Abductive reasoning

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

Abduction is a kind of logical inference described by Charles Sanders Peirce as "guessing". The term refers to the process of arriving at an explanatory hypothesis. Peirce said that to abduce a hypothetical explanation $a$ from an observed surprising circumstance $b$ is to surmise that $a$ may be true because then $b$ would be a matter of course. Thus, to abduce $a$ from $b$ involves determining that $a$ is sufficient (or nearly sufficient), but not necessary, for $b$.

For example, $b$: the lawn is wet. But if $a$: it rained last night, then it would be unsurprising that $b$ the lawn is wet. Therefore, by abductive reasoning, $a$: it rained last night. In that formulation the matter in the hypothetical explanation $a$ is introduced in a premise, while only the conclusion affirms that there is therefore reason to suspect that $a$ is true. (But note that Peirce did not remain convinced that a single logical form covers all abduction.) In any case, Peirce held that a new or outside idea is introduced to serve as a conclusion; this involvement of a new idea, along with the surprise which occasions it, helps distinguish abduction from induction (whose conclusion likewise can be sufficient but not necessary for the premises). Peirce's ideas about abduction, which changed somewhat over time to distinguish it from the induction of characteristics, stand at the beginning of others' developments and formulations, some of which diverge from Peirce or use his earlier ideas, and some of which are summarized below herein. There has been renewed interest in the subject, in computer science and artificial intelligence research. The term abduction itself is commonly presumed to mean the same thing as hypothesis; however, as above, it actually refers to the process of inference that produces a hypothesis as its end result. The term is used in both philosophy and computing to refer to such inferences.

In the broader methodological terms also brought into the discussion by Peirce, to abduce a hypothesis leads, in the context of rational inquiry, to judging whether trying the abduced hypothesis would help economize the inquiry. For example, abduction's guesses can have a strategy, and Peirce's example was the game of Twenty Questions. Simplification and economy, not only of explanation but of inquiry itself, are what call for the 'leap' of abduction. Meanwhile it is also the insecurity of the hypothesis that leads to its being judged for its suitability for being tested by experiment in thought, practice, or science as the case may warrant.

Deduction, induction, and abduction

Main article: Logical reasoning

Deduction

allows deriving $b$ from $a$ only where $b$ is a formal consequence of $a$. In other words, deduction is the process of deriving the consequences of what is assumed. Given the truth of the assumptions, a valid deduction guarantees the truth of the conclusion. For example, given that all bachelors are unmarried males, and given that this person is a bachelor, it can be deduced that this person is an unmarried male.

Induction

allows inferring $b$ from $a$, where $b$ does not follow necessarily from $a$. $a$ might give us very good reason to accept $b$, but it does not ensure that $b$. For example, if all of the swans that we have observed so far are white, we may induce that all swans are white. 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.)

Abduction

allows inferring $a$ as an explanation of $b$. Because of this, abduction allows the precondition $a$ to be inferred from the consequence $b$. Deduction and abduction thus differ in the direction in which a rule like "$a$ entails $b$" is used for inference. As such abduction is formally equivalent to the logical fallacy affirming the consequent or Post hoc ergo propter hoc, because there are multiple possible explanations for $b$. For example, after glancing up and seeing the eight ball moving towards us we may abduce that it was struck by the cue ball. The cue ball's strike would account for the eight ball's movement. It serves as a theory that explains our observation. There are in fact infinitely many possible explanations for the eight ball's movement, and so our abduction does not leave us certain that the cue ball did in fact strike the eight ball, but our abduction is still useful and can serve to orient us in our surroundings. This process of abduction is an instance of the scientific method. There are infinite possible explanations for any of the physical processes we observe, but we are inclined to abduce a single explanation (or a few explanations) for them in the hopes that we can better orient ourselves in our surroundings.

Formalizations of abduction

Logic-based abduction

In logic, explanation is done from a logical theory $T$ representing a domain and a set of observations $O$. Abduction is the process of deriving a set of explanations of $O$ according to $T$ and picking out one of those explanations. For $E$ to be an explanation of $O$ according to $T$, it should satisfy two conditions:

• $O$ 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 $O$ according to theory $T$ are formalized as:

$T\cup E\models O$;

$T\cup E$ is consistent.

Among the possible explanations $E$ satisfying these two conditions, some other condition of minimality is usually imposed to avoid irrelevant facts (not contributing to the entailment of $O$) being included in the explanations. Abduction is then the process that picks out some member of $E$. Criteria for picking out a member representing "the best" explanation include the simplicity, the prior probability, or the explanatory power of the explanation.

A proof theoretical abduction method for first order classical logic based on the sequent calculus and a dual one, based on semantic tableaux (analytic tableaux) have been proposed (Cialdea Mayer & Pirri 1993). The methods are sound and complete and work for full first order logic, without requiring any preliminary reduction of formulae into normal forms. These methods have also been extended to modal logic.

Abductive logic programming is a computational framework that extends normal logic programming with abduction. It separates the theory $T$ into two components, one of which is a normal logic program, used to generate $E$ by means of backward reasoning, the other of which is a set of integrity constraints, used to filter the set of candidate explanations.

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 $H$ and a set of manifestations $M$; they are related by the domain knowledge, represented by a function $e$ that takes as an argument a set of hypotheses and gives as a result the corresponding set of manifestations. In other words, for every subset of the hypotheses $H'\subseteq H$, their effects are known to be $e(H')$.

Abduction is performed by finding a set $H'\subseteq H$ such that $M\subseteq e(H')$. In other words, abduction is performed by finding a set of hypotheses $H'$ such that their effects $e(H')$ include all observations $M$.

A common assumption is that the effects of the hypotheses are independent, that is, for every $H'\subseteq H$, it holds that $e(H')=\bigcup _{h\in H'}e(\{h\})$. 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 Occam's razor). Abductive validation is common practice in hypothesis formation in science; moreover, Peirce argues it is a ubiquitous aspect of thought:

Looking out my window this lovely spring morning I see an azalea in full bloom. No, no! I do not see that; though that is the only way I can describe what I see. That is a proposition, a sentence, a fact; but what I perceive is not proposition, sentence, fact, but only an image, which I make intelligible in part by means of a statement of fact. This statement is abstract; but what I see is concrete. I perform an abduction when I so much as express in a sentence anything I see. The truth is that the whole fabric of our knowledge is one matted felt of pure hypothesis confirmed and refined by induction. Not the smallest advance can be made in knowledge beyond the stage of vacant staring, without making an abduction at every step.

It was Peirce's own maxim that "Facts cannot be explained by a hypothesis more extraordinary than these facts themselves; and of various hypotheses the least extraordinary must be adopted." After obtaining 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 letting a group of infected and a group of non-infected people undergo the test. Assume the statements $x$: "Positive test", ${\overline {x}}$: "Negative test", $y$: "Infected", and ${\overline {y}}$: "Not infected". The result of these trials will then determine the reliability of the test in terms of its sensitivity $p(x|y)$ and false positive rate $p(x|{\overline {y}})$. The interpretations of the conditionals are: $p(x|y)$: "The probability of positive test given infection", and $p(x|{\overline {y}})$: "The probability of positive test in the absence of infection". The problem with applying these conditionals in a practical setting is that they are expressed in the opposite direction to what the practitioner needs. The conditionals needed for making the diagnosis are: $p(y|x)$: "The probability of infection given positive test", and $p(y|{\overline {x}})$: "The probability of infection given negative test". The probability of infection could then have been conditionally deduced as $p(y\|x)=p(x)p(y|x)+p({\overline {x}})p(y|{\overline {x}})$, 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: ${\begin{cases}p(x|y)={\frac {p(x\land y)}{p(y)}}\\p(y|x)={\frac {p(x\land y)}{p(x)}}\end{cases}}\;\;\Rightarrow \;\;\;\;p(y|x)={\frac {p(y)p(x|y)}{p(x)}}\;.$ The term $p(y)$ on the right hand side of the equation expresses the base rate of the infection in the population. Similarly, the term $p(x)$ expresses the default likelihood of positive test on a random person in the population. In the expressions below $a(y)$ and $a({\overline {y}})=1-a(y)$ denote the base rates of $y$ and its complement ${\overline {y}}$ respectively, so that e.g. $p(x)=a(y)p(x|y)+a({\overline {y}})p(x|{\overline {y}})$. The full expression for the required conditionals $p(y|x)$ and $p(y|{\overline {x}})$ are then: ${\begin{cases}p(y|x)={\frac {a(y)p(x|y)}{a(y)p(x|y)+a({\overline {y}})p(x|{\overline {y}})}}\\p(y|{\overline {x}})={\frac {a(y)p({\overline {x}}|y)}{a(y)p({\overline {x}}|y)+a({\overline {y}})p({\overline {x}}|{\overline {y}})}}\end{cases}}$

The full expression for the conditionally abduced probability of infection in a tested person, expressed as $p(y{\overline {\|}}x)$, given the outcome of the test, the base rate of the infection, as well as the test's sensitivity and false positive rate, is then given by: $p(y{\overline {\|}}x)=p(x)\left({\frac {a(y)p(x|y)}{a(y)p(x|y)+a({\overline {y}})p(x|{\overline {y}})}}\right)+p({\overline {x}})\left({\frac {a(y)p({\overline {x}}|y)}{a(y)p({\overline {x}}|y)+a({\overline {y}})p({\overline {x}}|{\overline {y}})}}\right)$.

Probabilistic abduction can thus be described as a method for inverting conditionals in order to apply probabilistic deduction.

A medical test result is typically considered positive or negative, so when applying the above equation it can be assumed that either $p(x)=1$ (positive) or $p({\overline {x}})=1$ (negative). In case the patient tests positive, the above equation can be simplified to $p(y{\overline {\|}}x)=p(y|x)$ which will give the correct likelihood that the patient actually is infected.

The Base rate fallacy in medicine, or the Prosecutor's fallacy in legal reasoning, consists of making the erroneous assumption that $p(y|x)=p(x|y)$. While this reasoning error often can produce a relatively good approximation of the correct hypothesis probability value, it can lead to a completely wrong result and wrong conclusion in case the base rate is very low and the reliability of the test is not perfect. An extreme example of the base rate fallacy is to conclude that a male person is pregnant just because he tests positive in a pregnancy test. Obviously, the base rate of male pregnancy is zero, and assuming that the test is not perfect, it would be correct to conclude that the male person is not pregnant.

The expression for probabilistic abduction can be generalised to multinomial cases, i.e., with a state space $X$ of multiple $x_{i}$ and a state space $Y$ of multiple states $y_{j}$.

Subjective logic abduction

Subjective logic generalises probabilistic logic by including parameters for uncertainty in the input arguments. Abduction in subjective logic is thus similar to probabilistic abduction described above. The input arguments in subjective logic are composite functions called subjective opinions which can be binomial when the opinion applies to a single proposition or multinomial when it applies to a set of propositions. A multinomial opinion thus applies to a frame $X\,\!$ (i.e. a state space of exhaustive and mutually disjoint propositions $x_{i}\,\!$), and is denoted by the composite function $\omega _{X}=({\vec {b}},u,{\vec {a}})\,\!$, where ${\vec {b}}\,\!$ is a vector of belief masses over the propositions of $X\,\!$, $u\,\!$ is the uncertainty mass, and ${\vec {a}}\,\!$ is a vector of base rate values over the propositions of $X\,\!$. These components satisfy $u+\sum {\vec {b}}(x_{i})=1\,\!$ and $\sum {\vec {a}}(x_{i})=1\,\!$ as well as ${\vec {b}}(x_{i}),u,{\vec {a}}(x_{i})\in [0,1]\,\!$.

Assume the frames $X$ and $Y$, the sets of conditional opinions $\omega _{X|Y}$ and $\omega _{X|{\overline {Y}}}$, the opinion $\omega _{X}$ on $X$, and the base rate function $a_{Y}$ on $Y$. Based on these parameters, subjective logic provides a method for deriving the set of inverted conditionals $\omega _{Y|X}$ and $\omega _{Y|{\overline {X}}}$. 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 "${\overline {\|}}$", and the operator itself is denoted as ${\overline {\circledcirc }}$. The expression for subjective logic abduction is then: $\omega _{Y{\overline {\|}}X}=\omega _{X}\;{\overline {\circledcirc }}\;(\omega _{X|Y},\omega _{X|{\overline {Y}}},a_{Y})\,\!$.

The advantage of using subjective logic abduction compared to probabilistic abduction is that uncertainty about the probability values of the input arguments can be explicitly expressed and taken into account during the analysis. It is thus possible to perform abductive analysis in the presence of missing or incomplete input evidence, which normally results in degrees of uncertainty in the output conclusions.

History of the concept

The philosopher Charles Sanders Peirce introduced abduction into modern logic. Over the years he called such inference hypothesis, abduction, presumption, and retroduction. In his works before 1900, he mostly treated such inference as the use of a known rule to explain an observation, e.g., “if it rains the grass is wet” is a known rule used to explain that the grass is wet. In other words, it would be more technically correct to say, "If the grass is wet, the most probable explanation is that it recently rained."

Writing in 1910, Peirce admits that he himself, "in almost everything I printed before the beginning of this century I more or less mixed up hypothesis and induction" and he traces the confusion of these two types of reasoning to logicians' too "narrow and formalistic a conception of inference, as necessarily having formulated judgments from its premises." Peirce had tended in particular to characterize abduction in terms of induction of characters or traits (weighed, not counted like objects), explicitly so in an influential 1883 work. His old elementary example of abduction had made it seem to cover induction of characters (where from this bag and white are characters):

Rule: All the beans from this bag are white.
Result: These beans are white.
$\therefore$ Case: These beans are from this bag.

Starting around 1900 Peirce distinguished abduction from the induction of characters, and in 1903 offered the following form for abduction:

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.

Peirce did not remain quite convinced that one logical form covers all abduction. In any case he came to hold that abduction's validity is not related to "probability proper"; and that it seeks hypotheses plausible not in the sense of likeliness based on tests but in the sense of simplicity optimal in terms of the "facile and natural" (which he distinguished from "logical simplicity" and for which he cited Galileo). He tied abduction's aim at the simple and natural to abduction's reliance on inborn or developed instinctive attunement to nature and its function to expedite and economize inquiry. Simplicity in the sense of economy of research is such a concern that one sometimes best abduces to a hypothesis that, though simple, seems downright unlikely, but is cheap to test for falsity. Hence, in inquiry, abduction is not merely inference to the most natural and feasible explanation, but inference to the explanatory hypothesis best worth trying.

When induction and abduction are presented as principal, then the two stages of hypothesis development are easily collapsed into one overarching concept – the hypothesis. This is why, in the scientific method pioneered by Galileo and Bacon, the abductive stage of hypothesis formation is conceptualized simply as induction. In the twentieth century this collapse was reinforced by Karl Popper's explication of the Hypothetico-deductive model, where the hypothesis is considered to be just “a guess" (very much in the spirit of Peirce). However, when the formation of a hypothesis is considered the result of a process it becomes clear that this "guess" has already been tried and made more robust in thought as a necessary stage of its acquiring the status of hypothesis. Indeed many abductions are rejected or heavily modified by subsequent abductions before they ever reach this stage.

Peirce came to describe the process of science as a combination of abduction, deduction and induction, stressing that new truths, new abstract propositions concordant with the real, are created only by abduction.

Now, that the matter of no new truth can come from induction or from deduction, we have seen. It can only come from abduction; and abduction is, after all, nothing but guessing. We are therefore bound to hope that, although the possible explanations of our facts may be strictly innumerable, yet our mind will be able, in some finite number of guesses, to guess the sole true explanation of them. That we are bound to assume, independently of any evidence that it is true. Animated by that hope, we are to proceed to the construction of a hypothesis.

The step of adopting a hypothesis that predictively implies seemingly surprising facts is called abduction. Peirce came to regard abduction, deduction, and induction as separately making incomplete sense and as clarified by their cooperative functions in the cycle of inquiry, wherein abduction, constrained to prospective testability, generates explanatory hypotheses that deduction can explicate into practical implications through whose tests induction can evaluate the hypothesis. Induction seeks facts to test a hypothesis; abduction seeks a hypothesis to account for facts. This use differs from the common use of the term “abduction” in the social sciences and in artificial intelligence, where the old meaning (extending a known rule to explain an observation) is used. Peirce stated that reaching a hypothesis may involve placing a surprising observation under either a newly hypothesized rule or a hypothesized combination of a known rule with a peculiar state of facts, and is “very little hampered” by logical rules. He argued that humans guess with more success than if they guessed by sheer luck; and that having this ability is explained by the evolutionary advantage it gives.

Norwood Russell Hanson, a philosopher of science, wanted to grasp a logic explaining how scientific discoveries take place. He used Peirce's notion of abduction for this.

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

Applications

Applications in artificial intelligence include fault diagnosis, belief revision, and automated planning. The most direct application of abduction is that of automatically detecting faults in systems: given a theory relating faults with their effects and a set of observed effects, abduction can be used to derive sets of faults that are likely to be the cause of the problem.

Abduction can also be used to model automated planning. Given a logical theory relating action occurrences with their effects (for example, a formula of the event calculus), the problem of finding a plan for reaching a state can be modeled as the problem of abducting a set of literals implying that the final state is the goal state.

In intelligence analysis, Analysis of Competing Hypotheses and Bayesian networks, probabilistic abductive reasoning is used extensively. Similarly in medical diagnosis and legal reasoning, the same methods are being used, although there have been many examples of errors, especially caused by the base rate fallacy and the prosecutor's fallacy.

Belief revision, the process of adapting beliefs in view of new information, is another field in which abduction has been applied. The main problem of belief revision is that the new information may be inconsistent with the corpus of beliefs, while the result of the incorporation cannot be inconsistent. This process can be done by the use of abduction: once an explanation for the observation has been found, integrating it does not generate inconsistency. This use of abduction is not straightforward, as adding propositional formulae to other propositional formulae can only make inconsistencies worse. Instead, abduction is done at the level of the ordering of preference of the possible worlds. Preference models use fuzzy logic or utility models.

In the philosophy of science, abduction has been the key inference method to support scientific realism, and much of the debate about scientific realism is focused on whether abduction is an acceptable method of inference.

In historical linguistics, abduction during language acquisition is often taken to be an essential part of processes of language change such as reanalysis and analogy.

In anthropology, Alfred Gell in his influential book Art and Agency defined abduction, (after Eco) as “a case of synthetic inference 'where we find some very curious circumstances, which would be explained by the supposition that it was a case of some general rule, and thereupon adopt that supposition”. Gell criticizes existing 'anthropological' studies of art, for being too preoccupied with aesthetic value and not preoccupied enough with the central anthropological concern of uncovering 'social relationships' specifically the social contexts in which artworks are produced, circulated, and received. Abduction is used as the basis of one gets from art to agency in the sense of a theory of how works of art can inspire a sensus communis, or the commonly-held views that a characteristic of a given society because they are shared by everyone in that society. The question Gell asks in the book is, ‘how does initially to ‘speak’ to people?’ He answers by saying that “No reasonable person could suppose that art-like relations between people and things do not involve at least some form of semiosis.” However, he rejects any intimation that semiosis can be thought of as a language because then he would have to admit to some pre-established existence of the sensus communis that he wants to claim only emerges afterward out of art. Abduction is the answer to this conundrum because the tentative nature of the abduction concept (Pierce likened it to guessing) means that not only can it operate outside of any pre-existing framework, but moreover, it can actually intimate the existence of a framework. As Gell reasons in his analysis, the physical existence of the artwork prompts the viewer to perform an abduction that imbues the artwork with intentionality. A statue of a goddess, for example, in some senses actually becomes the goddess in the mind of the beholder; and represents not only the form of the deity but also her intentions (which are adduced from the feeling of her very presence). Therefore through abduction, Gell claims that art can have the kind of agency that plants the seeds that grow into cultural myths. The power of agency is the power to motivate actions and inspire ultimately the shared understanding that characterizes any given society.

See also

• Analogy
• Analysis of Competing Hypotheses
• Charles Sanders Peirce
• Charles Sanders Peirce bibliography
• Deductive reasoning
• Defeasible reasoning
• Doug Walton
• Gregory Bateson
• Inductive reasoning
• Inquiry
• List of thinking-related topics
• Logic
• Subjective logic
• Logical reasoning
• Maximum likelihood
• Scientific method
• Sherlock Holmes
• Sign relation

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: Critical Thinking Across the Disciplines, 15, 40-52. Eprint
• Cialdea Mayer, Marta and Pirri, Fiora (1993) "First order abduction via tableau and sequent calculi" Logic Jnl IGPL 1993 1: 99-117; doi:10.1093/jigpal/1.1.99.Oxford Journals
• Cialdea Mayer, Marta and Pirri, Fiora (1995) "Propositional Abduction in Modal Logic", Logic Jnl IGPL 1995 3: 907-919; doi:10.1093/jigpal/3.6.907 Oxford Journals
• Edwards, Paul (1967, eds.), "The Encyclopedia of Philosophy," Macmillan Publishing Co, Inc. & The Free Press, New York. Collier Macmillan Publishers, London.
• Eiter, T., and Gottlob, G. (1995), "The Complexity of Logic-Based Abduction, Journal of the ACM, 42.1, 3-42.
• Harman, Gilbert (1965). "The Inference to the Best Explanation," The Philosophical Review 74:1, 88-95.
• Josephson, John R., and Josephson, Susan G. (1995, eds.), Abductive Inference: Computation, Philosophy, Technology, Cambridge University Press, Cambridge, UK.
• Lipton, Peter. (2001). Inference to the Best Explanation, London: Routledge. ISBN 0-415-24202-9.
• Menzies, T. (1996), "Applications of Abduction: Knowledge-Level Modeling, International Journal of Human-Computer Studies, 45.3, 305-335.
• 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 Educational Research Association, New Orleans, LA, April, 1994. Website of Dr. Chong Ho (Alex) Yu

External links

• "Abductive Inference in Reasoning and Perception", John R. Josephson, Laboratory for Artificial Intelligence Research, Ohio State University.
• "Deduction, Induction, and Abduction", Chapter 3 in article "Charles Sanders Peirce" by Robert Burch, 2001 and 2006, in the Stanford Encyclopedia of Philosophy.
• "Instructional Technology Connections: Abduction", 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.
• "'You Know My Method': A Juxtaposition of Charles S. Peirce and Sherlock Holmes" (1981), by Thomas Sebeok with Jean Umiker-Sebeok, from The Play of Musement, Thomas Sebeok, Bloomington, IA: Indiana, pp. 17–52.
• Commens Dictionary of Peirce's Terms, Mats Bergman and Sami Paavola, editors, Helsinki U. Peirce's own definitions, often many per term across the decades. There, see "Hypothesis [as a form of reasoning]", "Abduction", "Retroduction", and "Presumption [as a form of reasoning]".