Abductive reasoning

Abductive reasoning 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 the American philosopher and logician Charles Sanders Peirce beginning in the latter half of the 19th century.
Abductive reasoning, unlike deductive reasoning, yields a plausible conclusion but does not definitively verify it. Abductive conclusions do not eliminate uncertainty or doubt, which is expressed in terms such as "best available" or "most likely". While inductive reasoning draws general conclusions that apply to many situations, abductive conclusions are confined to the particular observations in question.
In the 1990s, as computing power grew, the fields of law, computer science, and artificial intelligence research spurred renewed interest in the subject of abduction.
Diagnostic expert systems frequently employ abduction.

Deduction, induction, and abduction

Deduction

Deductive reasoning allows deriving from only where is a formal logical consequence of. 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 "all men are mortal" and "Socrates is a man", it follows that "Socrates is mortal".

Induction

Inductive reasoning is the process of inferring some general principle from a body of knowledge, where does not necessarily follow from. might give us very good reason to accept but does not ensure.
For example, if all swans that a person has observed so far are white, they may infer a universal categorical proposition of the form All swans are white. They have good reason to believe the conclusion from the premise because it is the best explanation for their observations, but the truth of the conclusion is not guaranteed. Indeed, it turns out that some swans are black.

Abduction

Abductive reasoning allows inferring as an explanation of. As a result of this inference, abduction allows the precondition to be abducted from the consequence. Deductive reasoning and abductive reasoning differ in which end, left or right, of the proposition " entails " serves as the conclusion. For example, with deductive reasoning, knowing that it rained heavily during the night you could deduce that the lawn will be wet in the morning, without looking outside. With abductive reasoning, a couple leaving their house in the morning and seeing that their lawn is wet might abduce that it rained while they were asleep. This serves as a hypothesis that "best explains" their observation. Given the many possible explanations for the lawn getting wet, their abduction does not establish certainty that it rained overnight, but it is still useful and can serve to orient them in their surroundings. Despite many possible explanations for any physical process we observe, we tend to abduce a single explanation 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.
One can understand abductive reasoning as inference to the best explanation, although the terms abduction and inference to the best explanation are not always used equivalently.

Formalizations of abduction

Logic-based abduction

In logic, explanation is accomplished through the use of a logical theory representing a domain and a set of observations. Abduction is the process of deriving a set of explanations of according to and picking out one of those explanations. For to be an explanation of according to, it should satisfy two conditions:

follows from and ;
is consistent with.

In formal logic, and are assumed to be sets of literals. The two conditions for being an explanation of according to theory are formalized as:
Among the possible explanations satisfying these two conditions, some other condition of minimality is usually imposed to avoid irrelevant facts being included in the explanations. Abduction is then the process that picks out some member of. 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 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.
Abductive logic programming is a computational framework that extends normal logic programming with abduction. It separates the theory into two components, one of which is a normal logic program, used to generate 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 and a set of manifestations ; they are related by the domain knowledge, represented by a function 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.
Abduction is performed by finding a set such that. In other words, abduction is performed by finding a set of hypotheses such that their effects include all observations.
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

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. Abductive validation is common practice in hypothesis formation in science; 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

generalises probabilistic logic by including degrees of epistemic uncertainty 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 which takes its values from a domain , and is denoted by the tuple, where is the belief mass distribution over, is the epistemic uncertainty mass, and is the base rate distribution over. These parameters satisfy and as well as.
Assume the domains and with respective variables and, the set of conditional opinions , and the base rate distribution. Based on these parameters, the subjective Bayes' theorem denoted with the operator produces the set of inverted conditionals expressed by:
Using these inverted conditionals together with the opinion subjective deduction denoted by the operator can be used to abduce the marginal opinion. The equality between the different expressions for subjective abduction is given below:
The symbolic notation for subjective abduction is "", and the operator itself is denoted as "". The operator for the subjective Bayes' theorem is denoted "", and subjective deduction is denoted "".
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, George Pólya, in his treatise on problem-solving, makes reference to the following Latin truism: simplex sigillum veri.

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, abduction and also induction are often collapsed into one overarching concept—the hypothesis. That is why, in the scientific method known from Galileo and Bacon, the abductive stage of hypothesis formation is conceptualized simply as induction. Thus, in the twentieth century this collapse was reinforced by Karl Popper's explication of the hypothetico-deductive model, where the hypothesis is considered to be just "a guess". 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 sciences and in 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 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. It is "very little hampered" by rules of logic. 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 instinct.
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 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, 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 testable and, through its trials, to expedite and economize inquiry. The economy of research is what calls for abduction and governs its art.

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 from the observed combined occurrence of multiple characters which its occurrence would necessarily involve; 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.
as aiming for a more or less probable hypothesis In a paper dated by editors as circa 1901, he discusses "instinct" and "naturalness", along with the kind of considerations that he later calls methodeutical.
as induction from characters
as citing a known rule in a premise rather than hypothesizing a rule in the conclusion
as basically a transformation of a deductive categorical syllogism.