 cellpadding=7 style="background-color:#999"
|- style="vertical-align:top;background-color:#fff"
|[Deduction].

[Any] M is P <br />
[Any] S is M <br />
<math>\therefore</math> [Any] S is P. 
| Induction.

''S′, S′′, S′′′'', &c. are taken at random as ''M'''s; <br />
''S′, S′′, S′′′'', &c. are ''P'': <br />
<math>\therefore</math> Any ''M'' is probably ''P''.
|Hypothesis.

Any ''M'' is, for instance, ''P′, P′′, P′′′,'' &c.; <br />
''S'' is ''P′, P′′, P′′′,'' &c.: <br />
<math>\therefore</math> ''S'' is probably ''M''.
|}
</blockquote>

====''Deduction, Induction, and Hypothesis'' (1878)====
In 1878, in "Deduction, Induction, and Hypothesis",<ref name="DIH" /> there is no longer a need for multiple characters or predicates in order for an inference to be hypothetical, although it is still helpful. Moreover, Peirce no longer poses hypothetical inference as concluding in a ''probable'' hypothesis. In the forms themselves, it is understood but not explicit that induction involves random selection and that hypothetical inference involves response to a "very curious circumstance". The forms instead emphasize the modes of inference as rearrangements of one another's propositions (without the bracketed hints shown below).

{| cellspacing=1 cellpadding=3 style="background-color:#999"
|- vAlign=top style="background-color:#fff"
|Deduction.

''Rule:'' All the beans from this bag are white. <br />
''Case:'' These beans are from this bag. <br />
<math>\therefore</math> ''Result:'' These beans are white.
| Induction.

''Case:'' These beans are [randomly selected] from this bag. <br />
''Result:'' These beans are white. <br />
<math>\therefore</math> ''Rule:'' All the beans from this bag are white.
|Hypothesis.

''Rule:'' All the beans from this bag are white. <br />
''Result:'' These beans [oddly] are white. <br />
<math>\therefore</math> ''Case:'' These beans are from this bag.
|}

====''A Theory of Probable Inference'' (1883)====
Peirce long treated abduction in terms of induction from characters or traits (weighed, not counted like objects), explicitly so in his influential 1883 "A theory of probable inference", in which he returns to involving probability in the hypothetical conclusion.<ref name=Pierce-1883>{{cite book |author-last=Peirce |author-first=Charles S. |date=1883 |chapter=A theory of probable inference |editor-last=Peirce |editor-first=Charles S. |title=Studies in Logic by Members of the Johns Hopkins University |location=Boston, MA |chapter-url=http://www.commens.org/bibliography/collection_article/peirce-charles-s-1883-theory-probable-inference-studies-logic |title-link=Charles Sanders Peirce bibliography#SIL |access-date=March 7, 2019 |archive-date=March 8, 2019 |archive-url=https://web.archive.org/web/20190308080846/http://www.commens.org/bibliography/collection_article/peirce-charles-s-1883-theory-probable-inference-studies-logic |url-status=dead }}</ref> Like "Deduction, Induction, and Hypothesis" in 1878, it was widely read (see the historical books on statistics by [[Biography:Stephen Stigler|Stephen Stigler]]), unlike his later amendments of his conception of abduction. Today abduction remains most commonly understood as induction from characters and extension of a known rule to cover unexplained circumstances.

Sherlock Holmes used this method of reasoning in the stories of [[Biography:Arthur Conan Doyle|Arthur Conan Doyle]], although Holmes refers to it as "[[Deductive reasoning|deductive reasoning]]".<ref>{{cite journal |last1=Sebeok |first1=Thomas A. |last2=Umiker-Sebeok |first2=Jean |date=1979 |title='You know my method': A juxtaposition of Charles S. Peirce and Sherlock Holmes |journal=Semiotica |volume=26 |issue=3–4 |pages=203–250 |doi=10.1515/semi.1979.26.3-4.203 |s2cid=170683439 |quote=Marcello Truzzi, in a searching article on Holmes's method (1973:93–126), anticipated our present work by pointing to the similarities between the detective's so-called deductions, or inductions, and Peirce's abductions, or conjectures. According to Peirce's system of logic, furthermore, Holmes's observations are themselves a form of abduction, and abduction is as legitimate a type of logical inference as either induction or deduction (Peirce 8.228).}}</ref><ref>{{cite journal |last=Niiniluoto |first=Ilkka |author-link=Ilkka Niiniluoto |date=September 1999 |title=Defending abduction |journal=Philosophy of Science |volume=66 |issue=Supplement&nbsp;1 |pages=S436–S451 (S440–S441) |doi=10.1086/392744 |s2cid=224841752 |quote=A historically interesting application of abduction as a heuristic method can be found in classical detective stories, as shown by the semiotical and logical essays collected in Eco and Sebeok 1983. C. Auguste Dupin, the hero of [[Biography:Edgar Allan Poe|Edgar Allan Poe]]'s novels in the 1840s, employed a method of 'ratiocination' or 'analysis' which has the structure of retroduction. Similarly, the logic of the 'deductions' of Sherlock Holmes is typically abductive.}}</ref><ref>{{cite journal |last=Carson |first=David |date=June 2009 |title=The abduction of Sherlock Holmes |journal=International Journal of Police Science & Management |volume=11 |issue=2 |pages=193–202 |doi=10.1350/ijps.2009.11.2.123 |s2cid=145337828 |quote=Sherlock Holmes, although a fictional character, remains renowned as a great detective. However, his methodology, which was ''abduction'' rather than deduction, and which is innocently used by many real detectives, is rarely described, discussed, or researched. This paper compares and contrasts the three forms of inferential reasoning, and makes a case for articulating and developing the role of abduction in the work, and training, of police officers. |url=https://researchportal.port.ac.uk/ws/files/84295/ijps.2009.11.2.pdf }}</ref>

====''Minute Logic'' (1902) and after====
In 1902 Peirce wrote that he now regarded the syllogistical forms and the doctrine of extension and comprehension (i.e., objects and characters as referenced by terms), as being less fundamental than he had earlier thought.<ref>In Peirce, C. S., 'Minute Logic' circa 1902, ''Collected Papers'' v. 2, paragraph 102. See under "[http://www.helsinki.fi/science/commens/terms/abduction.html Abduction]" at ''Commens Dictionary of Peirce's Terms''.</ref> In 1903 he offered the following form for abduction:<ref name="HL" /> 
{{blockquote|The surprising fact, C, is observed; 
:But if A were true, C would be a matter of course,
:Hence, there is reason to suspect that A is true.}}
The hypothesis is framed, but not asserted, in a premise, then asserted as rationally suspectable in the conclusion. Thus, as in the earlier categorical syllogistic form, the conclusion is formulated from some premise(s). But all the same the hypothesis consists more clearly than ever in a new or outside idea beyond what is known or observed. Induction in a sense goes beyond observations already reported in the premises, but it merely amplifies ideas already known to represent occurrences, or tests an idea supplied by hypothesis; either way it requires previous abductions in order to get such ideas in the first place. Induction seeks facts to test a hypothesis; abduction seeks a hypothesis to account for facts.

Note that the hypothesis ("A") could be of a rule. It need not even be a rule strictly necessitating the surprising observation ("C"), which needs to follow only as a "matter of course"; or the "course" itself could amount to some known rule, merely alluded to, and also not necessarily a rule of strict necessity. In the same year, Peirce wrote that reaching a hypothesis may involve placing a surprising observation under either a newly hypothesized rule or a hypothesized combination of a known rule with a peculiar state of facts, so that the phenomenon would be not surprising but instead either necessarily implied or at least likely.<ref name="newidea" />

Peirce did not remain quite convinced about any such form as the categorical syllogistic form or the 1903 form. In 1911, he wrote, "I do not, at present, feel quite convinced that any logical form can be assigned that will cover all 'Retroductions'. For what I mean by a Retroduction is simply a conjecture which arises in the mind."<ref name="kehler" />

====Pragmatism====
In 1901 Peirce wrote, "There would be no logic in imposing rules, and saying that they ought to be followed, until it is made out that the purpose of hypothesis requires them."<ref>Peirce, "On the Logic of drawing History from Ancient Documents", 1901 manuscript, ''Collected Papers'' v. 7, paragraphs 164–231, see 202, reprinted in ''Essential Peirce'' v. 2, pp. 75–114, see 95. See under "[http://www.helsinki.fi/science/commens/terms/abduction.html Abduction]" at ''Commens Dictionary of Peirce's Terms''.</ref> In 1903 Peirce called [[Philosophy:Pragmatism|pragmatism]] "the logic of abduction" and said that the [[Philosophy:Pragmatic maxim|pragmatic maxim]] gives the necessary and sufficient logical rule to abduction in general.<ref name="prag" /> The pragmatic maxim is:

{{blockquote|Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.}}

It is a method for fruitful clarification of conceptions by equating the meaning of a conception with the conceivable practical implications of its object's conceived effects. Peirce held that that is precisely tailored to abduction's purpose in inquiry, the forming of an idea that could conceivably shape informed conduct. In various writings in the 1900s<ref name="econ" /><ref>Peirce, "On the Logic of Drawing Ancient History from Documents", ''Essential Peirce'' v. 2, see pp. 107–9.</ref> he said that the conduct of abduction (or retroduction) is governed by considerations of economy, belonging in particular to the economics of research. He regarded economics as a normative science whose analytic portion might be part of logical methodeutic (that is, theory of inquiry).<ref>Peirce, Carnegie application, L75 (1902), Memoir 28: "On the Economics of Research", scroll down to Draft E. [http://www.cspeirce.com/menu/library/bycsp/l75/ver1/l75v1-08.htm#m28 Eprint] {{Webarchive|url=https://web.archive.org/web/20110524021101/http://www.cspeirce.com/menu/library/bycsp/l75/ver1/l75v1-08.htm#m28 |date=2011-05-24 }}.</ref>

====Three levels of logic about abduction====
Peirce came over the years to [[Classification of the sciences (Peirce)#Sciences|divide (philosophical) logic]] into three departments: 
# Stechiology, or speculative grammar, on the conditions for meaningfulness. Classification of signs (semblances, symptoms, symbols, etc.) and their combinations (as well as their objects and interpretants).
# Logical critic, or logic proper, on validity or justifiability of inference, the conditions for true representation. Critique of arguments in their various modes (deduction, induction, abduction).
# Methodeutic, or speculative rhetoric, on the conditions for determination of interpretations. Methodology of inquiry in its interplay of modes.
Peirce had, from the start, seen the modes of inference as being coordinated together in scientific inquiry and, by the 1900s, held that hypothetical inference in particular is inadequately treated at the level of critique of arguments.<ref name="L75" /><ref name="prag" /> To increase the assurance of a hypothetical conclusion, one needs to deduce implications about evidence to be found, predictions which induction can test through observation so as to evaluate the hypothesis. That is [[Biography:Charles Sanders Peirce#Scientific method|Peirce's outline of the scientific method]] of inquiry, as covered in his inquiry methodology, which includes [[Philosophy:Pragmatism|pragmatism]] or, as he later called it, [[Philosophy:Pragmaticism|pragmaticism]], the clarification of ideas in terms of their conceivable implications regarding informed practice.

=====Classification of signs=====
As early as 1866,<ref>Peirce, C. S., the 1866 Lowell Lectures on the Logic of Science, ''Writings of Charles S. Peirce'' v. 1, p. 485. See under "[http://www.helsinki.fi/science/commens/terms/hypothesis.html Hypothesis]" at ''Commens Dictionary of Peirce's Terms''.</ref> Peirce held that:

1. Hypothesis (abductive inference) is inference through an ''icon'' (also called a ''likeness''). <br />
2. Induction is inference through an ''index'' (a sign by factual connection); a sample is an index of the totality from which it is drawn. <br />
3. Deduction is inference through a ''symbol'' (a sign by interpretive habit irrespective of resemblance or connection to its object).

In 1902, Peirce wrote that, in abduction: "It is recognized that the phenomena are ''like'', i.e. constitute an Icon of, a replica of a general conception, or Symbol."<ref>Peirce, C. S., "A Syllabus of Certain Topics of Logic", written 1903. See ''The Essential Peirce'' v. 2, p. 287. Quote viewable under "[http://www.helsinki.fi/science/commens/terms/abduction.html Abduction]" at ''Commens Dictionary of Peirce's Terms''.</ref>

=====Critique of arguments=====
At the critical level Peirce examined the forms of abductive arguments (as discussed above), and came to hold that the hypothesis should economize explanation for plausibility in terms of the feasible and natural. In 1908 Peirce described this plausibility in some detail.<ref name="NA" /> It involves not likeliness based on observations (which is instead the inductive evaluation of a hypothesis), but instead optimal simplicity in the sense of the "facile and natural", as by Galileo's natural light of reason and as distinct from "logical simplicity" (Peirce does not dismiss logical simplicity entirely but sees it in a subordinate role; taken to its logical extreme it would favor adding no explanation to the observation at all). Even a well-prepared mind guesses oftener wrong than right, but our guesses succeed better than random luck at reaching the truth or at least advancing the inquiry, and that indicates to Peirce that they are based in instinctive attunement to nature, an affinity between the mind's processes and the processes of the real, which would account for why appealingly "natural" guesses are the ones that oftenest (or least seldom) succeed; to which Peirce added the argument that such guesses are to be preferred since, without "a natural bent like nature's", people would have no hope of understanding nature. In 1910 Peirce made a three-way distinction between probability, verisimilitude, and plausibility, and defined plausibility with a normative "ought": "By plausibility, I mean the degree to which a theory ought to recommend itself to our belief independently of any kind of evidence other than our instinct urging us to regard it favorably."<ref>Peirce, A Letter to Paul Carus 1910, ''Collected Papers'' v. 8, see paragraph 223.</ref> For Peirce, plausibility does not depend on observed frequencies or probabilities, or on verisimilitude, or even on testability, which is not a question of the critique of the hypothetical inference ''as'' an inference, but rather a question of the hypothesis's relation to the inquiry process.

The phrase "inference to the best explanation" (not used by Peirce but often applied to hypothetical inference) is not always understood as referring to the most simple and natural hypotheses (such as those with the [[Occam's razor|fewest assumptions]]). However, in other senses of "best", such as "standing up best to tests", it is hard to know which is the best explanation to form, since one has not tested it yet. Still, for Peirce, any justification of an abductive inference as "good" is not completed upon its formation as an argument (unlike with induction and deduction) and instead depends also on its methodological role and promise (such as its testability) in advancing inquiry.<ref name="L75" /><ref name="prag" /><ref>Peirce, C. S. (1902), Application to the Carnegie Institution, Memoir 27, [http://www.cspeirce.com/menu/library/bycsp/l75/ver1/l75v1-08.htm#m27 Eprint] {{Webarchive|url=https://web.archive.org/web/20110524021101/http://www.cspeirce.com/menu/library/bycsp/l75/ver1/l75v1-08.htm#m27 |date=2011-05-24 }}: "Of the different classes of arguments, abductions are the only ones in which after they have been admitted to be just, it still remains to inquire whether they are advantageous."</ref>

=====Methodology of inquiry=====
At the methodeutical level Peirce held that a hypothesis is judged and selected<ref name="L75" /> for testing because it offers, via its trial, to expedite and economize the [[Inquiry|inquiry]] process itself toward new truths, first of all by being testable and also by further economies,<ref name="econ" /> in terms of cost, value, and relationships among guesses (hypotheses). Here, considerations such as probability, absent from the treatment of abduction at the critical level, come into play. For examples:
* Cost: A simple but low-odds guess, if low in cost to test for falsity, may belong first in line for testing, to get it out of the way. If surprisingly it stands up to tests, that is worth knowing early in the inquiry, which otherwise might have stayed long on a wrong though seemingly likelier track.
* Value: A guess is intrinsically worth testing if it has instinctual plausibility or reasoned objective probability, while subjective likelihood, though reasoned, can be treacherous.
* Interrelationships: Guesses can be chosen for trial strategically for their
** ''caution'', for which Peirce gave as an example the game of Twenty Questions,
** ''breadth'' of applicability to explain various phenomena, and
** ''incomplexity'', that of a hypothesis that seems too simple but whose trial "may give a good 'leave', as the billiard-players say", and be instructive for the pursuit of various and conflicting hypotheses that are less simple.<ref name="econ2">Peirce, "On the Logic of Drawing Ancient History from Documents", ''Essential Peirce'' v. 2, see pp. 107–9 and 113. On Twenty Questions, p. 109, Peirce has pointed out that if each question eliminates half the possibilities, twenty questions can choose from among 2<sup>20</sup> or 1,048,576 objects, and goes on to say: {{blockquote|Thus, twenty skillful hypotheses will ascertain what 200,000 stupid ones might fail to do. The secret of the business lies in the caution which breaks a hypothesis up into its smallest logical components, and only risks one of them at a time.}}</ref>

====Uberty====
Peirce<ref>{{Cite web|title=An Essay toward Improving Our Reasoning in Security and in Uberty|url=http://www.commens.org/|access-date=2022-02-05|website=www.commens.org|language=en|archive-date=August 26, 2014|archive-url=https://web.archive.org/web/20140826222027/http://www.commens.org/|url-status=dead}}</ref> indicated that abductive reasoning is driven by the need for "economy in research"—the expected fact-based productivity of hypotheses, prior to deductive and inductive processes of verification. A key concept proposed by him in this regard is "uberty"<ref>{{Cite web|title=Peirce's last philosophic will and testament: Uberty in the logic of|url=https://paperzz.com/doc/7856381/peirce-s-last-philosophic-will-and-testament--uberty-in-t...|access-date=2022-02-05|website=paperzz.com}}</ref>—the expected fertility and pragmatic value of reasoning. This concept seems to be gaining support via association to the [[Biology:Free energy principle|Free Energy Principle]].<ref>{{Cite journal|last1=Beni|first1=Majid D.|last2=Pietarinen|first2=Ahti-Veikko|date=2021-09-10|title=Aligning the free-energy principle with Peirce's logic of science and economy of research|url=https://doi.org/10.1007/s13194-021-00408-y|journal=European Journal for Philosophy of Science|language=en|volume=11|issue=3|pages=94|doi=10.1007/s13194-021-00408-y|s2cid=237475038 |issn=1879-4920}}</ref>

===Gilbert Harman (1965)===
[[Biography:Gilbert Harman|Gilbert Harman]] is a professor of philosophy at [[Organization:Princeton University|Princeton University]]. Harman's 1965 account of the role of "inference to the best explanation" – inferring the existence of that which we need for the best explanation of observable phenomena – has been very influential.

===Stephen Jay Gould (1995)===
[[Biography:Stephen Jay Gould|Stephen Jay Gould]], in answering the [[Philosophy:Omphalos hypothesis|Omphalos hypothesis]], claimed that only hypotheses that can be proved incorrect lie within the domain of [[Science|science]] and only these hypotheses are good explanations of facts worth inferring to.<ref>Stephen Jay Gould, "Adam's Navel", in idem, Adam's Navel and Other Essays (London: Penguin, 1995), p.
3.</ref> {{Blockquote|text="[W]hat is so desperately wrong with Omphalos? Only this really (and perhaps paradoxically): that we can devise no way to find out whether it is wrong—or for that matter, right. Omphalos is the classic example of an utterly untestable notion, for the world will look exactly the same in all its intricate detail whether fossils and strata are prochronic [signs of a fictitious past] or products of an
extended history. . . . Science is a procedure for testing and rejecting hypotheses, not a compendium of certain knowledge. Claims that can be proved incorrect lie within its domain. . . . But theories that cannot be tested in principle are not part of science. . . . [W]e reject Omphalos as useless, not wrong."}}

==Applications==

=== Artificial intelligence ===
Applications in [[Artificial intelligence|artificial intelligence]] include [[Diagnosis (artificial intelligence)|fault diagnosis]], [[Belief revision|belief revision]], and automated planning. The most direct application of abduction is that of automatically detecting faults in systems: given a theory relating faults with their effects and a set of observed effects, abduction can be used to derive sets of faults that are likely to be the cause of the problem.<ref name="Josephson, Magnani"/>

=== Medicine ===
In [[Medicine:Medicine|medicine]], abduction can be seen as a component of clinical evaluation and judgment.<ref name="Rapezzi2005">{{cite journal|last=Rapezzi|first=C|author2=Ferrari, R|author3= Branzi, A|title=White coats and fingerprints: diagnostic reasoning in medicine and investigative methods of fictional detectives|journal=BMJ (Clinical Research Ed.)|date=24 December 2005| volume=331| issue=7531| pages=1491–4| pmid=16373725|pmc=1322237| doi=10.1136/bmj.331.7531.1491}}</ref><ref name="Rejon2012">{{cite journal|last=Rejón Altable|first=C|title=Logic structure of clinical judgment and its relation to medical and psychiatric semiology| journal=Psychopathology|date=October 2012| volume=45| issue=6| pages=344–51| pmid=22854297| url=http://www.karger.com/Article/Pdf/337968| access-date=17 January 2014| doi=10.1159/000337968| doi-access=free}}</ref>

=== Automated planning ===
Abduction can also be used to model automated planning.<ref>Kave Eshghi. Abductive planning with the event calculus. In Robert A. Kowalski, Kenneth A. Bowen editors: Logic Programming, Proceedings of the Fifth International Conference and Symposium, Seattle, Washington, August 15–19, 1988. MIT Press 1988, {{ISBN|0-262-61056-6}}</ref> Given a logical theory relating action occurrences with their effects (for example, a formula of the [[Event calculus|event calculus]]), the problem of finding a plan for reaching a state can be modeled as the problem of abducting a set of literals implying that the final state is the goal state.

=== Intelligence analysis ===
In intelligence analysis, [[Philosophy:Analysis of competing hypotheses|analysis of competing hypotheses]] and [[Bayesian network]]s, probabilistic abductive reasoning is used extensively. Similarly in [[Medicine:Medical diagnosis|medical diagnosis]] and legal reasoning, the same methods are being used, although there have been many examples of errors, especially caused by the [[Base rate fallacy|base rate fallacy]] and the [[Prosecutor's fallacy|prosecutor's fallacy]].

=== Belief revision ===
[[Belief revision]], the process of adapting beliefs in view of new information, is another field in which abduction has been applied. The main problem of belief revision is that the new information may be inconsistent with the prior [[Philosophy:Web of belief|web of belief]]s, while the result of the incorporation cannot be inconsistent. The process of updating the web of beliefs can be done by the use of abduction: once an explanation for the observation has been found, integrating it does not generate inconsistency.

Gärdenfors’ paper<ref name="Gärdenfors ">Gärdenfors, Peter. "Belief revision: A vade-mecum." Meta-Programming in Logic: Third International Workshop, META-92 Uppsala, Sweden, June 10–12, 1992 Proceedings 3. Springer Berlin Heidelberg, 1992.</ref> contains a brief survey of the area of belief revision and its relation to updating of logical databases, and explores the relationship between belief revision and nonmonotonic logic.

This use of abduction is not straightforward, as adding [[Propositional formula|propositional formula]]e to other propositional formulae can only make inconsistencies worse. Instead, abduction is done at the level of the ordering of preference of the [[Philosophy:Possible world|possible world]]s. Preference models use [[Fuzzy logic|fuzzy logic]] or utility models.

=== Philosophy of science ===
In the [[Philosophy:Philosophy of science|philosophy of science]], abduction has been the key inference method to support [[Philosophy:Scientific realism|scientific realism]], and much of the debate about scientific realism is focused on whether abduction is an acceptable method of inference.<ref>Lipton, Peter. (2001). Inference to the Best Explanation, London: Routledge. {{ISBN|0-415-24202-9}}.</ref>

=== Historical linguistics ===
In historical linguistics, abduction during language acquisition is often taken to be an essential part of processes of [[Social:Language change|language change]] such as reanalysis and [[Analogy|analogy]].<ref>April M. S. McMahon (1994): Understanding language change. Cambridge: Cambridge University Press. {{ISBN|0-521-44665-1}}</ref>

=== Applied linguistics ===
In [[Social:Applied linguistics|applied linguistics]] research, abductive reasoning is starting to be used as an alternative explanation to inductive reasoning, in recognition of anticipated outcomes of qualitative inquiry playing a role in shaping the direction of analysis. It is defined as "The use of an unclear premise based on observations, pursuing theories to try to explain it" (Rose et al., 2020, p.&nbsp;258)<ref>{{cite book |last1=Rose, McKinley, & Briggs Baffoe-Djan |title=Data Collection Research Methods in Applied Linguistics |date=2020 |publisher=Bloomsbury |isbn=9781350025851}}</ref><ref>{{cite book |last1=McKinley |first1=J |editor1-last=McKinley & Rose |title=The Routledge Handbook of Research Methods in Applied Linguistics |publisher=Routledge |location=Abingdon |isbn=9780367824471 |pages=1–13 |chapter-url=https://www.englishappliedlinguistics.com/uploads/2/4/1/9/2419477/mckinley__2020__theorizing_applied_linguistics_research.pdf |chapter=Introduction: Theorizing research methods in the 'golden age' of applied linguistics research |date=2019-12-06 |access-date=2020-02-15 |archive-date=2020-02-15 |archive-url=https://web.archive.org/web/20200215113328/https://www.englishappliedlinguistics.com/uploads/2/4/1/9/2419477/mckinley__2020__theorizing_applied_linguistics_research.pdf |url-status=dead }}</ref>

=== Anthropology ===
In [[Social:Anthropology|anthropology]], Alfred Gell in his influential book ''Art and Agency'' defined abduction (after Eco<ref>{{cite book|last=Eco |first= Umberto |date=1976|title =A Theory of Semiotics|publisher = Indiana University Press|page= 131|isbn =9780253359551|url = https://books.google.com/books?id=BoXO4ItsuaMC&pg=PA131}}</ref>) as "a case of synthetic inference 'where we find some very curious circumstances, which would be explained by the supposition that it was a case of some general rule, and thereupon adopt that supposition{{'"}}.<ref name="Gell, A 1984, p 14">{{cite book|last=Gell|first= A.|date = 1998|title = Art and Agency|location = Oxford |publisher = Clarendon Press|page =14|isbn = 9780191037450}}</ref> Gell criticizes existing "anthropological" studies of art for being too preoccupied with aesthetic value and not preoccupied enough with the central anthropological concern of uncovering "social relationships", specifically the social contexts in which artworks are produced, circulated, and received.<ref>Bowden, R. (2004) A critique of Alfred Gell on Art and Agency. Retrieved Sept 2007 from: [https://web.archive.org/web/20050327073225/http://www.findarticles.com/p/articles/mi_qa3654/is_200406/ai_n9453295 Find Articles at BNET]</ref> Abduction is used as the mechanism for getting from art to agency. That is, abduction can explain how works of art inspire a ''sensus communis:'' the commonly held views shared by members that characterize a given society.<ref name="University of California, Berkeley">Whitney D. (2006) "Abduction the agency of art". Retrieved May 2009 from: [http://arthistory.berkeley.edu/davis/Gell.pdf University of California, Berkeley] {{webarchive |url=https://web.archive.org/web/20081120152801/http://arthistory.berkeley.edu/davis/Gell.pdf |date=2008-11-20 }}</ref>

The question Gell asks in the book is, "how does it initially 'speak' to people?" He answers by saying that "No reasonable person could suppose that art-like relations between people and things do not involve at least some form of semiosis."<ref name="Gell, A 1984, p 14" /> However, he rejects any intimation that semiosis can be thought of as a language because then he would have to admit to some pre-established existence of the ''sensus communis'' that he wants to claim only emerges afterwards out of art. Abduction is the answer to this conundrum because the tentative nature of the abduction concept (Peirce likened it to guessing) means that not only can it operate outside of any pre-existing framework, but moreover, it can actually intimate the existence of a framework. As Gell reasons in his analysis, the physical existence of the artwork prompts the viewer to perform an abduction that imbues the artwork with intentionality. A statue of a goddess, for example, in some senses actually becomes the goddess in the mind of the beholder; and represents not only the form of the deity but also her intentions (which are adduced from the feeling of her very presence). Therefore, through abduction, Gell claims that art can have the kind of agency that plants the seeds that grow into cultural myths. The power of agency is the power to motivate actions and inspire ultimately the shared understanding that characterizes any given society.<ref name= "University of California, Berkeley" />

===Computer programming===
In [[Formal methods|formal methods]], logic is used to specify and prove properties of computer programs. Abduction has been used in mechanized reasoning tools to increase the level of automation of the proof activity.

A technique known as bi-abduction, which mixes abduction and the [[Frame problem|frame problem]], was used to scale reasoning techniques for memory properties to millions of lines of code;<ref>{{Cite journal|first1=Cristiano|last1=Calcagno|first2=Dino|last2=Distefano|last3=O'Hearn|first3=Peter W.|first4=Hongseok|last4=Yang|date=2011-12-01|title=Compositional Shape Analysis by Means of Bi-Abduction|journal=[[Journal of the ACM]]|volume=58|issue=6|pages=1–66|language=EN|doi=10.1145/2049697.2049700|s2cid=52808268|url=https://ora.ox.ac.uk/objects/uuid:bcfefe74-a79c-4155-8160-c51f92f05466}}</ref> logic-based abduction was used to infer pre-conditions for individual functions in a program, relieving the human of the need to do so. It led to a program-proof startup company, which was acquired by Facebook,<ref>{{Cite web |date=July 18, 2013 |title=Facebook Acquires Assets Of UK Mobile Bug-Checking Software Developer Monoidics |url=https://techcrunch.com/2013/07/18/facebook-monoidics/ |access-date=2020-02-22 |website=TechCrunch |language=en-US}}{{Dead link|date=July 2023|bot=InternetArchiveBot|fix-attempted=yes}}</ref> and the Infer program analysis tool, which led to thousands of bugs being prevented in industrial codebases.<ref>{{Cite journal|last1=Distefano|first1=Dino|last2=Fähndrich|first2=Manuel|last3=Logozzo|first3=Francesco|last4=O'Hearn|first4=Peter W.|date=2019-07-24|title=Scaling static analyses at Facebook|journal=Communications of the ACM|volume=62|issue=8|pages=62–70|language=EN|doi=10.1145/3338112|doi-access=free}}</ref>

In addition to inference of function preconditions, abduction has been used to automate inference of invariants for program loops,<ref>{{Cite book|last1=Dillig|first1=Isil|last2=Dillig|first2=Thomas|last3=Li|first3=Boyang|last4=McMillan|first4=Ken|title=Proceedings of the 2013 ACM SIGPLAN international conference on Object oriented programming systems languages & applications |chapter=Inductive invariant generation via abductive inference |date=2013-10-29|chapter-url=https://dl.acm.org/doi/abs/10.1145/2544173.2509511|series=ACM SIGPLAN Notices|volume=48|issue=10|pages=443–456|language=EN|doi=10.1145/2509136.2509511|isbn=9781450323741|s2cid=16518775}}</ref> inference of specifications of unknown code,<ref>{{Cite journal|last=Giacobazzi|first=Roberto|date=1998-08-01|title=Abductive Analysis of Modular Logic Programs|url=https://academic.oup.com/logcom/article/8/4/457/1063792|journal=Journal of Logic and Computation|language=en|volume=8|issue=4|pages=457–483|doi=10.1093/logcom/8.4.457|issn=0955-792X}}</ref> and in synthesis of the programs themselves.<ref>{{Cite journal|last1=Polikarpova|first1=Nadia|last2=Sergey, Ilya|date=2019-01-02|title=Structuring the synthesis of heap-manipulating programs|journal=Proceedings of the ACM on Programming Languages|language=EN|volume=3|pages=1–30|doi=10.1145/3290385|doi-access=free|arxiv=1807.07022}}</ref>

==See also==
{{div col}}
*{{annotated link|Philosophy:Argument}}
*{{annotated link|Philosophy:Argumentation theory}}
*{{annotated link|Philosophy:Attribution (psychology)}}
*{{annotated link|Critical thinking}}
*{{annotated link|Defeasible reasoning}}
*{{annotated link|Philosophy:Duck test}}
*{{annotated link|Philosophy:Falsifiability}}
*{{annotated link|Biography:Gregory Bateson}}
*{{annotated link|Heuristic}}
*{{annotated link|Inductive probability}}
*{{annotated link|Philosophy:Logical reasoning}}
*{{annotated link|Maximum likelihood estimation}}
*{{annotated link|Occam's razor}}
*{{annotated link|Sensemaking}}
*{{annotated link|Statistical model}}
{{div col end}}

==Notes==
{{Reflist}}

==References==
{{Refbegin|colwidth=25em}}
* {{citation | author-last= Akaike | author-first= Hirotugu |  chapter= Implications of informational point of view on the development of statistical science | title= Proceedings of the First US/JAPAN Conference on The Frontiers of Statistical Modeling: An Informational Approach—Volume 3 | editor-first= H. | editor-last= Bozdogan | publisher= Kluwer Academic Publishers | year= 1994 | pages= 27–38}}.
*Awbrey, Jon, and Awbrey, Susan (1995), "Interpretation as Action:  The Risk of Inquiry", ''Inquiry:  Critical Thinking Across the Disciplines'', 15, 40–52. [https://web.archive.org/web/20071006102012/http://www.chss.montclair.edu/inquiry/fall95/awbrey.html Eprint]
*Cialdea Mayer, Marta and Pirri, Fiora (1993) "First order abduction via tableau and sequent calculi" Logic Jnl IGPL 1993 1: 99–117; {{doi|10.1093/jigpal/1.1.99}}. [https://web.archive.org/web/20080828081630/http://jigpal.oxfordjournals.org/content/vol1/issue1/index.dtl#ARTICLES Oxford Journals]
*Cialdea Mayer, Marta and Pirri, Fiora (1995) "Propositional Abduction in Modal Logic", Logic Jnl IGPL 1995 3: 907–919; {{doi|10.1093/jigpal/3.6.907}} [https://web.archive.org/web/20090210191135/http://jigpal.oxfordjournals.org/content/vol3/issue6/ Oxford Journals]
*Edwards, Paul (1967, eds.), "The Encyclopedia of Philosophy," Macmillan Publishing Co, Inc. & The Free Press, New York. Collier Macmillan Publishers, London.
* , T., and Gottlob, G. (1995), "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.37.2425&rep=rep1&type=pdf The Complexity of Logic-Based Abduction], ''Journal of the ACM'', 42.1, 3–42.
*Hanson, N. R. (1958). ''Patterns of Discovery: An Inquiry into the Conceptual Foundations of Science'', Cambridge: Cambridge University Press.  {{ISBN|978-0-521-09261-6}}.
*{{cite journal | last1 = Harman | first1 = Gilbert | year = 1965 | title = The Inference to the Best Explanation | journal = The Philosophical Review | volume = 74 | issue = 1| pages = 88–95 | doi=10.2307/2183532| jstor = 2183532 }}
*Josephson, John R., and Josephson, Susan G. (1995, eds.), ''[https://books.google.com/books?id=uu6zXrogwWAC Abductive Inference:  Computation, Philosophy, Technology]'', Cambridge University Press, Cambridge, UK.
*Lipton, Peter.  (2001).  ''Inference to the Best Explanation'', London: Routledge.  {{ISBN|0-415-24202-9}}.
*Magnani, Lorenzo (2014), "Understanding abduction", ''Model-Based Reasoning in Science and Technology: Theoretical and Cognitive Issues'' (editor—Magnani L.) Springer, p.&nbsp;173-205.
*McKaughan, Daniel J. (2008), "From Ugly Duckling to Swan: C. S. Peirce, Abduction, and the Pursuit of Scientific Theories", ''Transactions of the Charles S. Peirce Society'', v. 44, no. 3 (summer), 446–468.
*{{cite journal | last1 = Menzies | first1 = T | year = 1996 | title = Applications of Abduction: Knowledge-Level Modeling | url = http://menzies.us/pdf/96abkl.pdf | journal = International Journal of Human-Computer Studies | volume = 45 | issue = 3| pages = 305–335 | doi=10.1006/ijhc.1996.0054| citeseerx = 10.1.1.352.8159 }}
*Queiroz, Joao & Merrell, Floyd (guest eds.). (2005). "Abduction - between subjectivity and objectivity". (special issue on abductive inference) ''Semiotica'' 153 (1/4). [http://www.degruyter.com/view/j/semi.2005.2005.issue-153-1-4/issue-files/semi.2005.2005.issue-153-1-4.xml].
*Santaella, Lucia (1997) "The Development of Peirce's Three Types of Reasoning: Abduction, Deduction, and Induction", 6th Congress of the IASS. [http://www.pucsp.br/~lbraga/epap_peir1.htm Eprint].
*Sebeok, T. (1981) "You Know My Method". In Sebeok, T. "The Play of Musement". Indiana. Bloomington, IA.
*Yu, Chong Ho (1994), "Is There a Logic of Exploratory Data Analysis?", ''Annual Meeting of American Educational Research Association'', New Orleans, LA, April, 1994. [http://www.creative-wisdom.com/pub/Peirce/Logic_of_EDA.html Website of Dr. Chong Ho (Alex) Yu]
{{Refend}}

==External links==
*{{cite SEP |url-id=abduction |title=Abduction |last=Douven |first=Igor}}
*{{InPho|idea|5421}}
*{{PhilPapers|category|inference-to-the-best-explanation}}
*"[https://web.archive.org/web/20131214005416/http://www.cse.ohio-state.edu/lair/research.html Abductive Inference]" (once there, scroll down), John R. Josephson, Laboratory for Artificial Intelligence Research, Ohio State University. ([https://web.archive.org/web/20110720020440/http://www.cse.ohio-state.edu/lair/Projects/Abduction/abduction.html Former webpage] via the Wayback Machine.)
*"[http://plato.stanford.edu/entries/peirce/#dia Deduction, Induction, and Abduction]", Chapter 3 in article "[http://plato.stanford.edu/entries/peirce/ Charles Sanders Peirce]" by Robert W. Burch, 2001 and 2006, in the [http://plato.stanford.edu/ Stanford Encyclopedia of Philosophy].
*"[https://web.archive.org/web/20100609064615/http://carbon.ucdenver.edu/~mryder/itc/abduction.html Abduction]", links to articles and websites on abductive inference, [https://web.archive.org/web/20100914010915/http://carbon.ucdenver.edu/~mryder/martin.html Martin Ryder].
*[https://web.archive.org/web/20091028001425/http://user.uni-frankfurt.de/~wirth/ International Research Group on Abductive Inference], Uwe Wirth and Alexander Roesler, eds. Uses frames. Click on link at bottom of its home page for English. Wirth moved to U. of Gießen, Germany, and set up [https://archive.today/20130210054159/http://www.abduktionsforschung.de/ Abduktionsforschung], home page not in English but see Artikel section there. [https://translate.google.com/translate?js=n&layout=2&eotf=1&sl=de&tl=en&u=http%3A%2F%2Fwww.abduktionsforschung.de%2Fabduktionsforschung.html Abduktionsforschung home page via Google translation].
*"[http://www.visual-memory.co.uk/b_resources/abduction.html 'You Know My Method': A Juxtaposition of Charles S. Peirce and Sherlock Holmes]" (1981), by Thomas Sebeok with Jean Umiker-Sebeok, from ''The Play of Musement'', Thomas Sebeok, Bloomington, Indiana: Indiana University Press, pp.&nbsp;17–52.
*[http://www.helsinki.fi/science/commens/dictionary.html Commens Dictionary of Peirce's Terms], Mats Bergman and Sami Paavola, editors, Helsinki U. Peirce's own definitions, often many per term across the decades. There, see "Hypothesis [as a form of reasoning]", "Abduction", "Retroduction", and "Presumption [as a form of reasoning]".
*[https://philsci-archive.pitt.edu/22699/ "Touching Reality"], a critique of abductive reasoning in the context of cosmology.

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