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Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of
logical inference Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word ''infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in ...
formulated and advanced by American
philosopher A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for ...
beginning in the last third of the 19th century. It starts with an observation or set of observations and then seeks the simplest and most likely conclusion from the observations. This process, unlike
deductive reasoning Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fal ...
, yields a plausible conclusion but does not positively verify it. Abductive conclusions are thus qualified as having a remnant of 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, although not all usages of the terms ''abduction'' and ''inference to the best explanation'' are exactly equivalent. In the 1990s, as computing power grew, the fields of law,
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, and
artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech ...
researchFor examples, see
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 Lorenzo Magnani (born 1952), is an Italian philosopher who teaches philosophy of science in the Department of Humanities, Philosophy Section, at the University of Pavia, where he is full professor and directs the Computational Philosophy Laboratory ...
(Kluwer Academic/Plenum Publishers, New York, 2001).
spurred renewed interest in the subject of abduction. Diagnostic
expert system In artificial intelligence, an expert system is a computer system emulating the decision-making ability of a human expert. Expert systems are designed to solve complex problems by reasoning through bodies of knowledge, represented mainly as if� ...
s frequently employ abduction.


Deduction, induction, and abduction


Deduction

Deductive reasoning allows deriving b from a only where b is a formal
logical consequence Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is on ...
of a. 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" (a_1) and "Wikipedia is a wiki" (a_2), it follows that "Wikipedia can be edited by anyone" (b).


Induction

Inductive reasoning is the process of inferring some general principle b from a body of knowledge a, where b does not necessarily follow from a. a might give us very good reason to accept b, but does not ensure b. 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 a as an explanation of b. As a result of this inference, abduction allows the precondition a to be abducted from the consequence b.
Deductive reasoning Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fal ...
and abductive reasoning thus differ in which end, left or right, of the proposition "a entails b" serves as conclusion. As such, abduction is formally equivalent to the logical fallacy of affirming the consequent because of multiple possible explanations for b. 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 priors in Bayesian statistics.


Formalizations of abduction


Logic-based abduction

In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
,
explanation An explanation is a set of statements usually constructed to describe a set of facts which clarifies the causes, context, and consequences of those facts. It may establish rules or laws, and may clarify the existing rules or laws in relatio ...
is accomplished through the use of a logical theory T representing a
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
and a set of observations O. Abduction is the process of deriving a set of explanations of O according to T and picking out one of those explanations. For E to be an explanation of O according to T, it should satisfy two conditions: * O follows from E and T; * E is consistent with T. In formal logic, O and E are assumed to be sets of
literal Literal may refer to: * Interpretation of legal concepts: ** Strict constructionism ** The plain meaning rule The plain meaning rule, also known as the literal rule, is one of three rules of statutory construction traditionally applied by ...
s. The two conditions for E being an explanation of O according to theory T are formalized as: :T \cup E \models O; :T \cup E is consistent. Among the possible explanations E satisfying these two conditions, some other condition of minimality is usually imposed to avoid irrelevant facts (not contributing to the entailment of O) being included in the explanations. Abduction is then the process that picks out some member of E. Criteria for picking out a member representing "the best" explanation include the simplicity, the prior probability, or the explanatory power of the explanation. A proof-theoretical abduction method for first order classical logic based on the sequent calculus and a dual one, based on semantic tableaux (
analytic tableaux In proof theory, the semantic tableau (; plural: tableaux, also called truth tree) is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic. An analytic tableau is a tree structure computed ...
) have been proposed. 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 is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend ot ...
.
Abductive logic programming Abductive logic programming (ALP) is a high-level knowledge-representation framework that can be used to solve problems declaratively based on abductive reasoning. It extends normal logic programming by allowing some predicates to be incompletel ...
is a computational framework that extends normal logic programming with abduction. It separates the theory T into two components, one of which is a normal logic program, used to generate E by means of
backward reasoning Backward chaining (or backward reasoning) is an inference method described colloquially as working backward from the goal. It is used in automated theorem provers, inference engines, proof assistants, and other artificial intelligence applications ...
, 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 H and a set of manifestations M; they are related by the domain knowledge, represented by a function e 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 H' \subseteq H, their effects are known to be e(H'). Abduction is performed by finding a set H' \subseteq H such that M \subseteq e(H'). In other words, abduction is performed by finding a set of hypotheses H' such that their effects e(H') include all observations M. A common assumption is that the effects of the hypotheses are independent, that is, for every H' \subseteq H, it holds that e(H') = \bigcup_ e(\). If this condition is met, abduction can be seen as a form of
set covering The set cover problem is a classical question in combinatorics, computer science, operations research, and complexity theory. It is one of Karp's 21 NP-complete problems shown to be NP-complete in 1972. Given a set of elements (called the uni ...
.


Abductive validation

Abductive validation is the process of validating a given hypothesis through abductive reasoning. This can also be called reasoning through successive approximation. 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 is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence ...
; moreover, Peirce claims that it is a ubiquitous aspect of thought: 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." 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 by including degrees of epistemic
uncertainty Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable ...
in the input arguments, i.e. instead of probabilities, the analyst can express arguments as subjective opinions. Abduction in subjective logic is thus a generalization of probabilistic abduction described above. 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 X which takes its values from a domain \mathbf (i.e. a state space of exhaustive and mutually disjoint state values x), and is denoted by the tuple \omega_=(b_, u_, a_)\,\!, where b_\,\! is the belief mass distribution over \mathbf, u_\,\! is the epistemic uncertainty mass, and a_\,\! is the base rate distribution over \mathbf. These parameters satisfy u_+\sum b_(x) = 1\,\! and \sum a_(x) = 1\,\! as well as b_(x),u_,a_(x) \in ,1,\!. Assume the domains \mathbf and \mathbf with respective variables X and Y, the set of conditional opinions \omega_ (i.e. one conditional opinion for each value y), and the base rate distribution a_. Based on these parameters, the subjective Bayes' theorem denoted with the operator \;\widetilde produces the set of inverted conditionals \omega_ (i.e. one inverted conditional for each value x) expressed by: :\omega_=\omega_\;\widetilde\;a_. Using these inverted conditionals together with the opinion \omega_ subjective deduction denoted by the operator \circledcirc can be used to abduce the marginal opinion \omega_. The equality between the different expressions for subjective abduction is given below: :\begin \omega_ &= \omega_ \;\widetilde\; \omega_\\ &= (\omega_ \;\widetilde\; a_) \;\circledcirc\;\omega_\\ &= \omega_ \;\circledcirc\;\omega_\;. \end The symbolic notation for subjective abduction is "\widetilde", and the operator itself is denoted as "\widetilde". The operator for the subjective Bayes' theorem is denoted "\widetilde", and subjective deduction is denoted "\circledcirc".A. Jøsang. ''Subjective Logic: A Formalism for Reasoning Under Uncertainty'', Springer 2016, 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


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 An inquiry (also spelled as enquiry in British English) is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ...
, abduction and also induction are often collapsed into one overarching concept — the hypothesis. That is why, in the
scientific method The scientific method is an empirical method for acquiring knowledge that has characterized the development of science since at least the 17th century (with notable practitioners in previous centuries; see the article history of scientifi ...
known from
Galileo Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
and
Bacon Bacon is a type of salt-cured pork made from various cuts, typically the belly or less fatty parts of the back. It is eaten as a side dish (particularly in breakfasts), used as a central ingredient (e.g., the bacon, lettuce, and tomato sand ...
, the abductive stage of hypothesis formation is conceptualized simply as induction. Thus, in the twentieth century this collapse was reinforced by
Karl Popper Sir Karl Raimund Popper (28 July 1902 – 17 September 1994) was an Austrian-British philosopher, academic and social commentator. One of the 20th century's most influential philosophers of science, Popper is known for his rejection of the ...
's explication of the
hypothetico-deductive model The hypothetico-deductive model or method is a proposed description of the scientific method. According to it, scientific inquiry proceeds by formulating a hypothesis in a form that can be falsifiable, using a test on observable data where the ...
, where the hypothesis is considered to be just "a guess" (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 accounte.g. the grass could be wet from dew. This remains the common use of the term "abduction" in the
social science Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of s ...
s and in
artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech ...
. 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. In later years his view came to be: * Abduction is guessing.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
Abduction
and

at ''Commens Dictionary of Peirce's Terms''.)
It is "very little hampered" by rules of logic.Peirce, C. S. (1903), Harvard lectures on pragmatism, ''Collected Papers'' v. 5

Even a well-prepared mind's individual guesses are more frequently wrong than right. But the success of our guesses far exceeds that of random luck and seems born of attunement to nature by instinctPeirce, 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. (some speak of intuition in such contexts). * 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. * Its longer aim is to economize
inquiry An inquiry (also spelled as enquiry in British English) is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ...
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. 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. * Pragmatism is the logic of abduction. Upon the generation of an explanation (which he came to regard as instinctively guided), the pragmatic maxim gives the necessary and sufficient logical rule to abduction in general. The hypothesis, being insecure, needs to have conceivable implications for informed practice, so as to be testablePeirce, C. S., Carnegie Application (L75, 1902, ''New Elements of Mathematics'' v. 4, pp. 37–38. See under
Abduction
at the ''Commens Dictionary of Peirce's Terms'':
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

and in full in ''Essential Peirce'' v. 2, pp. 226–241 (see sections III and IV).
and, through its trials, to expedite and economize inquiry. The economy of research is what calls for abduction and governs its art.Peirce, C.S. (1902), application to the Carnegie Institution, see MS L75.329-330, fro

of Memoir 27:
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." 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;(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
p. 284 P. is an abbreviation or acronym that may refer to: * Page (paper), where the abbreviation comes from Latin ''pagina'' * Paris Herbarium, at the ''Muséum national d'histoire naturelle'' * ''Pani'' (Polish), translating as Mrs. * The ''Pacific Rep ...
in Part III §1. Reprinted in ''Collected Papers v. 2, paragraphs 461–516 and ''Writings'' v. 2, pp. 23–49.''
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.Peirce, C. S. (1878), "Deduction, Induction, and Hypothesis", ''Popular Science Monthly'', v. 13, pp. 470–82, see 472 ''Collected Papers'' 2.619–44, see 623.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;A letter to Langley, 1900, published in ''Historical Perspectives on Peirce's Logic of Science''. See excerpts under

at the ''Commens Dictionary of Peirce's Terms''.
by 1903 he speaks of the "likely" in the sense of nearing the truth in an "indefinite sense"; by 1908 he discusses ''plausibility'' as instinctive appeal.) 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. * as induction from characters (but as early as 1900 he characterized abduction as guessing) * as citing a known rule in a premise rather than hypothesizing a rule in the conclusion (but by 1903 he allowed either approachPeirce, C. S., "A Syllabus of Certain Topics of Logic" (1903), ''Essential Peirce'' v. 2, p. 287: ) * as basically a transformation of a deductive categorical syllogism (but in 1903 he offered a variation on ''modus ponens'' instead, and by 1911 he was unconvinced that any one form covers all hypothetical inferenceA Letter to J. H. Kehler (1911), ''New Elements of Mathematics'' v. 3, pp. 203–4, see under

at ''Commens Dictionary of Peirce's Terms''.
).


''The Natural Classification of Arguments'' (1867)

In 1867, Peirce's "On the Natural Classification of Arguments", 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'' iddleare ''mortal'' redicate ''Socrates'' ubjectis a ''man'' iddle ergo ''Socrates'' ubjectis ''mortal'' redicate. 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
syllogism A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be tru ...
Barbara (AAA-1).


''Deduction, Induction, and Hypothesis'' (1878)

In 1878, in "Deduction, Induction, and Hypothesis", 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).


''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. Like "Deduction, Induction, and Hypothesis" in 1878, it was widely read (see the historical books on statistics by 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 Sherlock Holmes () is a fictional detective created by British author Arthur Conan Doyle. Referring to himself as a " consulting detective" in the stories, Holmes is known for his proficiency with observation, deduction, forensic science and ...
used this method of reasoning in the stories of
Arthur Conan Doyle Sir Arthur Ignatius Conan Doyle (22 May 1859 – 7 July 1930) was a British writer and physician. He created the character Sherlock Holmes in 1887 for ''A Study in Scarlet'', the first of four novels and fifty-six short stories about Ho ...
, although Holmes refers to it as "
deductive reasoning Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fal ...
".


''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. In 1903 he offered the following form for abduction: 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. 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."


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." In 1903 Peirce called pragmatism "the logic of abduction" and said that the pragmatic maxim gives the necessary and sufficient logical rule to abduction in general. The pragmatic maxim is: 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 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).


Three levels of logic about abduction

Peirce came over the years to 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. 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 Peirce's outline of the scientific method of inquiry, as covered in his inquiry methodology, which includes pragmatism or, as he later called it, pragmaticism, the clarification of ideas in terms of their conceivable implications regarding informed practice.


=Classification of signs

= As early as 1866, Peirce held that: 1. Hypothesis (abductive inference) is inference through an ''icon'' (also called a ''likeness'').
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.
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."


=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. 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." 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 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.


=Methodology of inquiry

= At the methodeutical level Peirce held that a hypothesis is judged and selected for testing because it offers, via its trial, to expedite and economize the
inquiry An inquiry (also spelled as enquiry in British English) is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ...
process itself toward new truths, first of all by being testable and also by further economies, 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.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 220 or 1,048,576 objects, and goes on to say:


Uberty

Peirce 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"—the expected fertility and pragmatic value of reasoning. This concept seems to be gaining support via association to the
Free Energy Principle The free energy principle is a mathematical principle in biophysics and cognitive science that provides a formal account of the representational capacities of physical systems: that is, why things that exist look as if they track properties of the ...
.


Gilbert Harman (1965)

Gilbert Harman is a professor of philosophy at
Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...
. 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)

Stephen Jay Gould Stephen Jay Gould (; September 10, 1941 – May 20, 2002) was an American paleontologist, evolutionary biologist, and historian of science. He was one of the most influential and widely read authors of popular science of his generation. Goul ...
, in answering the
Omphalos hypothesis The Omphalos hypothesis is one attempt to reconcile the scientific evidence that the Earth is billions of years old with a literal interpretation of the Genesis creation narrative, which implies that the Earth is only a few thousand years old. I ...
, claimed that only hypotheses that can be proved incorrect lie within the domain of
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence ...
and only these hypotheses are good explanations of facts worth inferring to.


Applications


Artificial intelligence

Applications in
artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech ...
include
fault diagnosis Diagnosis is the identification of the nature and cause of a certain phenomenon. Diagnosis is used in many different disciplines, with variations in the use of logic, analytics, and experience, to determine "cause and effect". In systems enginee ...
, 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.


Medicine

In
medicine Medicine is the science and practice of caring for a patient, managing the diagnosis, prognosis, prevention, treatment, palliation of their injury or disease, and promoting their health. Medicine encompasses a variety of health care pr ...
, abduction can be seen as a component of clinical evaluation and judgment.


Automated planning

Abduction can also be used to model automated planning. Given a logical theory relating action occurrences with their effects (for example, a formula of the 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, analysis of competing hypotheses and
Bayesian network A Bayesian network (also known as a Bayes network, Bayes net, belief network, or decision network) is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Ba ...
s, probabilistic abductive reasoning is used extensively. Similarly in
medical diagnosis Medical diagnosis (abbreviated Dx, Dx, or Ds) is the process of determining which disease or condition explains a person's symptoms and signs. It is most often referred to as diagnosis with the medical context being implicit. The information r ...
and legal reasoning, the same methods are being used, although there have been many examples of errors, especially caused by the base rate fallacy and the 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
web of belief Web most often refers to: * Spider web, a silken structure created by the animal * World Wide Web or the Web, an Internet-based hypertext system Web, WEB, or the Web may also refer to: Computing * WEB, a literate programming system created by ...
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. This use of abduction is not straightforward, as adding propositional formulae to other propositional formulae can only make inconsistencies worse. Instead, abduction is done at the level of the ordering of preference of the possible worlds. Preference models use
fuzzy logic Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completel ...
or utility models.


Philosophy of science

In the
philosophy of science Philosophy of science is a branch of philosophy concerned with the foundations, methods, and implications of science. The central questions of this study concern what qualifies as science, the reliability of scientific theories, and the ultim ...
, abduction has been the key inference method to support
scientific realism Scientific realism is the view that the universe described by science is real regardless of how it may be interpreted. Within philosophy of science, this view is often an answer to the question "how is the success of science to be explained?" T ...
, and much of the debate about scientific realism is focused on whether abduction is an acceptable method of inference.


Historical linguistics

In
historical linguistics Historical linguistics, also termed diachronic linguistics, is the scientific study of language change over time. Principal concerns of historical linguistics include: # to describe and account for observed changes in particular languages # ...
, abduction during language acquisition is often taken to be an essential part of processes of
language change Language change is variation over time in a language's features. It is studied in several subfields of linguistics: historical linguistics, sociolinguistics, and evolutionary linguistics. Traditional theories of historical linguistics identif ...
such as reanalysis and
analogy Analogy (from Greek ''analogia'', "proportion", from ''ana-'' "upon, according to" lso "against", "anew"+ ''logos'' "ratio" lso "word, speech, reckoning" is a cognitive process of transferring information or meaning from a particular subject ...
.


Applied linguistics

In
applied linguistics Applied linguistics is an interdisciplinary field which identifies, investigates, and offers solutions to language-related real-life problems. Some of the academic fields related to applied linguistics are education, psychology, communication res ...
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. 258)


Anthropology

In
anthropology Anthropology is the scientific study of humanity, concerned with human behavior, human biology, cultures, societies, and linguistics, in both the present and past, including past human species. Social anthropology studies patterns of be ...
,
Alfred Gell Alfred Antony Francis Gell, (; June 12, 1945 – January 28, 1997) was a British social anthropologist whose most influential work concerned art, language, symbolism and ritual. He was trained by Edmund Leach (MPhil, Cambridge University) and R ...
in his influential book ''Art and Agency'' defined abduction (after Eco) 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. 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. 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.Whitney D. (2006) "Abduction the agency of art". Retrieved May 2009 from
University of California, Berkeley
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 Semiosis (, ), or sign process, is any form of activity, conduct, or process that involves signs, including the production of meaning. A sign is anything that communicates a meaning, that is not the sign itself, to the interpreter of the sign ...
." 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.


Computer programming

In 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, was used to scale reasoning techniques for memory properties to millions of lines of code; 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, and the Infer program analysis tool which led to thousands of bugs being prevented in industrial codebases. In addition to inference of function preconditions, abduction has been used to automate inference of invariants for program loops, inference of specifications of unknown code, and in synthesis of the programs themselves.


See also

* * * * * * * * * * * * * * * * * *


Notes


References

* . *Awbrey, Jon, and Awbrey, Susan (1995), "Interpretation as Action: The Risk of Inquiry", ''Inquiry: Critical Thinking Across the Disciplines'', 15, 40–52
Eprint
*Cialdea Mayer, Marta and Pirri, Fiora (1993) "First order abduction via tableau and sequent calculi" Logic Jnl IGPL 1993 1: 99–117;
Oxford Journals
*Cialdea Mayer, Marta and Pirri, Fiora (1995) "Propositional Abduction in Modal Logic", Logic Jnl IGPL 1995 3: 907–919;
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),
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. . * *Josephson, John R., and Josephson, Susan G. (1995, eds.),
Abductive Inference: Computation, Philosophy, Technology
', Cambridge University Press, Cambridge, UK. *Lipton, Peter. (2001). ''Inference to the Best Explanation'', London: Routledge. . * Magnani, Lorenzo (2014), "Understanding abduction", ''Model-Based Reasoning in Science and Technology: Theoretical and Cognitive Issues'' (editor—Magnani L.) Springer, p. 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. * *Queiroz, Joao & Merrell, Floyd (guest eds.). (2005). "Abduction - between subjectivity and objectivity". (special issue on abductive inference) '' Semiotica'' 153 (1/4)

*Santaella, Lucia (1997) "The Development of Peirce's Three Types of Reasoning: Abduction, Deduction, and Induction", 6th Congress of the
IASS International Association for Semiotic Studies (''Association Internationale de Sémiotique'', IASS-AIS) is the major world organisation of semioticians, established in 1969. Members of the association include Algirdas Julien Greimas, Roman Jakobs ...

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


External links

* * *

(once there, scroll down), John R. Josephson, Laboratory for Artificial Intelligence Research, Ohio State University.

via the Wayback Machine.)
Deduction, Induction, and Abduction
, Chapter 3 in article
Charles Sanders Peirce
by Robert Burch, 2001 and 2006, in th
Stanford Encyclopedia of Philosophy


, links to articles and websites on abductive inference


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 u
Abduktionsforschung
home page not in English but see Artikel section there



(1981), by Thomas Sebeok with Jean Umiker-Sebeok, from ''The Play of Musement'', Thomas Sebeok, Bloomington, Indiana: Indiana University Press, pp. 17–52.
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 s a form of reasoning, "Abduction", "Retroduction", and "Presumption s a form of reasoning. {{DEFAULTSORT:Abductive Reasoning Belief revision Bayesian statistics Epistemology Reasoning Charles Sanders Peirce