Abduction is a method of logical inference introduced by Charles Sanders Peirce which comes prior to induction and deduction for which the colloquial name is to have a "hunch". Abductive reasoning starts when an inquirer considers of a set of seemingly unrelated facts, armed with an intuition that they are somehow connected. The term abduction is commonly presumed to mean the same thing as hypothesis; however, an abduction is actually the process of inference that produces a hypothesis as its end result^{[1]}. It is used in both philosophy and computing.
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Unlike deduction and in some sense induction, abduction can produce results that are incorrect within its formal system. Hence the conclusions of abduction can only be made valid by separately checking them with a different method, either by deduction or exhaustive induction. However, it can still be useful as a heuristic, especially when something is known about the likelihood of different causes for b.
In logic, explanation is done from a logical theory T representing a domain 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:
In formal logic, O and E are assumed to be sets of literals. The two conditions for E being an explanation of O according to theory T are formalized as:
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) have been proposed (Cialdea Mayer & Pirri 1993). 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.
Abductive logic programming 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, the other of which is a set of integrity constraints, used to filter the set of candidate explanations.
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 , their effects are known to be e(H').
Abduction is performed by finding a set such that . 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 , it holds that . If this condition is met, abduction can be seen as a form of set covering.
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; moreover, Peirce argues it is a ubiquitous aspect of thought:
Looking out my window this lovely spring morning I see an azalea in full bloom. No, no! I do not 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.^{[2]}
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." ^{[3]} After obtaining results from an inference procedure, we may be left with multiple assumptions, some of which may be contradictory. Abductive validation is a method for identifying the assumptions that will lead to your goal.
Probabilistic abductive reasoning is a form of abductive validation, and is used extensively in areas where conclusions about possible hypotheses need to be derived, such as for making diagnoses from medical tests. For example, a pharmaceutical company that develops a test for a particular infectious disease will typically determine the reliability of the test by letting a group of infected and a group of noninfected people undergo the test. Assume the statements x: "Positive test", : "Negative test", y: "Infected", and : "Not infected". The result of these trials will then determine the reliability of the test in terms of its sensitivity p(x  y) and false positive rate . The interpretations of the conditionals are: p(x  y): "The probability of positive test given infection", and : "The probability of positive test in the absence of infection". The problem with applying these conditionals in a practical setting is that they are expressed in the opposite direction to what the practitioner needs. The conditionals needed for making the diagnosis are: p(y  x): "The probability of infection given positive test", and : "The probability of infection given negative test". The probability of infection could then have been conditionally deduced as , where "" denotes conditional deduction. Unfortunately the required conditionals are usually not directly available to the medical practitioner, but they can be obtained if the base rate of the infection in the population is known.
The required conditionals can be correctly derived by inverting the available conditionals using Bayes rule. The inverted conditionals are obtained as follows: The term p(y) on the right hand side of the equation expresses the base rate of the infection in the population. Similarly, the term p(x) expresses the default likelihood of positive test on a random person in the population. In the expressions below a(y) and denote the base rates of y and its complement respectively, so that e.g. . The full expression for the required conditionals p(y  x) and are then:
The full expression for the conditionally abduced probability of infection in a tested person, expressed as , given the outcome of the test, the base rate of the infection, as well as the test's sensitivity and false positive rate, is then given by: .
Probabilistic abduction can thus be described as a method for inverting conditionals in order to apply probabilistic deduction.
A medical test result is typically considered positive or negative, so when applying the above equation it can be assumed that either p(x) = 1 (positive) or (negative). In case the patient tests positive, the above equation can be simplified to which will give the correct likelihood that the patient actually is infected.
The Base rate fallacy^{[4]} in medicine, or the Prosecutor's fallacy^{[5]} in legal reasoning, consists of making the erroneous assumption that p(y  x) = p(x  y). While this reasoning error often can produce a relatively good approximation of the correct hypothesis probability value, it can lead to a completely wrong result and wrong conclusion in case the base rate is very low and the reliability of the test is not perfect. An extreme example of the base rate fallacy is to conclude that a male person is pregnant just because he tests positive in a pregnancy test. Obviously, the base rate of male pregnancy is zero, and assuming that the test is not perfect, it would be correct to conclude that the male person is not pregnant.
The expression for probabilistic abduction can be generalised to multinomial cases^{[6]}, i.e. with a state space X of multiple x_{i} and a state space Y of multiple states y_{j}.
Subjective logic generalises probabilistic logic by including parameters for uncertainty in the input arguments. Abduction in subjective logic^{[6]} is thus similar to probabilistic abduction described above. The input arguments in subjective logic are composite functions called subjective opinions which can be binomial when the opinion applies to a single proposition or multinomial when it applies to a set of propositions. A multinomial opinion thus applies to a frame (i.e. a state space of exhaustive and mutually disjoint propositions ), and is denoted by the composite function , where is a vector of belief masses over the propositions of , is the uncertainty mass, and is a vector of base rate values over the propositions of . These components satisfy and as well as .
Assume the frames X and Y, the sets of conditional opinions ω_{X  Y} and , the opinion ω_{X} on X, and the base rate function a_{Y} on Y. Based on these parameters, subjective logic provides a method for deriving the set of inverted conditionals ω_{Y  X} and . Using these inverted conditionals, subjective logic also provides a method for deduction. Abduction in subjective logic consists of inverting the conditionals and then applying deduction.
The symbolic notation for conditional abduction is "", and the operator itself is denoted as . The expression for subjective logic abduction is then^{[6]}: .
The advantage of using subjective logic abduction compared to probabilistic abduction is that uncertainty about the probability values of the input arguments can be explicitly expressed and taken into account during the analysis. It is thus possible to perform abductive analysis in the presence of missing or incomplete input evidence, which normally results in degrees of uncertainty in the output conclusions.
The philosopher Charles Sanders Peirce introduced abduction into modern logic. 'The processes by which we form hunches about the world are, in Peirce's conception, dependent on perceptual judgments, which contain general elements such that universal propositions may be deduced from them.'^{[7]} In his works before 1900, he mostly used the term to mean the use of a known rule to explain an observation, e.g., “if it rains the grass is wet” is a known rule used to explain that the grass is wet. In other words, it would be more technically correct to say, "If the grass is wet, the most probable explanation is that it recently rained."
Writing in 1910, Peirce admits that he himself, "in almost everything I printed before the beginning of this century I more or less mixed up hypothesis and induction"^{[8]} 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."^{[9]} Peirce had tended in particular to characterize abduction in terms of induction of characters or traits (weighed, not counted like objects), explicitly so in an influential 1883 work.^{[10]} Peirce's view came to be that induction of characters was a branch of induction, and that abduction's general validity was related not to "probability proper" but to simplicity optimal in terms of the "facile and natural" (which he distinguished from "logical simplicity" and for which he cited Galileo), because of abduction's aim to expedite and economize inquiry and its reliance on inborn or developed instinctive attunement to nature.^{[11]}
When induction and abduction are presented as principal, then the two stages of hypothesis formation are easily collapsed into one overarching concept – the hypothesis. This is why, in the scientific method pioneered by Galileo and Bacon, the abductive stage of hypothesis formation is conceptualized simply as induction. In the twentieth century this collapse was reinforced by Karl Popper's explication of the Hypotheticodeductive model, where the hypothesis is considered to be just “a guess"^{[12]} (very much 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 tested 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.
Peirce came to describe the process of science as a combination of abduction, deduction and induction, stressing that new knowledge is only created by abduction.
Now, that the matter of no new truth can come from induction or from deduction, we have seen. It can only come from abduction; and abduction is, after all, nothing but guessing. We are therefore bound to hope that, although the possible explanations of our facts may be strictly innumerable, yet our mind will be able, in some finite number of guesses, to guess the sole true explanation of them. That we are bound to assume, independently of any evidence that it is true. Animated by that hope, we are to proceed to the construction of a hypothesis.^{[13]}
The step of adopting a hypothesis or a proposition which would lead to the prediction of what appear to be surprising facts is called abduction. Peirce came to regard abduction, deduction, and induction as clarified in their nature by their distinctive functions in the cycle of inquiry, wherein abduction generates the kind of explanatory hypotheses that deduction can explicate into implied consequences through whose tests induction can evaluate the hypothesis. Induction seeks facts to test a hypothesis; abduction seeks a hypothesis to account for facts. This use is contrary to the common use of the term “abduction” in the social sciences and in artificial intelligence, where the old meaning (as a rule to explain an observation) is used. Contrary to this use, Peirce stated that the actual process of generating a new rule is not “hampered” by logic rules. Rather, he pointed out, that humans have an innate ability to guess better than sheer luck would achieve; and that having this ability is explained by the evolutionary advantage it gives.
Norwood Russell Hanson, a philosopher of science, wanted to grasp a logic explaining how scientific discoveries take place. He used Peirce's notion of abduction for this ^{[14]}.
Further development of the concept can be found in Peter Lipton's "Inference to the Best Explanation" (Lipton, 1991).
Applications in artificial intelligence include fault diagnosis, 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.
Abduction can also be used to model automated planning ^{[15]}. 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.
In intelligence analysis, Analysis of Competing Hypotheses and Bayesian networks, probabilistic abductive reasoning is used extensively. Similarly in 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 and the prosecutor's fallacy.
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 corpus of beliefs, while the result of the incorporation cannot be inconsistent. This process 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 or utility models.
In the philosophy of science, abduction has been the key inference method to support scientific realism, and much of the debate about scientific realism is focused on whether abduction is an acceptable method of inference.
In historical linguistics, abduction during language acquisition is often taken to be an essential part of processes of language change such as reanalysis and analogy ^{[16]}.
In anthropology, Alfred Gell in his influential book Art and Agency defined abduction, (after Eco^{[17]}) 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”^{[18]}. 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 ^{[19]} Abduction is used as the basis of one gets from art to agency in the sense of a theory of how works of art can inspire a sensus communis, or the commonlyheld views that a characteristic of a given society because they are shared by everyone in that society.^{[20]} The question Gell asks in the book is, ‘how does initially to ‘speak’ to people?’ He answers by saying that “No reasonable person could suppose that artlike relations between people and things do not involve at least some form of semiosis.” ^{[18]} However, he rejects any intimation that semiosis can be thought of as a language because then he would have to admit to some preestablished existence of the sensus communis that he wants to claim only emerges afterward out of art. Abduction is the answer to this conundrum because the tentative nature of the abduction concept (Pierce likened it to guessing) means that not only can it operate outside of any preexisting 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.^{[20]}
