of logical inference that goes from observation to a hypothesis that accounts for the reliable data (observation) and seeks to explain relevant evidence. The American philosopher Charles Sanders Peirce(1839–1914) first introduced the term as guessing.[2] Peirce said that to abduce a hypothetical explanation from an observed surprising circumstance is to surmise that may be true because then would be a matter of course.[3] Thus, to abduce from involves determining that is sufficient (or nearly sufficient), but not necessary, for . For example, the lawn is wet. But if it rained last night, then it would be unsurprising that the lawn is wet. Therefore, by abductive reasoning, the possibility that it rained last night is reasonable (but note that Peirce did not remain convinced that a single logical form covers all abduction).[4] Moreover, abducing rain last night from the observation of the wet lawn can lead to a false conclusion. In this example, dew, lawn sprinklers, or some other process may have resulted in the wet lawn, even in the absence of rain. Peirce argues that good abductive reasoning from P to Q involves not simply a determination that, e.g., Q is sufficient for P, but also that Q is among the most economical explanations for P. Simplification and economy both call for that leap of abduction.[5] In abductive reasoning, unlike in deductive reasoning, the premises do not guarantee the conclusion. One can understand abductive reasoning as inference to the best explanation.[6] The fields of law,[7] computer science, and artificial intelligence research[8] renewed interest in the subject of abduction. Diagnostic expert systems frequently employ abduction. Deduction, induction, and abductionEdit Main article: Logical reasoning Deductive reasoning (deduction) 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 bachelors are unmarried males, and given that this person is a bachelor, one can deduce that this person is an unmarried male. Inductive reasoning (induction) allows inferring from , where does not follow necessarily from . might give us very good reason to accept , but it does not ensure . 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.) Abductive reasoning (abduction) allows inferring as an explanation of . Because of this inference, abduction allows the precondition to be abduced from the consequence . Deductive reasoning and abductive reasoning thus differ in the direction in which a rule like entails is used for inference. As such, abduction is formally equivalent to the logical fallacy of affirming the consequent (or Post hoc ergo propter hoc) because of multiple possible explanations for . For example, 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 fact of infinitely 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 infinite 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 orient better ourselves in our surroundings and disregard some possibilities. Formalizations of abductionEdit Logic-based abduction In logic, explanation is done from 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: ; is consistent. Among the possible explanations satisfying these two conditions, some other condition of minimality is usually imposed to avoid irrelevant facts (not contributing to the entailment of ) 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 (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 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 (see Occams 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 dont 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.[9] It was Peirces 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.[10] 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 abduction 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 hiring a group of infected and a group of non-infected people to undergo the test. Assume the statements : Positive test, : Negative test, : Infected, and : Not infected. The result of these trials will then determine the reliability of the test in terms of its sensitivity and false positive rate . The interpretations of the conditionals are: : 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: : 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 on the right hand side of the equation expresses the base rate of the infection in the population. Similarly, the term expresses the default likelihood of positive test on a random person in the population. In the expressions below and denote the base rates of and its complement respectively, so that e.g. . The full expression for the required conditionals 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 tests sensitivity and false positive rate, is then given by . This further simplifies to . 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 (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 in medicine,[11] or the Prosecutors fallacy[12] in legal reasoning, consists of making the erroneous assumption that . 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,[13]i.e., with a state space of multiple and a state space of multiple states . Subjective logic abduction Subjective logic generalises probabilistic logicby including parameters for uncertainty in the input arguments. Abduction in subjective logic is thus similar to probabilistic abduction described above.[13] 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 and , the sets of conditional opinions and , the opinion on , and the base rate function on . Based on these parameters, subjective logic provides a method for deriving the set of inverted conditionals 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:[13] . 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 concl
Posted on: Thu, 05 Jun 2014 04:41:50 +0000
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