Abductive Reasoning

Abductive reasoningFrom Wikipedia, the free encyclopedia

Contents

1 Abductive reasoning 11.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Deduction, induction, and abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Formalizations of abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Logic-based abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Set-cover abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.3 Abductive validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.4 Probabilistic abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.5 Subjective logic abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.1 1867 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.2 1878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.3 1883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.4 1902 and after . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.5 Pragmatism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.6 Three levels of logic about abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.7 Other writers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Deductive reasoning 152.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Law of detachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Law of syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Law of contrapositive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Validity and soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.7 Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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2.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Inductive reasoning 203.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Inductive vs. deductive reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 Biases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4.1 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4.2 Statistical syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4.3 Simple induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4.4 Causal inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4.5 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.5 Bayesian inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.6 Inductive inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Inference 274.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 Example for denition #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Incorrect inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Automatic logical inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3.1 Example using Prolog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3.2 Use with the semantic web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3.3 Bayesian statistics and probability logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3.4 Nonmonotonic logic[2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Logic 335.1 The study of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1.1 Logical form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.1.2 Deductive and inductive reasoning, and abductive inference . . . . . . . . . . . . . . . . . 345.1.3 Consistency, validity, soundness, and completeness . . . . . . . . . . . . . . . . . . . . . . 355.1.4 Rival conceptions of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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5.3 Types of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.3.1 Syllogistic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.3.2 Propositional logic (sentential logic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.3.3 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.3.4 Modal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.3.5 Informal reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.3.6 Mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3.7 Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3.8 Computational logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3.9 Bivalence and the law of the excluded middle; non-classical logics . . . . . . . . . . . . . 405.3.10 Is logic empirical?" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3.11 Implication: strict or material? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3.12 Tolerating the impossible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3.13 Rejection of logical truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Chapter 1

Abductive reasoning

Abductive redirects here. For other uses, see Abduction (disambiguation).

Abductive reasoning (also called abduction,[1] abductive inference[2] or retroduction[3]) is a form of logicalinference that goes from an observation to a hypothesis that accounts for the observation, ideally seeking to nd thesimplest and most likely explanation. In abductive reasoning, unlike in deductive reasoning, the premises do notguarantee the conclusion. One can understand abductive reasoning as inference to the best explanation.[4]

The elds of law,[5] computer science, and articial intelligence research[6] renewed interest in the subject of abduc-tion. Diagnostic expert systems frequently employ abduction.

1.1 HistoryThe American philosopher Charles Sanders Peirce (18391914) rst introduced the term as guessing.[7] Peircesaid that to abduce a hypothetical explanation a from an observed circumstance b is to surmise that a may be truebecause then b would be a matter of course.[8] Thus, to abduce a from b involves determining that a is sucient, butnot necessary, for b .For example, suppose we observe that the lawn is wet. If it rained last night, then it would be unsurprising that thelawn is wet. Therefore, by abductive reasoning, the possibility that it rained last night is reasonable (but note thatPeirce did not remain convinced that a single logical form covers all abduction);[9]however, some other process mayhave also resulted in a wet lawn, i.e. dew or lawn sprinklers. Moreover, abducing that it rained last night from theobservation of a wet lawn can lead to false conclusion(s).Peirce argues that good abductive reasoning from P to Q involves not simply a determination that Q is sucient forP, but also that Q is among the most economical explanations for P. Simplication and economy both call for thatleap of abduction.[10]

1.2 Deduction, induction, and abductionMain article: Logical reasoning

Deductive reasoning (deduction) allows deriving b from a only where b is a formal logical consequence of a . Inother words, deduction derives the consequences of the assumed. Given the truth of the assumptions, a validdeduction 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 b from a , where b does not follow necessarily from a . a mightgive us very good reason to accept b , but it does not ensure b . For example, if all swans that we have observedso far are white, we may induce that the possibility that all swans are white is reasonable. We have good reason

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2 CHAPTER 1. ABDUCTIVE REASONING

to believe the conclusion from the premise, but the truth of the conclusion is not guaranteed. (Indeed, it turnsout that some swans are black.)

Abductive reasoning (abduction) allows inferring a as an explanation of b . Because of this inference, abductionallows the precondition a to be abduced from the consequence b . Deductive reasoning and abductive reasoningthus dier in the direction in which a rule like " a entails b " is used for inference. As such, abduction is formallyequivalent to the logical fallacy of arming the consequent (or Post hoc ergo propter hoc) because of multiplepossible explanations for b . For example, in a billiard game, after glancing and seeing the eight ball movingtowards us, we may abduce that the cue ball struck the eight ball. The strike of the cue ball would account forthe movement of the eight ball. It serves as a hypothesis that explains our observation. Given the many possibleexplanations for the movement of the eight ball, our abduction does not leave us certain that the cue ball in factstruck the eight ball, but our abduction, still useful, can serve to orient us in our surroundings. Despite manypossible explanations for any physical process that we observe, we tend to abduce a single explanation (or afew explanations) for this process in the expectation that we can better orient ourselves in our surroundings anddisregard some possibilities. Properly used, abductive reasoning can be a useful source of priors in Bayesianstatistics.

1.3 Formalizations of abduction

1.3.1 Logic-based abductionIn logic, explanation is done from a logical theory T representing a domain and a set of observations O . Abductionis the process of deriving a set of explanations of O according to T and picking out one of those explanations. ForE 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 literals. The two conditions for E being an explanation of Oaccording to theory T are formalized as:

T [ E j= OT [ EAmong the possible explanations E satisfying these two conditions, some other condition of minimality is usuallyimposed to avoid irrelevant facts (not contributing to the entailment of O ) being included in the explanations. Ab-duction is then the process that picks out some member of E . Criteria for picking out a member representing thebest explanation include the simplicity, the prior probability, or the explanatory power of the explanation.A proof theoretical abduction method for rst 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 aresound and complete and work for full rst order logic, without requiring any preliminary reduction of formulae intonormal 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 bymeans of backward reasoning, the other of which is a set of integrity constraints, used to lter the set of candidateexplanations.

1.3.2 Set-cover abductionA dierent formalization of abduction is based on inverting the function that calculates the visible eects of thehypotheses. Formally, we are given a set of hypotheses H and a set of manifestations M ; they are related by thedomain knowledge, represented by a function e that takes as an argument a set of hypotheses and gives as a result the

1.3. FORMALIZATIONS OF ABDUCTION 3

corresponding set of manifestations. In other words, for every subset of the hypotheses H 0 H , their eects areknown to be e(H 0) .Abduction is performed by nding a set H 0 H such that M e(H 0) . In other words, abduction is performed bynding a set of hypotheses H 0 such that their eects e(H 0) include all observations M .A common assumption is that the eects of the hypotheses are independent, that is, for every H 0 H , it holds thate(H 0) =

Sh2H0 e(fhg) . If this condition is met, abduction can be seen as a form of set covering.

1.3.3 Abductive validationAbductive validation is the process of validating a given hypothesis through abductive reasoning. This can also becalled reasoning through successive approximation. Under this principle, an explanation is valid if it is the bestpossible explanation of a set of known data. The best possible explanation is often dened in terms of simplicity andelegance (see Occams razor). Abductive validation is common practice in hypothesis formation in science; moreover,Peirce claims that 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 don'tsee 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 partby means of a statement of fact. This statement is abstract; but what I see is concrete. I perform anabduction when I so much as express in a sentence anything I see. The truth is that the whole fabric ofour knowledge is one matted felt of pure hypothesis conrmed and rened by induction. Not the smallestadvance can be made in knowledge beyond the stage of vacant staring, without making an abduction atevery step.[11]

It was Peirces own maxim that Facts cannot be explained by a hypothesis more extraordinary than these factsthemselves; and of various hypotheses the least extraordinary must be adopted.[12] After obtaining results from aninference procedure, we may be left with multiple assumptions, some of which may be contradictory. Abductivevalidation is a method for identifying the assumptions that will lead to your goal.

1.3.4 Probabilistic abductionProbabilistic abductive reasoning is a form of abductive validation, and is used extensively in areas where conclusionsabout possible hypotheses need to be derived, such as for making diagnoses from medical tests. For example, apharmaceutical company that develops a test for a particular infectious disease will typically determine the reliabilityof the test by hiring a group of infected and a group of non-infected people to undergo the test. Assume the statementsx : Positive test, x : Negative test, y : Infected, and y : Not infected. The result of these trials will thendetermine the reliability of the test in terms of its sensitivity p(xjy) and false positive rate p(xjy) . The interpretationsof the conditionals are: p(xjy) : The probability of positive test given infection, and p(xjy) : The probabilityof positive test in the absence of infection. The problem with applying these conditionals in a practical settingis that they are expressed in the opposite direction to what the practitioner needs. The conditionals needed formaking the diagnosis are: p(yjx) : The probability of infection given positive test, and p(yjx) : The probability ofinfection given negative test. The probability of infection could then have been conditionally deduced as p(ykx) =p(x)p(yjx) + p(x)p(yjx) , where " k " denotes conditional deduction. Unfortunately the required conditionals areusually not directly available to the medical practitioner, but they can be obtained if the base rate of the infection inthe 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:(p(xjy) = p(x^y)p(y)p(yjx) = p(x^y)p(x)

) p(yjx) = p(y)p(xjy)p(x) : The termp(y) on the right hand side of the equation expresses the base rate of the infection in the population. Similarly, theterm p(x) expresses the default likelihood of positive test on a random person in the population. In the expressionsbelow a(y) and a(y) = 1 a(y) denote the base rates of y and its complement y respectively, so that e.g. p(x) =a(y)p(xjy) + a(y)p(xjy) . The full expression for the required conditionals p(yjx) and p(yjx) are then(p(yjx) = a(y)p(xjy)a(y)p(xjy)+a(y)p(xjy)p(yjx) = a(y)p(xjy)a(y)p(xjy)+a(y)p(xjy)


The full expression for the conditionally abduced probability of infection in a tested person, expressed as p(ykx) ,given the outcome of the test, the base rate of the infection, as well as the tests sensitivity and false positive rate, isthen given byp(ykx) = p(x)

a(y)p(xjy)

a(y)p(xjy)+a(y)p(xjy)+ p(x)

a(y)p(xjy)

a(y)p(xjy)+a(y)p(xjy)

.This further simplies top(ykx) = a(y) (p(xjy) + p(xjy)) .Probabilistic abduction can thus be described as a method for inverting conditionals in order to apply probabilisticdeduction.A medical test result is typically considered positive or negative, so when applying the above equation it can beassumed that either p(x) = 1 (positive) or p(x) = 1 (negative). In case the patient tests positive, the above equationcan be simplied to p(ykx) = p(yjx) which will give the correct likelihood that the patient actually is infected.The Base rate fallacy in medicine,[13] or the Prosecutors fallacy[14] in legal reasoning, consists of making the erroneousassumption that p(yjx) = p(xjy) . While this reasoning error often can produce a relatively good approximation ofthe correct hypothesis probability value, it can lead to a completely wrong result and wrong conclusion in case thebase rate is very low and the reliability of the test is not perfect. An extreme example of the base rate fallacy is toconclude that a male person is pregnant just because he tests positive in a pregnancy test. Obviously, the base rate ofmale pregnancy is zero, and assuming that the test is not perfect, it would be correct to conclude that the male personis not pregnant.The expression for probabilistic abduction can be generalised to multinomial cases,[15] i.e., with a state space X ofmultiple xi and a state space Y of multiple states yj .

1.3.5 Subjective logic abductionSubjective logic generalises probabilistic logic by including parameters for uncertainty in the input arguments. Ab-duction in subjective logic is thus similar to probabilistic abduction described above.[15] The input arguments insubjective logic are composite functions called subjective opinions which can be binomial when the opinion appliesto a single proposition or multinomial when it applies to a set of propositions. A multinomial opinion thus applies toa frame X (i.e. a state space of exhaustive and mutually disjoint propositions xi ), and is denoted by the compositefunction !X = (~b; u;~a) , where~b is a vector of belief masses over the propositions of X , u is the uncertainty mass,and ~a is a vector of base rate values over the propositions of X . These components satisfy u +P~b(xi) = 1 andP~a(xi) = 1 as well as~b(xi); u;~a(xi) 2 [0; 1] .

Assume the frames X and Y , the sets of conditional opinions !XjY and !XjY , the opinion !X on X , and thebase rate function aY on Y . Based on these parameters, subjective logic provides a method for deriving the set ofinverted conditionals !Y jX and !Y jX . Using these inverted conditionals, subjective logic also provides a method fordeduction. Abduction in subjective logic consists of inverting the conditionals and then applying deduction.The symbolic notation for conditional abduction is " k ", and the operator itself is denoted as } . The expression forsubjective logic abduction is then:[15] !

Y kX = !X } (!XjY ; !XjY ; aY ) .The advantage of using subjective logic abduction compared to probabilistic abduction is that uncertainty about theprobability values of the input arguments can be explicitly expressed and taken into account during the analysis. It isthus possible to perform abductive analysis in the presence of missing or incomplete input evidence, which normallyresults in degrees of uncertainty in the output conclusions.

1.4 HistoryThe philosopher Charles Sanders Peirce (/prs/; 18391914) introduced abduction into modern logic. Over the yearshe called such inference hypothesis, abduction, presumption, and retroduction. He considered it a topic in logic as anormative eld in philosophy, not in purely formal or mathematical logic, and eventually as a topic also in economicsof research.As two stages of the development, extension, etc., of a hypothesis in scientic inquiry, abduction and also inductionare often collapsed into one overarching concept the hypothesis. That is why, in the scientic method pioneeredby Galileo and Bacon, the abductive stage of hypothesis formation is conceptualized simply as induction. Thus, in

1.4. HISTORY 5

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[16] (in the spirit of Peirce). However, when the formation ofa hypothesis is considered the result of a process it becomes clear that this guess has already been tried and mademore robust in thought as a necessary stage of its acquiring the status of hypothesis. Indeed many abductions arerejected or heavily modied 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, e.g., it is a known rulethat if it rains the grass is wet; so, to explain the fact that the grass is wet; one infers that it has rained. This remainsthe common use of the term abduction in the social sciences and in articial intelligence.Peirce consistently characterized it as the kind of inference that originates a hypothesis by concluding in an expla-nation, 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 the1900s he wrote that all explanatory content of theories is reached through abduction. In other respects Peirce revisedhis view of abduction over the years.[17]

In later years his view came to be:

Abduction is guessing.[7] It is very little hampered by rules of logic.[8] Even a well-prepared minds individualguesses are more frequently wrong than right.[18] But the success of our guesses far exceeds that of randomluck and seems born of attunement to nature by instinct[19] (some speak of intuition in such contexts[20]).

Abduction guesses a new or outside idea so as to account in a plausible, instinctive, economical way for asurprising or very complicated phenomenon. That is its proximate aim.[19]

Its longer aim is to economize inquiry itself. Its rationale is inductive: it works often enough, is the only sourceof new ideas, and has no substitute in expediting the discovery of new truths.[21] Its rationale especially involvesits role in coordination with other modes of inference in inquiry. It is inference to explanatory hypotheses forselection of those best worth trying.

Pragmatism is the logic of abduction. Upon the generation of an explanation (which he came to regard asinstinctively guided), the pragmatic maxim gives the necessary and sucient logical rule to abduction in general.The hypothesis, being insecure, needs to have conceivable[22] implications for informed practice, so as to betestable[23][24] and, through its trials, to expedite and economize inquiry. The economy of research is what callsfor abduction and governs its art.[10]

Writing in 1910, Peirce admits that in almost everything I printed before the beginning of this century I more orless mixed up hypothesis and induction and he traces the confusion of these two types of reasoning to logicians toonarrow and formalistic a conception of inference, as necessarily having formulated judgments from its premises.[25]

He started out in the 1860s treating hypothetical inference in a number of ways which he eventually peeled away asinessential or, in some cases, mistaken:

as inferring the occurrence of a character (a characteristic) from the observed combined occurrence of multiplecharacters which its occurrence would necessarily involve;[26] for example, if any occurrence of A is knownto necessitate occurrence of B, C, D, E, then the observation of B, C, D, E suggests by way of explanationthe occurrence of A. (But by 1878 he no longer regarded such multiplicity as common to all hypotheticalinference.[27])

as aiming for a more or less probable hypothesis (in 1867 and 1883 but not in 1878; anyway by 1900 thejustication is not probability but the lack of alternatives to guessing and the fact that guessing is fruitful;[28]by 1903 he speaks of the likely in the sense of nearing the truth in an indenite sense";[29] by 1908 hediscusses plausibility as instinctive appeal.[19]) In a paper dated by editors as circa 1901, he discusses instinctand naturalness, along with the kind of considerations (low cost of testing, logical caution, breadth, andincomplexity) that he later calls methodeutical.[30]

as induction from characters (but as early as 1900 he characterized abduction as guessing[28]) as citing a known rule in a premise rather than hypothesizing a rule in the conclusion (but by 1903 he allowed

either approach[8][31]) as basically a transformation of a deductive categorical syllogism[27] (but in 1903 he oered a variation onmodusponens instead,[8] and by 1911 he was unconvinced that any one form covers all hypothetical inference[9]).


1.4.1 1867In 1867, in The Natural Classication of Arguments,[26] hypothetical inference always deals with a cluster of char-acters (call them P, P, P, etc.) known to occur at least whenever a certain character (M) occurs. Note thatcategorical syllogisms have elements traditionally called middles, predicates, and subjects. For example: All men[middle] are mortal [predicate]; Socrates [subject] is a man [middle]; ergo Socrates [subject] is mortal [predicate]".Below, 'M' stands for a middle; 'P' for a predicate; 'S' for a subject. Note also that Peirce held that all deduction canbe put into the form of the categorical syllogism Barbara (AAA-1).

1.4.2 1878In 1878, in Deduction, Induction, and Hypothesis,[27] there is no longer a need for multiple characters or predicatesin order for an inference to be hypothetical, although it is still helpful. Moreover Peirce no longer poses hypotheticalinference as concluding in a probable hypothesis. In the forms themselves, it is understood but not explicit that induc-tion 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 anothers propositions (without thebracketed hints shown below).

1.4.3 1883Peirce long treated abduction in terms of induction from characters or traits (weighed, not counted like objects),explicitly so in his inuential 1883 A Theory of Probable Inference, in which he returns to involving probabilityin the hypothetical conclusion.[32] Like Deduction, Induction, and Hypothesis in 1878, it was widely read (see thehistorical books on statistics by Stephen Stigler), unlike his later amendments of his conception of abduction. Todayabduction remains most commonly understood as induction from characters and extension of a known rule to coverunexplained circumstances.Sherlock Holmes uses this method of reasoning in the stories of Arthur Conan Doyle, although Holmes refers to itas deductive reasoning.

1.4.4 1902 and afterIn 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.[33] In 1903he oered the following form for abduction:[8]

The surprising fact, C, is observed;But if A were true, C would be a matter of course,Hence, there is reason to suspect that A is true.

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 thesame 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 amplies ideas alreadyknown to represent occurrences, or tests an idea supplied by hypothesis; either way it requires previous abductions inorder to get such ideas in the rst place. Induction seeks facts to test a hypothesis; abduction seeks a hypothesis toaccount for facts.Note that the hypothesis (A) could be of a rule. It need not even be a rule strictly necessitating the surprisingobservation (C), which needs to follow only as a matter of course"; or the course itself could amount to someknown rule, merely alluded to, and also not necessarily a rule of strict necessity. In the same year, Peirce wrotethat reaching a hypothesis may involve placing a surprising observation under either a newly hypothesized rule ora hypothesized combination of a known rule with a peculiar state of facts, so that the phenomenon would be notsurprising but instead either necessarily implied or at least likely.[31]

1.4. HISTORY 7

Peirce did not remain quite convinced about any such form as the categorical syllogistic form or the 1903 form. In1911, 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.[9]

1.4.5 PragmatismIn 1901 Peirce wrote, There would be no logic in imposing rules, and saying that they ought to be followed, untilit is made out that the purpose of hypothesis requires them.[34] In 1903 Peirce called pragmatism the logic of ab-duction and said that the pragmatic maxim gives the necessary and sucient logical rule to abduction in general.[24]The pragmatic maxim is: Consider what eects, that might conceivably have practical bearings, we conceive theobject of our conception to have. Then, our conception of these eects is the whole of our conception of the object.It is a method for fruitful clarication of conceptions by equating the meaning of a conception with the conceiv-able practical implications of its objects conceived eects. Peirce held that that is precisely tailored to abductionspurpose in inquiry, the forming of an idea that could conceivably shape informed conduct. In various writings inthe 1900s[10][35] 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 analyticportion might be part of logical methodeutic (that is, theory of inquiry).[36]

1.4.6 Three levels of logic about abductionPeirce came over the years to divide (philosophical) logic into three departments:

1. Stechiology, or speculative grammar, on the conditions for meaningfulness. Classication of signs (semblances,symptoms, symbols, etc.) and their combinations (as well as their objects and interpretants).

2. Logical critic, or logic proper, on validity or justiability of inference, the conditions for true representation.Critique of arguments in their various modes (deduction, induction, abduction).

3. Methodeutic, or speculative rhetoric, on the conditions for determination of interpretations. Methodology ofinquiry in its interplay of modes.

Peirce had, from the start, seen the modes of inference as being coordinated together in scientic inquiry and, by the1900s, held that hypothetical inference in particular is inadequately treated at the level of critique of arguments.[23][24]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 Peirces outlineof the scientic method of inquiry, as covered in his inquiry methodology, which includes pragmatism or, as helater called it, pragmaticism, the clarication of ideas in terms of their conceivable implications regarding informedpractice.

Classication of signs

As early as 1866,[37] 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 fromwhich it is drawn.3. Deduction is inference through a symbol (a sign by interpretive habit irrespective of resemblance or connection toits object).In 1902, Peirce wrote that, in abduction: It is recognized that the phenomena are like, i.e. constitute an Icon of, areplica of a general conception, or Symbol.[38]

Critique of arguments

At the critical level Peirce examined the forms of abductive arguments (as discussed above), and came to hold thatthe hypothesis should economize explanation for plausibility in terms of the feasible and natural. In 1908 Peircedescribed this plausibility in some detail.[19] 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 byGalileos natural light of reason and as distinct from logical simplicity (Peirce does not dismiss logical simplicityentirely but sees it in a subordinate role; taken to its logical extreme it would favor adding no explanation to theobservation at all). Even a well-prepared mind guesses oftener wrong than right, but our guesses succeed better thanrandom luck at reaching the truth or at least advancing the inquiry, and that indicates to Peirce that they are basedin instinctive attunement to nature, an anity between the minds processes and the processes of the real, whichwould account for why appealingly natural guesses are the ones that oftenest (or least seldom) succeed; to whichPeirce added the argument that such guesses are to be preferred since, without a natural bent like natures, peoplewould have no hope of understanding nature. In 1910 Peirce made a three-way distinction between probability,verisimilitude, and plausibility, and dened plausibility with a normative ought": By plausibility, I mean the degreeto which a theory ought to recommend itself to our belief independently of any kind of evidence other than our instincturging us to regard it favorably.[39] 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 aninference, but rather a question of the hypothesiss relation to the inquiry process.The phrase inference to the best explanation (not used by Peirce but often applied to hypothetical inference) is notalways understood as referring to the most simple and natural. However, in other senses of best, such as standingup best to tests, it is hard to know which is the best explanation to form, since one has not tested it yet. Still, forPeirce, any justication of an abductive inference as good is not completed upon its formation as an argument (unlikewith induction and deduction) and instead depends also on its methodological role and promise (such as its testability)in advancing inquiry.[23][24][40]

Methodology of inquiry

At the methodeutical level Peirce held that a hypothesis is judged and selected[23] for testing because it oers, via itstrial, to expedite and economize the inquiry process itself toward new truths, rst of all by being testable and also byfurther economies,[10] in terms of cost, value, and relationships among guesses (hypotheses). Here, considerationssuch 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 rst in line for testing, to get itout of the way. If surprisingly it stands up to tests, that is worth knowing early in the inquiry, which otherwisemight 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 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 conicting hypotheses that are lesssimple.[41]

1.4.7 Other writersNorwood Russell Hanson, a philosopher of science, wanted to grasp a logic explaining how scientic discoveries takeplace. He used Peirces notion of abduction for this.[42]

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

1.5 ApplicationsApplications in articial intelligence include fault diagnosis, belief revision, and automated planning. The most directapplication of abduction is that of automatically detecting faults in systems: given a theory relating faults with theireects and a set of observed eects, abduction can be used to derive sets of faults that are likely to be the cause ofthe problem.

1.6. SEE ALSO 9

In medicine, abduction can be seen as a component of clinical evaluation and judgment.[43][44]

Abduction can also be used to model automated planning.[45] Given a logical theory relating action occurrences withtheir eects (for example, a formula of the event calculus), the problem of nding a plan for reaching a state can bemodeled as the problem of abducting a set of literals implying that the nal state is the goal state.In intelligence analysis, Analysis of Competing Hypotheses and Bayesian networks, probabilistic abductive reasoningis used extensively. Similarly in medical diagnosis and legal reasoning, the same methods are being used, althoughthere have been many examples of errors, especially caused by the base rate fallacy and the prosecutors fallacy.Belief revision, the process of adapting beliefs in view of new information, is another eld in which abduction hasbeen applied. The main problem of belief revision is that the new information may be inconsistent with the corpus ofbeliefs, 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 ofabduction is not straightforward, as adding propositional formulae to other propositional formulae can only makeinconsistencies 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 scientic realism, and much ofthe debate about scientic 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 oflanguage change such as reanalysis and analogy.[46]

In anthropology, Alfred Gell in his inuential book Art and Agency dened abduction (after Eco[47]) as a case ofsynthetic inference 'where we nd some very curious circumstances, which would be explained by the suppositionthat it was a case of some general rule, and thereupon adopt that supposition.[48] Gell criticizes existing 'anthropo-logical' studies of art, for being too preoccupied with aesthetic value and not preoccupied enough with the centralanthropological concern of uncovering 'social relationships,' specically the social contexts in which artworks areproduced, circulated, and received.[49] 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 membersthat characterize a given society.[50] 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 donot involve at least some form of semiosis.[48] However, he rejects any intimation that semiosis can be thought ofas a language because then he would have to admit to some pre-established existence of the sensus communis that hewants to claim only emerges afterwards out of art. Abduction is the answer to this conundrum because the tentativenature of the abduction concept (Peirce likened it to guessing) means that not only can it operate outside of anypre-existing framework, but moreover, it can actually intimate the existence of a framework. As Gell reasons in hisanalysis, the physical existence of the artwork prompts the viewer to perform an abduction that imbues the artworkwith intentionality. A statue of a goddess, for example, in some senses actually becomes the goddess in the mind ofthe beholder; and represents not only the form of the deity but also her intentions (which are adduced from the feelingof her very presence). Therefore through abduction, Gell claims that art can have the kind of agency that plants theseeds that grow into cultural myths. The power of agency is the power to motivate actions and inspire ultimately theshared understanding that characterizes any given society.[50]

1.6 See also Abductive logic programming Analogy Analysis of Competing Hypotheses Charles Sanders Peirce Charles Sanders Peirce bibliography Deductive reasoning Defeasible reasoning Doug Walton


Gregory Bateson Inductive inference Inductive probability Inductive reasoning Inquiry List of thinking-related topics Practopoiesis Logic Subjective logic Logical reasoning Maximum likelihood Scientic method Sherlock Holmes Sign relation

1.7 References This article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November

2008 and incorporated under the relicensing terms of the GFDL, version 1.3 or later.

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

Cialdea Mayer, Marta and Pirri, Fiora (1993) First order abduction via tableau and sequent calculi Logic JnlIGPL 1993 1: 99-117; doi:10.1093/jigpal/1.1.99. Oxford Journals

Cialdea Mayer, Marta and Pirri, Fiora (1995) Propositional Abduction in Modal Logic, Logic Jnl IGPL 19953: 907-919; doi:10.1093/jigpal/3.6.907 Oxford Journals

Edwards, Paul (1967, eds.), The Encyclopedia of Philosophy, Macmillan Publishing Co, Inc. & The FreePress, 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.

Hanson, N. R. (1958). Patterns of Discovery: An Inquiry into the Conceptual Foundations of Science, Cam-bridge: Cambridge University Press. ISBN 978-0-521-09261-6.

Harman, Gilbert (1965). The Inference to the Best Explanation. The Philosophical Review 74 (1): 8895.doi:10.2307/2183532.

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. McKaughan, Daniel J. (2008), From Ugly Duckling to Swan: C. S. Peirce, Abduction, and the Pursuit of

Scientic Theories, Transactions of the Charles S. Peirce Society, v. 44, no. 3 (summer), 446468. Abstract.

Menzies, T (1996). Applications of Abduction: Knowledge-Level Modeling (PDF). International Journalof Human-Computer Studies 45 (3): 305335. doi:10.1006/ijhc.1996.0054.

1.8. NOTES 11

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 Peirces Three Types of Reasoning: Abduction, Deduction, andInduction, 6th Congress of the IASS. 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 Edu-cational Research Association, New Orleans, LA, April, 1994. Website of Dr. Chong Ho (Alex) Yu

1.8 Notes[1] Magnani, L. Abduction, Reason, and Science: Processes of Discovery and Explanation. Kluwer Academic Plenum

Publishers, New York, 2001. xvii. 205 pages. Hard cover, ISBN 0-306-46514-0. R. Josephson, J. & G. Josephson, S. Abductive Inference: Computation, Philosophy, Technology Cambridge Uni-

versity Press, New York & Cambridge (U.K.). viii. 306 pages. Hard cover (1994), ISBN 0-521-43461-0, Paperback(1996), ISBN 0-521-57545-1.

Bunt, H. & Black, W. Abduction, Belief and Context in Dialogue: Studies in Computational Pragmatics (NaturalLanguage Processing, 1.) John Benjamins, Amsterdam & Philadelphia, 2000. vi. 471 pages. Hard cover, ISBN90-272-4983-0 (Europe), 1-58619-794-2 (U.S.)

[2] R. Josephson, J. & G. Josephson, S. Abductive Inference: Computation, Philosophy, Technology Cambridge UniversityPress, New York & Cambridge (U.K.). viii. 306 pages. Hard cover (1994), ISBN 0-521-43461-0, Paperback (1996), ISBN0-521-57545-1.

[3] Retroduction | Dictionary | Commens. Commens Digital Companion to C. S. Peirce. Mats Bergman, Sami Paavola &Joo Queiroz. Retrieved 2014-08-24.

[4] Sober, Elliot. Core Questions in Philosophy,5th edition.

[5] See, e.g. Analysis of Evidence, 2d ed. by Terence Anderson (Cambridge University Press, 2005)

[6] For examples, see "Abductive Inference in Reasoning and Perception", John R. Josephson, Laboratory for Articial Intel-ligence Research, Ohio State University, and Abduction, Reason, and Science. Processes of Discovery and Explanation byLorenzo Magnani (Kluwer Academic/Plenum Publishers, New York, 2001).

[7] Peirce, C. S.

On the Logic of drawing History from Ancient Documents especially from Testimonies (1901), Collected Papersv. 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 "Retroduction" at Commens Dictionary of Peirces Terms.)

[8] Peirce, C. S. (1903), Harvard lectures on pragmatism, Collected Papers v. 5, paragraphs 188189.

[9] A Letter to J. H. Kehler (1911), New Elements of Mathematics v. 3, pp. 2034, see under "Retroduction" at CommensDictionary of Peirces Terms.

[10] Peirce, C.S. (1902), application to the Carnegie Institution, see MS L75.329-330, from Draft D of Memoir 27:

Consequently, to discover is simply to expedite an event that would occur sooner or later, if we had nottroubled ourselves to make the discovery. Consequently, the art of discovery is purely a question of economics.The economics of research is, so far as logic is concerned, the leading doctrine with reference to the art ofdiscovery. Consequently, the conduct of abduction, which is chiey a question of heuristic and is the rstquestion of heuristic, is to be governed by economical considerations.

[11] Peirce MS. 692, quoted in Sebeok, T. (1981) "You Know My Method" in Sebeok, T., The Play of Musement, Bloomington,IA: Indiana, page 24.


[12] Peirce MS. 696, quoted in Sebeok, T. (1981) "You Know My Method" in Sebeok, T., The Play of Musement, Bloomington,IA: Indiana, page 31.

[13] Jonathan Koehler. The Base Rate Fallacy Reconsidered: Descriptive, Normative and Methodological Challenges. Behav-ioral and Brain Sciences. 19, 1996.

[14] Robertson, B., & Vignaux, G. A. (1995). Interpreting evidence: Evaluating forensic evidence in the courtroom. Chichester:John Wiley and Sons.

[15] A. Jsang. Conditional Reasoning with Subjective Logic. Journal of multiple valued logic and soft computing. 15(1),pp.5-38, 2008.PDF

[16] Popper, Karl (2002), Conjectures and Refutations: The Growth of Scientic Knowledge, London, UK: Routledge. p 536

[17] See Santaella, Lucia (1997) The Development of Peirces Three Types of Reasoning: Abduction, Deduction, and Induc-tion, 6th Congress of the IASS. Eprint.

[18] Peirce, C. S. (1908), "A Neglected Argument for the Reality of God", Hibbert Journal v. 7, pp. 90112, see 4. InCollected Papers v. 6, see paragraph 476. In The Essential Peirce v. 2, see p. 444.

[19] Peirce, C. S. (1908), "A Neglected Argument for the Reality of God", Hibbert Journal v. 7, pp. 90112. See both partIII and part IV. Reprinted, including originally unpublished portion, in Collected Papers v. 6, paragraphs 45285, EssentialPeirce v. 2, pp. 43450, and elsewhere.

[20] Peirce used the term intuition not in the sense of an instinctive or anyway half-conscious inference as people oftendo currently. Instead he used intuition usually in the sense of a cognition devoid of logical determination by previouscognitions. He said, We have no power of Intuition in that sense. See his Some Consequences of Four Incapacities(1868), Eprint.

[21] For a relevant discussion of Peirce and the aims of abductive inference, see McKaughan, Daniel J. (2008), From UglyDuckling to Swan: C. S. Peirce, Abduction, and the Pursuit of Scientic Theories, Transactions of the Charles S. PeirceSociety, v. 44, no. 3 (summer), 446468.

[22] Peirce means conceivable very broadly. See Collected Papers v. 5, paragraph 196, or Essential Peirce v. 2, p. 235,Pragmatism as the Logic of Abduction (Lecture VII of the 1903 Harvard lectures on pragmatism):

It allows any ight of imagination, provided this imagination ultimately alights upon a possible practicaleect; and thus many hypotheses may seem at rst glance to be excluded by the pragmatical maxim that arenot really so excluded.

[23] Peirce, C. S., Carnegie Application (L75, 1902, New Elements of Mathematics v. 4, pp. 3738. See under "Abduction" atthe Commens Dictionary of Peirces Terms:

Methodeutic has a special interest in Abduction, or the inference which starts a scientic hypothesis. Forit is not sucient that a hypothesis should be a justiable one. Any hypothesis which explains the facts isjustied critically. But among justiable hypotheses we have to select that one which is suitable for beingtested by experiment.

[24] Peirce, Pragmatism as the Logic of Abduction (Lecture VII of the 1903 Harvard lectures on pragmatism), see parts IIIand IV. Published in part in Collected Papers v. 5, paragraphs 180212 (see 196200, Eprint and in full in Essential Peircev. 2, pp. 226241 (see sections III and IV).

.... What is good abduction? What should an explanatory hypothesis be to be worthy to rank as a hypoth-esis? Of course, it must explain the facts. But what other conditions ought it to fulll to be good? .... Anyhypothesis, therefore, may be admissible, in the absence of any special reasons to the contrary, provided it becapable of experimental verication, and only insofar as it is capable of such verication. This is approximatelythe doctrine of pragmatism.

[25] Peirce, A Letter to Paul Carus circa 1910, Collected Papers v. 8, paragraphs 227228. See under "Hypothesis" at theCommens Dictionary of Peirces Terms.

[26] (1867), On the Natural Classication of Arguments, Proceedings of the American Academy of Arts and Sciences v. 7,pp. 261287. Presented April 9, 1867. See especially starting at p. 284 in Part III 1. Reprinted in Collected Papers v. 2,paragraphs 461516 and Writings v. 2, pp. 2349.

[27] Peirce, C. S. (1878), Deduction, Induction, and Hypothesis, Popular Science Monthly, v. 13, pp. 47082, see 472.Collected Papers 2.61944, see 623.

1.8. NOTES 13

[28] A letter to Langley, 1900, published in Historical Perspectives on Peirces Logic of Science. See excerpts under "Abduction"at the Commens Dictionary of Peirces Terms.

[29] A Syllabus of Certain Topics of Logic'" (1903 manuscript), Essential Peirce v. 2, see p. 287. See under "Abduction" atthe Commens Dictionary of Peirces Terms.

[30] Peirce, C. S., On the Logic of Drawing History from Ancient Documents, dated as circa 1901 both by the editors ofCollected Papers (see CP v. 7, bk 2, ch. 3, footnote 1) and by those of the Essential Peirce (EP) (Eprint. The articlesdiscussion of abduction is in CP v. 7, paragraphs 21831 and in EP v. 2, pp. 10714.

[31] Peirce, C. S., A Syllabus of Certain Topics of Logic (1903), Essential Peirce v. 2, p. 287:

The mind seeks to bring the facts, as modied by the new discovery, into order; that is, to form a generalconception embracing them. In some cases, it does this by an act of generalization. In other cases, no new lawis suggested, but only a peculiar state of facts that will explain the surprising phenomenon; and a law alreadyknown is recognized as applicable to the suggested hypothesis, so that the phenomenon, under that assumption,would not be surprising, but quite likely, or even would be a necessary result. This synthesis suggesting a newconception or hypothesis, is the Abduction.

[32] Peirce, C. S. (1883), A Theory of Probable Inference in Studies in Logic).

[33] In Peirce, C. S., 'Minute Logic' circa 1902, Collected Papers v. 2, paragraph 102. See under "Abduction" at CommensDictionary of Peirces Terms.

[34] Peirce, On the Logic of drawing History from Ancient Documents, 1901 manuscript, Collected Papers v. 7, paragraphs164231, see 202, reprinted in Essential Peirce v. 2, pp. 75114, see 95. See under "Abduction" at Commens Dictionaryof Peirces Terms.

[35] Peirce, On the Logic of Drawing Ancient History from Documents, Essential Peirce v. 2, see pp. 1079.

[36] Peirce, Carnegie application, L75 (1902), Memoir 28: On the Economics of Research, scroll down to Draft E. Eprint.

[37] Peirce, C. S., the 1866 Lowell Lectures on the Logic of Science, Writings of Charles S. Peirce v. 1, p. 485. See under"Hypothesis" at Commens Dictionary of Peirces Terms.

[38] Peirce, C. S., A Syllabus of Certain Topics of Logic, written 1903. See The Essential Peirce v. 2, p. 287. Quote viewableunder "Abduction" at Commens Dictionary of Peirces Terms.

[39] Peirce, A Letter to Paul Carus 1910, Collected Papers v. 8, see paragraph 223.

[40] Peirce, C. S. (1902), Application to the Carnegie Institution, Memoir 27, Eprint: Of the dierent classes of arguments,abductions are the only ones in which after they have been admitted to be just, it still remains to inquire whether they areadvantageous.

[41] Peirce, On the Logic of Drawing Ancient History from Documents, Essential Peirce v. 2, see pp. 1079 and 113. OnTwenty Questions, p. 109, Peirce has pointed out that if each question eliminates half the possibilities, twenty questionscan choose from among 220 or 1,048,576 objects, and goes on to say:

Thus, twenty skillful hypotheses will ascertain what 200,000 stupid ones might fail to do. The secret of thebusiness lies in the caution which breaks a hypothesis up into its smallest logical components, and only risksone of them at a time.

[42] Schwendtner, Tibor and Ropolyi, Lszl and Kiss, Olga (eds): Hermeneutika s a termszettudomnyok. ron Kiad,Budapest, 2001. It is written in Hungarian. Meaning of the title: Hermeneutics and the natural sciences. See, e.g.,Hansons Patterns of Discovery (Hanson, 1958), especially pp. 85-92

[43] Rapezzi, C; Ferrari, R; Branzi, A (24 December 2005). White coats and ngerprints: diagnostic reasoning in medicine andinvestigative methods of ctional detectives. BMJ (Clinical research ed.) 331 (7531): 14914. doi:10.1136/bmj.331.7531.1491.PMC 1322237. PMID 16373725. Retrieved 17 January 2014.

[44] Rejn Altable, C (October 2012). Logic structure of clinical judgment and its relation to medical and psychiatric semi-ology. Psychopathology 45 (6): 34451. doi:10.1159/000337968. PMID 22854297. Retrieved 17 January 2014.

[45] Kave Eshghi. Abductive planning with the event calculus. In Robert A. Kowalski, Kenneth A. Bowen editors: LogicProgramming, Proceedings of the Fifth International Conference and Symposium, Seattle, Washington, August 1519,1988. MIT Press 1988, ISBN 0-262-61056-6

[46] April M. S. McMahon (1994): Understanding language change. Cambridge: Cambridge University Press. ISBN 0-521-44665-1


[47] Eco, U. (1976). A theory of Semiotics. Bloomington, IA: Indiana. p 131

[48] Gell, A. 1984, Art and Agency. Oxford: Oxford. p 14

[49] Bowden, R. (2004) A critique of Alfred Gell on Art and Agency. Retrieved Sept 2007 from: Find Articles at BNET

[50] Whitney D. (2006) 'Abduction the agency of art.' Retrieved May 2009 from: University of California, Berkeley

1.9 External links Abduction entry by Igor Douven in the Stanford Encyclopedia of Philosophy Abductive reasoning at the Indiana Philosophy Ontology Project Abductive reasoning at PhilPapers "Abductive Inference" (once there, scroll down), John R. Josephson, Laboratory for Articial Intelligence

Research, Ohio State University. (Former webpage via the Wayback Machine.) "Deduction, Induction, and Abduction", Chapter 3 in article "Charles Sanders Peirce" by Robert Burch, 2001

and 2006, in the Stanford Encyclopedia of Philosophy. "Abduction", links to articles and websites on abductive inference, Martin Ryder. 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 Gieen, Germany, and set upAbduktionsforschung, home page not in English but see Artikel section there. Abduktionsforschunghomepage via Google translation.

"'You Know My Method': A Juxtaposition of Charles S. Peirce and Sherlock Holmes" (1981), by Thomas Se-beok with Jean Umiker-Sebeok, from The Play of Musement, Thomas Sebeok, Bloomington, Indiana: IndianaUniversity Press, pp. 1752.

Commens Dictionary of Peirces Terms, Mats Bergman and Sami Paavola, editors, Helsinki U. Peirces owndenitions, often many per term across the decades. There, see Hypothesis [as a form of reasoning]", Ab-duction, Retroduction, and Presumption [as a form of reasoning]".

Chapter 2

Deductive reasoning

Deductive reasoning, also deductive logic or logical deduction or, informally, "top-down" logic,[1] is the processof reasoning from one or more statements (premises) to reach a logically certain conclusion.[2] It diers from inductivereasoning or abductive reasoning.Deductive reasoning links premises with conclusions. If all premises are true, the terms are clear, and the rules ofdeductive logic are followed, then the conclusion reached is necessarily true.Deductive reasoning (top-down logic) contrasts with inductive reasoning (bottom-up logic) in the following way: Indeductive reasoning, a conclusion is reached reductively by applying general rules that hold over the entirety of aclosed domain of discourse, narrowing the range under consideration until only the conclusions is left. In inductivereasoning, the conclusion is reached by generalizing or extrapolating from, i.e., there is epistemic uncertainty. Note,however, that the inductive reasoning mentioned here is not the same as induction used in mathematical proofs mathematical induction is actually a form of deductive reasoning.

2.1 Simple exampleAn example of a deductive argument:

1. All men are mortal.2. Kass is a man.3. Therefore, Kass is mortal.

The rst premise states that all objects classied as men have the attribute mortal. The second premise states thatKass is classied as a man a member of the set men. The conclusion then states that Kass must be mortalbecause he inherits this attribute from his classication as a man.

2.2 Law of detachmentMain article: Modus ponens

The law of detachment (also known as arming the antecedent and Modus ponens) is the rst form of deductivereasoning. A single conditional statement is made, and a hypothesis (P) is stated. The conclusion (Q) is then deducedfrom the statement and the hypothesis. The most basic form is listed below:

1. P Q (conditional statement)2. P (hypothesis stated)3. Q (conclusion deduced)

15

16 CHAPTER 2. DEDUCTIVE REASONING

In deductive reasoning, we can conclude Q from P by using the law of detachment.[3] However, if the conclusion (Q)is given instead of the hypothesis (P) then there is no denitive conclusion.The following is an example of an argument using the law of detachment in the form of an if-then statement:

1. If an angle satises 90 < A < 180, then A is an obtuse angle.2. A = 120.3. A is an obtuse angle.

Since the measurement of angle A is greater than 90 and less than 180, we can deduce that A is an obtuse angle.If however, we are given the conclusion that A is an obtuse angle we cannot deduce the premise that A = 120.

2.3 Law of syllogismThe law of syllogism takes two conditional statements and forms a conclusion by combining the hypothesis of onestatement with the conclusion of another. Here is the general form:

1. P Q2. Q R3. Therefore, P R.

The following is an example:

1. If Larry is sick, then he will be absent.2. If Larry is absent, then he will miss his classwork.3. Therefore, if Larry is sick, then he will miss his classwork.

We deduced the nal statement by combining the hypothesis of the rst statement with the conclusion of the secondstatement. We also allow that this could be a false statement. This is an example of the Transitive Property inmathematics. The Transitive Property is sometimes phrased in this form:

1. A = B.2. B = C.3. Therefore A = C.

2.4 Law of contrapositiveMain article: Modus tollens

The law of contrapositive states that, in a conditional, if the conclusion is false, then the hypothesis must be false also.The general form is the following:

1. P Q.2. ~Q.3. Therefore we can conclude ~P.

The following are examples:

1. If it is raining, then there are clouds in the sky.2. There are no clouds in the sky.3. Thus, it is not raining.

2.5. VALIDITY AND SOUNDNESS 17

2.5 Validity and soundnessDeductive arguments are evaluated in terms of their validity and soundness.An argument is valid if it is impossible for its premises to be true while its conclusion is false. In other words, theconclusion must be true if the premises are true. An argument can be valid even though the premises are false.An argument is sound if it is valid and the premises are true.It is possible to have a deductive argument that is logically valid but is not sound. Fallacious arguments often takethat form.The following is an example of an argument that is valid, but not sound:

1. Everyone who eats carrots is a quarterback.2. John eats carrots.3. Therefore, John is a quarterback.

The examples rst premise is false there are people who eat carrots and are not quarterbacks but the conclusionmust be true, so long as the premises are true (i.e. it is impossible for the premises to be true and the conclusionfalse). Therefore the argument is valid, but not sound. Generalizations are often used to make invalid arguments,such as everyone who eats carrots is a quarterback. Not everyone who eats carrots is a quarterback, thus provingthe aw of such arguments.In this example, the rst statement uses categorical reasoning, saying that all carrot-eaters are denitely quarterbacks.This theory of deductive reasoning also known as term logic was developed by Aristotle, but was superseded bypropositional (sentential) logic and predicate logic.Deductive reasoning can be contrasted with inductive reasoning, in regards to validity and soundness. In cases ofinductive reasoning, even though the premises are true and the argument is valid, it is possible for the conclusionto be false (determined to be false with a counterexample or other means).

2.6 HistoryAristotle started documenting deductive reasoning in the 4th century BC.[4]

2.7 EducationDeductive reasoning is generally thought of as a skill that develops without any formal teaching or training. As aresult of this belief, deductive reasoning skills are not taught in secondary schools, where students are expected to usereasoning more often and at a higher level.[5] It is in high school, for example, that students have an abrupt introductionto mathematical proofs which rely heavily on deductive reasoning.[5]

2.8 See also Argument (logic) Logic Mathematical logic Abductive reasoning Analogical reasoning Correspondence theory of truth Defeasible reasoning

18 CHAPTER 2. DEDUCTIVE REASONING

Decision making Decision theory Fallacy Fault Tree Analysis Geometry Hypothetico-deductive method Inquiry Mathematical induction Inductive reasoning Inference Logical consequence Natural deduction Propositional calculus Retroductive reasoning Scientic method Theory of justication Soundness Syllogism

2.9 References[1] Deduction & Induction, Research Methods Knowledge Base

[2] Sternberg, R. J. (2009). Cognitive Psychology. Belmont, CA: Wadsworth. p. 578. ISBN 978-0-495-50629-4.

[3] Guide to Logic

[4] Evans, Jonathan St. B. T.; Newstead, Stephen E.; Byrne, Ruth M. J., eds. (1993). Human Reasoning: The Psychology ofDeduction (Reprint ed.). Psychology Press. p. 4. ISBN 9780863773136. Retrieved 2015-01-26. In one sense [...] onecan see the psychology of deductive reasoning as being as old as the study of logic, which originated in the writings ofAristotle.

[5] Stylianides, G. J.; Stylianides (2008). A. J.. Mathematical Thinking and Learning 10 (2): 103133. doi:10.1080/10986060701854425.

2.10 Further reading Vincent F. Hendricks, Thought 2 Talk: A Crash Course in Reection and Expression, New York: Automatic

Press / VIP, 2005, ISBN 87-991013-7-8

Philip Johnson-Laird, Ruth M. J. Byrne, Deduction, Psychology Press 1991, ISBN 978-0-86377-149-1 Zarefsky, David, Argumentation: The Study of Eective Reasoning Parts I and II, The Teaching Company 2002 Bullemore, Thomas, * The Pragmatic Problem of Induction.

2.11. EXTERNAL LINKS 19

2.11 External links Deductive reasoning at PhilPapers Deductive reasoning at the Indiana Philosophy Ontology Project

Deductive reasoning entry in the Internet Encyclopedia of Philosophy

Chapter 3

Inductive reasoning

Inductive inference redirects here. For the technique in mathematical proof, see Mathematical induction. For thetheory introduced by Ray Solomono, see Solomonos theory of inductive inference.

Inductive reasoning (as opposed to deductive reasoning or abductive reasoning) is reasoning in which the premisesseek to supply strong evidence for (not absolute proof of) the truth of the conclusion. While the conclusion of adeductive argument is certain, the truth of the conclusion of an inductive argument is probable, based upon theevidence given.[1]

The philosophical denition of inductive reasoning is more nuanced than simple progression from particular/individualinstances to broader generalizations. Rather, the premises of an inductive logical argument indicate some degree ofsupport (inductive probability) for the conclusion but do not entail it; that is, they suggest truth but do not ensureit. In this manner, there is the possibility of moving from general statements to individual instances (for example,statistical syllogisms, discussed below).Many dictionaries dene inductive reasoning as reasoning that derives general principles from specic observations,though some sources disagree with this usage.[2]

3.1 Description

Inductive reasoning is inherently uncertain. It only deals in degrees to which, given the premises, the conclusion iscredible according to some theory of evidence. Examples include a many-valued logic, DempsterShafer theory,or probability theory with rules for inference such as Bayes rule. Unlike deductive reasoning, it does not rely onuniversals holding over a closed domain of discourse to draw conclusions, so it can be applicable even in cases ofepistemic uncertainty (technical issues with this may arise however; for example, the second axiom of probability isa closed-world assumption).[3]

An example of an inductive argument:

100% of biological life forms that we know of depend on liquid water to exist.

Therefore, if we discover a new biological life form it will probably depend on liquid water to exist.

This argument could have been made every time a new biological life form was found, and would have been correctevery time; however, it is still possible that in the future a biological life form not requiring water could be discovered.As a result, the argument may be stated less formally as:

All biological life forms that we know of depend on liquid water to exist.

All biological life probably depends on liquid water to exist.

20

3.2. INDUCTIVE VS. DEDUCTIVE REASONING 21

3.2 Inductive vs. deductive reasoningUnlike deductive arguments, inductive reasoning allows for the possibility that the conclusion is false, even if all of thepremises are true.[4] Instead of being valid or invalid, inductive arguments are either strong or weak, which describeshow probable it is that the conclusion is true.[5]

Given that if A is true then B, C, and D are true, an example of deduction would be "A is true therefore we candeduce that B, C, and D are true. An example of induction would be "B, C, and D are observed to be true thereforeA may be true. A is a reasonable explanation for B, C, and D being true.For example:

A large enough asteroid impact would create a very large crater and cause a severe impact winter thatcould drive the non-avian dinosaurs to extinction.We observe that there is a very large crater in the gulf of Mexico dating to very near the time the theextinction of the non-avian dinosaursTherefore it is possible that this impact could explain why the non-avian dinosaurs went extinct.

Note however that this is not necesarily the case. Other events also coincide with the extinction of the non-aviandinosaurs. For example the Deccan Traps in India.A classical example of an incorrect inductive argument was presented by John Vickers:

All of the swans we have seen are white.Therefore, all swans are white.

Note that this denition of inductive reasoning excludes mathematical induction, which is a form of deductive rea-soning.

3.3 CriticismMain article: Problem of induction

Inductive reasoning has been criticized by thinkers as diverse as Sextus Empiricus[6] and Karl Popper.[7]

The classic philosophical treatment of the problem of induction was given by the Scottish philosopher David Hume.[8]

Although the use of inductive reasoning demonstrates considerable success, its application has been questionable.Recognizing this, Hume highlighted the fact that our mind draws uncertain conclusions from relatively limited expe-riences. In deduction, the truth value of the conclusion is based on the truth of the premise. In induction, however,the dependence on the premise is always uncertain. As an example, lets assume all ravens are black. The fact thatthere are numerous black ravens supports the assumption. However, the assumption becomes inconsistent with thefact that there are white ravens. Therefore, the general rule of all ravens are black is inconsistent with the existenceof the white raven. Hume further argued that it is impossible to justify inductive reasoning: specically, that it cannotbe justied deductively, so our only option is to justify it inductively. Since this is circular he concluded that our useof induction is unjustiable with the help of Humes Fork.[9]

However, Hume then stated that even if induction were proved unreliable, we would still have to rely on it. Soinstead of a position of severe skepticism, Hume advocated a practical skepticism based on common sense, wherethe inevitability of induction is accepted.[10]

3.3.1 BiasesInductive reasoning is also known as hypothesis construction because any conclusions made are based on currentknowledge and predictions. As with deductive arguments, biases can distort the proper application of inductiveargument, thereby preventing the reasoner from forming the most logical conclusion based on the clues. Examplesof these biases include the availability heuristic, conrmation bias, and the predictable-world bias.

22 CHAPTER 3. INDUCTIVE REASONING

The availability heuristic causes the reasoner to depend primarily upon information that is readily available to him/her.People have a tendency to rely on information that is easily accessible in the world around them. For example, insurveys, when people are asked to estimate the percentage of people who died from various causes, most respondentswould choose the causes that have been most prevalent in the media such as terrorism, and murders, and airplaneaccidents rather than causes such as disease and trac accidents, which have been technically less accessible to theindividual since they are not emphasized as heavily in the world around him/her.The conrmation bias is based on the natural tendency to conrm rather than to deny a current hypothesis. Researchhas demonstrated that people are inclined to seek solutions to problems that are more consistent with known hy-potheses rather than attempt to refute those hypotheses. Often, in experiments, subjects will ask questions that seekanswers that t established hypotheses, thus conrming these hypotheses. For example, if it is hypothesized that Sallyis a sociable individual, subjects will naturally seek to conrm the premise by asking questions that would produceanswers conrming that Sally is in fact a sociable individual.The predictable-world bias revolves around the inclination to perceive order where it has not been proved to exist,either at all or at a particular level of abstraction. Gambling, for example, is one of the most popular examples ofpredictable-world bias. Gamblers often begin to think that they see simple and obvious patterns in the outcomes and,therefore, believe that they are able to predict outcomes based upon what they have witnessed. In reality, however,the outcomes of these games are dicult to predict and highly complex in nature. However, in general, people tendto seek some type of simplistic order to explain or justify their beliefs and experiences, and it is often dicult forthem to realise that their perceptions of order may be entirely dierent from the truth.[11]

3.4 Types

3.4.1 GeneralizationA generalization (more accurately, an inductive generalization) proceeds from a premise about a sample to a conclusionabout the population.

The proportion Q of the sample has attribute A.Therefore:The proportion Q of the population has attribute A.

Example

There are 20 ballseither black or whitein an urn. To estimate their respective numbers, you draw a sample offour balls and nd that three are black and one is white. A good inductive generalization would be that there are 15black, and ve white, balls in the urn.How much the premises support the conclusion depends upon (a) the number in the sample group, (b) the number inthe population, and (c) the degree to which the sample represents the population (which may be achieved by takinga random sample). The hasty generalization and the biased sample are generalization fallacies.

3.4.2 Statistical syllogismMain article: Statistical syllogism

A statistical syllogism proceeds from a generalization to a conclusion about an individual.

A proportion Q of population P has attribute A.An individual X is a member of P.Therefore:There is a probability which corresponds to Q that X has A.

The proportion in the rst premise would be something like 3/5ths of, all, few, etc. Two dicto simpliciterfallacies can occur in statistical syllogisms: "accident" and "converse accident".

3.5. BAYESIAN INFERENCE 23

3.4.3 Simple induction

Simple induction proceeds from a premise about a sample group to a conclusion about another individual.

Proportion Q of the known instances of population P has attribute A.Individual I is another member of P.Therefore:There is a probability corresponding to Q that I has A.

This is a combination of a generalization and a statistical syllogism, where the conclusion of the generalization is alsothe rst premise of the statistical syllogism.

Argument from analogy

Main article: Argument from analogy

The process of analogical inference involves noting the shared properties of two or more things, and from this basisinferring that they also share some further property:[12]

P and Q are similar in respect to properties a, b, and c.Object P has been observed to have further property x.Therefore, Q probably has property x also.

Analogical reasoning is very frequent in common sense, science, philosophy and the humanities, but sometimes it isaccepted only as an auxiliary method. A rened approach is case-based reasoning.[13]

3.4.4 Causal inference

A causal inference draws a conclusion about a causal connection based on the conditions of the occurrence of aneect. Premises about the correlation of two things can indicate a causal relationship between them, but additionalfactors must be conrmed to establish the exact form of the causal relationship.

3.4.5 Prediction

A prediction draws a conclusion about a future individual from a past sample.

Proportion Q of observed members of group G have had attribute A.Therefore:There is a probability corresponding to Q that other members of group G will have attribute A whennext observed.

3.5 Bayesian inferenceAs a logic of induction rather than a theory of belief, Bayesian inference does not determine which beliefs are a priorirational, but rather determines how we should rationally change the beliefs we have when presented with evidence.We begin by committing to a prior probability for a hypothesis based on logic or previous experience, and when facedwith evidence, we adjust the strength of our belief in that hypothesis in a precise manner using Bayesian logic.


3.6 Inductive inferenceAround 1960, Ray Solomono founded the theory of universal inductive inference, the theory of prediction based onobservations; for example, predicting the next symbol based upon a given series of symbols. This is a formal inductiveframework that combines algorithmic information theory with the Bayesian framework. Universal inductive inferenceis based on solid philosophical foundations,[14] and can be considered as a mathematically formalized Occams razor.Fundamental ingredients of the theory are the concepts of algorithmic probability and Kolmogorov complexity.

3.7 See also Abductive reasoning Algorithmic information theory Algorithmic probability Analogy Bayesian probability Counterinduction Deductive reasoning Explanation Failure mode and eects analysis Falsiability Grammar induction Inductive inference Inductive logic programming Inductive probability Inductive programming Inductive reasoning aptitude Inquiry Kolmogorov complexity Lateral thinking Laurence Jonathan Cohen Logic Logical positivism Machine learning Mathematical induction Mills Methods Minimum description length Minimum message length Open world assumption

3.8. REFERENCES 25

Raven paradox Recursive Bayesian estimation Retroduction Solomonos theory of inductive inference Statistical inference Stephen Toulmin Universal articial intelligence

3.8 References[1] Copi, I. M.; Cohen, C.; Flage, D. E. (2007). Essentials of Logic (Second ed.). Upper Saddle River, NJ: Pearson Education.

ISBN 978-0-13-238034-8.

[2] Deductive and Inductive Arguments, Internet Encyclopedia of Philosophy, Some dictionaries dene deduction as rea-soning from the general to specic and induction as reasoning from the specic to the general. While this usage is stillsometimes found even in philosophical and mathematical contexts, for the most part, it is outdated.

[3] Kosko, Bart (1990). Fuzziness vs. Probability. International Journal ofGeneral Systems 17 (1): 211240. doi:10.1080/03081079008935108.

[4] John Vickers. The Problem of Induction. The Stanford Encyclopedia of Philosophy.

[5] Herms, D. Logical Basis of Hypothesis Testing in Scientic Research (PDF).

[6] Sextus Empiricus, Outlines Of Pyrrhonism. Trans. R.G. Bury, Harvard University Press, Cambridge, Massachusetts, 1933,p. 283.

[7] Popper, Karl R.; Miller, David W. (1983). A proof of the impossibility of inductive probability. Nature 302 (5910):687688. doi:10.1038/302687a0.

[8] David Hume (1910) [1748]. An Enquiry concerning Human Understanding. P.F. Collier & Son. ISBN 0-19-825060-6.

[9] Vickers, John. The Problem of Induction (Section 2). Stanford Encyclopedia of Philosophy. 21 June 2010

[10] Vickers, John. The Problem of Induction (Section 2.1). Stanford Encyclopedia of Philosophy. 21 June 2010.

[11] Gray, Peter (2011). Psychology (Sixth ed.). New York: Worth. ISBN 978-1-4292-1947-1.

[12] Baronett, Stan (2008). Logic. Upper Saddle River, NJ: Pearson Prentice Hall. pp. 321325.

[13] For more information on inferences by analogy, see Juthe, 2005.

[14] Rathmanner, Samuel; Hutter, Marcus (2011). A Philosophical Treatise of Universal Induction. Entropy 13 (6): 10761136. doi:10.3390/e13061076.

3.9 Further reading Cushan, Anna-Marie (1983/2014). Investigation into Facts and Values: Groundwork for a theory of moralconict resolution. [Thesis, Melbourne University], Ondwelle Publications (online): Melbourne.

Herms, D. Logical Basis of Hypothesis Testing in Scientic Research (PDF). Kemerling, G. (27 October 2001). Causal Reasoning. Holland, J. H.; Holyoak, K. J.; Nisbett, R. E.; Thagard, P. R. (1989). Induction: Processes of Inference,Learning, and Discovery. Cambridge, MA, USA: MIT Press. ISBN 0-262-58096-9.

Holyoak, K.; Morrison, R. (2005). The CambridgeHandbook of Thinking and Reasoning. New York: CambridgeUniversity Press. ISBN 978-0-521-82417-0.


3.10 External links Conrmation and Induction entry in the Internet Encyclopedia of Philosophy Inductive Logic entry in the Stanford Encyclopedia of Philosophy Inductive reasoning at PhilPapers Inductive reasoning at the Indiana Philosophy Ontology Project Four Varieties of Inductive Argument from the Department of Philosophy, University of North Carolina at

Greensboro.

Properties of Inductive Reasoning PDF (166 KiB), a psychological review by Evan Heit of the University ofCalifornia, Merced.

The Mind, Limber An article which employs the lm The Big Lebowski to explain the value of inductivereasoning.

The Pragmatic Problem of Induction, by Thomas Bullemore

Chapter 4

Inference

Inference is the act or process of deriving logical conclusions from premises known or assumed to be true.[1] Theconclusion drawn is also called an idiomatic. The laws of valid inference are studied in the eld of logic.Alternatively, inference may be dened as the non-logical, but rational means, through observation of patterns of facts,to indirectly see new meanings and contexts for understanding. Of particular use to this application of inference areanomalies and symbols. Inference, in this sense, does not draw conclusions but opens new paths for inquiry. (Seesecond set of Examples.) In this denition of inference, there are two types of inference: inductive inference anddeductive inference. Unlike the denition of inference in the rst paragraph above, meaning of word meanings arenot tested but meaningful relationships are articulated.Human inference (i.e. how humans draw conclusions) is traditiona

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