Abductive reasoning

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Abduction, or inference to the best explanation, is a method of reasoning in which one chooses the hypothesis that would, if true, best explain the relevant evidence. Abductive reasoning starts from a set of accepted facts and infers their most likely, or best, explanations. The term abduction is also sometimes used to just mean the generation of hypotheses to explain observations or conclusions, but the former definition is more common both in philosophy and computing.

Deduction, induction, and abduction

(see also logical reasoning)

Deduction 

allows deriving Failed to parse (Missing texvc executable; please see math/README to configure.): b

as a consequence of Failed to parse (Missing texvc executable; please see math/README to configure.): a

. In other words, deduction is the process of deriving the consequences of what is assumed. Given the truth of the assumptions, a valid deduction guarantees the truth of the conclusion.

Induction 

allows inferring some Failed to parse (Missing texvc executable; please see math/README to configure.): a

from multiple instantiations of Failed to parse (Missing texvc executable; please see math/README to configure.): b
when Failed to parse (Missing texvc executable; please see math/README to configure.): a
entails Failed to parse (Missing texvc executable; please see math/README to configure.): b

. Induction is the process of inferring probable antecedents as a result of observing multiple consequents.

Abduction 

allows inferring Failed to parse (Missing texvc executable; please see math/README to configure.): a

as an explanation of Failed to parse (Missing texvc executable; please see math/README to configure.): b

. Because of this, abduction allows the precondition Failed to parse (Missing texvc executable; please see math/README to configure.): a

of “Failed to parse (Missing texvc executable; please see math/README to configure.): a
entails Failed to parse (Missing texvc executable; please see math/README to configure.): b

” to be inferred from the consequence Failed to parse (Missing texvc executable; please see math/README to configure.): b . Deduction and abduction thus differ in the direction in which a rule like “Failed to parse (Missing texvc executable; please see math/README to configure.): a

entails Failed to parse (Missing texvc executable; please see math/README to configure.): b

” is used for inference. As such abduction is formally equivalent to the logical fallacy affirming the consequent or Post hoc ergo propter hoc, because there are multiple possible explanations for Failed to parse (Missing texvc executable; please see math/README to configure.): b .

Unlike deduction and induction, abduction can produce results that are incorrect within its formal system. However, it can still be useful as a heuristic, especially when something is known about the likelihood of different causes for Failed to parse (Missing texvc executable; please see math/README to configure.): b .

Logic-based abduction

In logic, explanation is done from a logical theory Failed to parse (Missing texvc executable; please see math/README to configure.): T

representing a domain and a set of observations Failed to parse (Missing texvc executable; please see math/README to configure.): O

. Abduction is the process of deriving a set of explanations of Failed to parse (Missing texvc executable; please see math/README to configure.): O

according to Failed to parse (Missing texvc executable; please see math/README to configure.): T
and picking out one of those explanations.  For Failed to parse (Missing texvc executable; please see math/README to configure.): E
to be an explanation of Failed to parse (Missing texvc executable; please see math/README to configure.): O
according to Failed to parse (Missing texvc executable; please see math/README to configure.): T

, it should satisfy two conditions:

• Failed to parse (Missing texvc executable; please see math/README to configure.): O

follows from Failed to parse (Missing texvc executable; please see math/README to configure.): E
and Failed to parse (Missing texvc executable; please see math/README to configure.): T

• Failed to parse (Missing texvc executable; please see math/README to configure.): E

is consistent with Failed to parse (Missing texvc executable; please see math/README to configure.): T

.

In formal logic, Failed to parse (Missing texvc executable; please see math/README to configure.): O

and Failed to parse (Missing texvc executable; please see math/README to configure.): E
are assumed to be sets of literals. The two conditions for Failed to parse (Missing texvc executable; please see math/README to configure.): E
being an explanation of Failed to parse (Missing texvc executable; please see math/README to configure.): O
according to theory Failed to parse (Missing texvc executable; please see math/README to configure.): T
are formalized as:

Failed to parse (Missing texvc executable; please see math/README to configure.): T \cup E \models O

Failed to parse (Missing texvc executable; please see math/README to configure.): T \cup E

is consistent.

Among the possible explanations Failed to parse (Missing texvc executable; please see math/README to configure.): E

satisfying these two conditions, some other condition of minimality is usually imposed to avoid irrelevant facts (not contributing to the entailment of Failed to parse (Missing texvc executable; please see math/README to configure.): O

) being included in the explanations. Abduction is then the process that picks out some member of Failed to parse (Missing texvc executable; please see math/README to configure.): 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. 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 Failed to parse (Missing texvc executable; please see math/README to configure.): T

into two components, one of which is a normal logic program, used to generate Failed to parse (Missing texvc executable; please see math/README to configure.): E
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 Failed to parse (Missing texvc executable; please see math/README to configure.): H

and a set of manifestations Failed to parse (Missing texvc executable; please see math/README to configure.): M

they are related by the domain knowledge, represented by a function Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): H' \subseteq H

, their effects are known to be Failed to parse (Missing texvc executable; please see math/README to configure.): e(H') .

Abduction is performed by finding a set Failed to parse (Missing texvc executable; please see math/README to configure.): H' \subseteq H

such that Failed to parse (Missing texvc executable; please see math/README to configure.): M \subseteq e(H')

. In other words, abduction is performed by finding a set of hypotheses Failed to parse (Missing texvc executable; please see math/README to configure.): H'

such that their effects Failed to parse (Missing texvc executable; please see math/README to configure.): e(H')
include all observations Failed to parse (Missing texvc executable; please see math/README to configure.): M

.

A common assumption is that the effects of the hypotheses are independent, that is, for every Failed to parse (Missing texvc executable; please see math/README to configure.): H' \subseteq H , it holds that Failed to parse (Missing texvc executable; please see math/README to configure.): e(H') = \bigcup_{h \in H'} e(\{h\}) . If this condition is met, abduction can be seen as a form of set covering.

History of the concept

Historically, Aristotle's use of the term epagoge has referred to a syllogism in which the major premise is known to be true, but the minor premise is only probable (Edwards, 1967).

The philosopher Charles Peirce introduced abduction into modern logic. In his works before 1900, he mostly uses 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."

He later used the term to mean creating new rules to explain new observations, emphasizing that abduction is the only logical process that actually creates anything new. Namely, he described the process of science as a combination of abduction, deduction and induction, stressing that new knowledge is only created by abduction.

This is contrary to the common use of abduction in the social sciences and in artificial intelligence, where the old meaning 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 infer correctly; possessing this ability is explained by the evolutionary advantage it gives. Peirce's second use of 'abduction' is most similar to induction.

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 ^[1].

Further development of the concept can be found in Peter Lipton's "Inference to the Best Explanation" (Lipton, 1991).

Applications

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 ^[2]. 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.

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.

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 ^[3].

References

• Awbrey, Jon, and Awbrey, Susan (1995), "Interpretation as Action: The Risk of Inquiry", Inquiry: Critical Thinking Across the Disciplines, 15, 40-52. Eprint

• Edwards, Paul (1967, eds.), "The Encyclopedia of Philosophy," Macmillan Publishing Co, Inc. & The Free Press, New York. Collier Macmillan Publishers, London.

• Eiter, T., and Gottlob, G. (1995), "The Complexity of Logic-Based Abduction, Journal of the ACM, 42.1, 3-42.

• Harman, Gilbert (1965). "The Inference to the Best Explanation," The Philosophical Review 74:1, 88-95.

• 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. ISBN 0-415-24202-9.

• Menzies, T. (1996), "Applications of Abduction: Knowledge-Level Modeling, International Journal of Human-Computer Studies, 45.3, 305-335.

• Yu, Chong Ho (1994), "Is There a Logic of Exploratory Data Analysis?", Annual Meeting of American Educational Research Association, New Orleans, LA, April, 1994. :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.[1]

• 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 [2]

Notes

1. ^ Schwendtner, Tibor and Ropolyi, László and Kiss, Olga (eds): Hermeneutika és a természettudományok. Áron Kiadó, Budapest, 2001. It is written in Hungarian. Meaning of the title: Hermeneutics and the natural sciences.
2. ^ 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
3. ^ April M. S. McMahon (1994): Understanding language change. Cambridge: Cambridge University Press. ISBN 0-521-44665-1

See also

External links

• Josephson, John, "Abductive Inference in Reasoning and Perception", Webpage
• Ryder, Martin, Instructional Technology Connections: Abduction, Webpage
• Magnani, Lorenzo, Abduction, Reason, and Science. Processes of Discovery and Explanation, Webpage
• [it] Magnani, Lorenzo
• Chapter 3. Deduction, Induction, and Abduction in article Charles Sanders Peirce of the Stanford Encyclopedia of Philosophy

This article was originally based on material from the Free On-line Dictionary of Computing, which is licensed under the GFDL.

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