Abduction as an aspect of retroduction
Adding to the confusion is Peirces identification of logic as one of three branches of normative science (aesthetics and ethics are the other two), meaning that the category of logic provides the norms (standards, rules, methods) by which each of the methods of reasoning are to be performed. In addition, Peirce insists that reasoning is a form of deliberate conduct, and is therefore subject to praise and blamemaking logic a form of ethical behavior and thus subject to the rules of right conduct. In addition, Peirce placed ethics into a subordinate relationship to aesthetics (the science of the ideal), saying that ethics must be informed by aesthetics.
Part one of the abduction dilemma involves reconciling the first three senses of abduction:
Part two of the abduction dilemma brings up only one question here, but that question provides the strongest basis for separating the concepts of "abduction" and "retroduction." This second part of the abduction dilemma arises from Peirces statement that methodeutic (which is "nothing but heuretic") "concerns abduction alone."
Because the two parts of the abduction/retroduction dilemma are so intertwined, we will not be dealing with them separately as "part one" and "part two." Instead, we will examine together clues and contradictions which point up these two parts of the dilemma, and which will lead us to the basis for rescuing the terms "abduction" and "retroduction" from their mistaken synonymy and place them into service appropriately.
DEFINITION OF TERMS
Therefore, based upon their Latin derivations (to which Peirce was partial, as he was for Greek roots) our four terms have the following meanings:
ABDUCTION AND RETRODUCTION ARE NOT INTERCHANGEABLE
In 1911, just three years before his death, Peirce wrote:
"I am just now trying to get a small book written in which I positively prove just what the justification of each of the three types of reasoning consists in and showing the real nature of Retroduction.
Since Peirce never completed this "small book," we cannot be entirely certain of what he exactly meant by the third of these "types of reasoning" (the other two are deduction and induction), or by his comment about "the real nature of Retroduction." However, although he never completed the book proposed in 1911, Peirce did publish an essay in 1908 that seems to have heuristically fulfilled that goal. This essay provides strong support for a different set of meanings for the terms "abduction" and "retroduction" (although there he applied the term "retroduction" for both meanings). In "A Neglected Argument for the Reality of God," Peirce describes a process (musement) that neatly corresponds to the process of abduction as "leading away from" a point of interest into "a petite bouchée with the Universes" of experience. There, he provides a detailed description of how the abductive process (as musement) engages within and among the three categories. Then Peirce tells us that:
Every inquiry whatsoever takes its rise in the observation, in one or another of the three Universes, as some surprising phenomenon, some experience which either disappoints an expectation, or breaks in upon some habit of expectation . The whole series of mental performances between the notice of the wonderful phenomenon and the acceptance of the hypothesis, during which the usually docile understanding seems to hold the bit between its teeth and to have us at its mercy, the search for pertinent circumstances and the laying hold of them, the dark laboring, the bursting out of startling conjecture, the remarking of its smooth fitting to the anomaly, as it is turned back and forth like a key in a lock, and the final estimation of its Plausibility, I reckon as composing the First Stage of Inquiry.
The "First Stage" of inquiry that Peirce is referring to above is the stage of hypothesis construction. The "Second Stage" of inquiry is explication and demonstration of the hypothesis (by means of deduction) and the "Third Stage" is comprised of classification, testing, and evaluation. However, in actual practice, the reasoning forms of deduction and induction are not just applied during their specific stages. The chasm Peirce describes above between the noticing of an anomaly and the "acceptance of the hypothesis" cannot be bridged by the making of abductive inferences alone, but rather by the recursive interplay of abduction, deduction, and induction. For, Peirce said that "the whole series of mental performances between the notice of the wonderful phenomenon and the acceptance of the hypothesis compos[es] the First Stage of Inquiry." He did not say that merely the "noticing of an anomaly" and "the getting of a hunch" composes the First Stage of inquiry. For, within the full process of "engendering a hypothesis (which is a retroductive process)," resides the subordinate process of noticing an anomaly and getting an explanatory hunch (by means of abduction). Thus, for the "first stage" of reasoning to occur by means of retroduction, abduction must operate in recursive interplay with the other reasoning forms (deduction and induction) to engender a hypothesis worthy of acceptance and scientific inquiry.
"Retroduction," Peirce then states, "does not afford security. The proposition must be tested." He follows his description of musement with explanations of the processes and purposes of deduction and induction as well. After making this thorough description of the roles and processes of abduction as musement (as well as of deduction and induction), Peirce paradoxically claims that these three inferencing methods interact to engender a hypothesis (in this case, for "the Reality of God"). He specifically asserts that "the N. A. [Neglected Argument] is the First Stage of scientific inquiry, resulting in a hypothesis of the very highest plausibility ." His "Neglected Argument," then, is comprised of the deliberate and recursive use of abduction, deduction, and induction, for engendering " a hypothesis of the very highest plausibility [the hypothesis of the reality of God] whose ultimate test must lie in its value in the self-controlled growth of a mans conduct of life." Peirces Neglected Argument lends credibility to the proposition that one meaning of abduction (the one here referred to as "retroduction") includes the processes of deduction and induction, as well as abduction, for its performance. Peirces claim for the requirement of the deliberate control of abduction, deduction, and induction (as described in "Neglected Argument") for ultimately engendering the "hypothesis of God," corresponds to the deliberate process of pre-and post reflection and analysisthe "backward leading" recursive analysis, which we are here defining as "retroduction."
His description of the process of musement in "Neglected Argument," and the implication that this process is abductive (not retroductive) brings up again the questions of the first part of our abduction dilemma:
Perhaps the first two questions above can be answered together. For a "surprising fact" to be noticed and for a subsequent "guess" as to its cause to be made according to normative principles (without any reference to psychology), there must be a normative category which (though seeming psychological) is dependent upon mathematics, as are all of the normative sciences. Since the category of aesthetics fits that description, we can explore the apparent paradox of norming the apprehension of a "surprise" and the making of "guesses" by addressing the aesthetic process which Peirce called "musing." Peirce borrowed the term "musement" from Friedrich Von Schillers 1794 book Letters Upon The Aesthetic Education of a Young Man. Peirce studied Schiller intently in his early twenties, then put aside this work for many years. Yet, one cannot help wonder at Schillers influence upon the emergence of Peirces concept of abduction. For late in life, Peirce returned to Schillers aesthetic concept of musement as a heuristic device for explaining the way in which new possibilities are discovered and exploredthe same process which he elsewhere refers to variously as abduction and retroduction. Many scholars have addressed this apparent relation of abduction to aesthetics, even in the face of Peirces insistence that abduction is a distinct logical form subject to the strictures of ethical norms.
In "Neglected Argument," Peirce emphasizes that "musement" (the state from which an abductive inference is generated) must be an entirely unfettered processfollowing only the laws of "Pure Play." "Now Play, as we all know," wrote Peirce, "is a lively exercise of ones powers. Pure Play has no rules, except this very law of liberty. It bloweth where it listeth. It has no purpose, unless recreation " Peirce even ventures so far as to insist that pessimists cannot properly perform abductive reasoning, since pessimism closes off entire categories of possibilities and is thus a hindrance to obeying the "law of liberty." 
On the other hand, Peirce insisted that logic is not only a normative science (as are aesthetics and ethics), it is also a "mirror of ethics" (the science of conduct) and, thus, subject to standards by which one should properly conduct rational thought. In his 1905 essay "What Pragmatism Is," Peirce wrote:
Now, thinking is a species of conduct which is largely subject to self-control. In all their features (which there is no room to describe here), logical self-control is a perfect mirror of ethical self-controlunless it be a species under that genus.
Let us take a closer look now at the apparent contradiction between Peirces notion of abduction as deriving from the unfettered activity of Pure Play and his insistence that logic is a species of conduct.
Therein lies the dilemma. Peirce claims abduction as the crown jewel of logic. He insists that logic is a normative science, dependent upon ethicsmeaning that it must follow certain standards of right conduct for its proper performance. Yet, how can abduction be subject to self-control (as are ethics and logic according to Peirce) if its performance must have no rules, "except the very law of liberty?" Can there be self-control without norms (or rules) from which to deliberately control oneselffrom which to make deliberate choices? If abductive inferences result from the entirely unfettered process of "Pure Play," and if "logical self-control is a perfect mirror of ethical self-control," how can abductive inferences be made while logic is "fettered" by the standards (rules or norms) of "ethical self-control?"
By turning to Peirces Classification of the Sciences, we can begin to get a view of Peirces "big picture" concerning all of theoretical science and its sub-branch, the "science of discovery," within which logic dwells. Perhaps from this larger view, we can see how his concepts of abduction/retroduction should fit in. Peirce tells us that each division in his classification of the sciences depends upon the one which precedes it.
The "first" science of discovery, according to Peirce, is mathematicswhich studies "what is and is not logically possible, without making itself responsible for its actual existence." Philosophy (which is the second branch of the science of discovery and, thus, dependent upon mathematics ) is comprised of three branches: phenomenology, normative science, and metaphysics. Phenomenology studies the sorts of "elements universally present" in phenomena (meaning the elements universally present in whatever is present in the mind at anytime in anyway). Peirce writes that:
Normative science "distinguishes what ought to be from what ought not to be . Metaphysics "seeks to give an account of the universe of mind and matter. Normative science rests largely on phenomenology and on mathematics; metaphysics on phenomenology and on normative science."
Peirces division of normative science is comprised of aesthetics, ethics, and logic (in that order), with ethics dependent upon aesthetics for its ideals, and logic dependent upon ethics for its principles of conduct. Logic, in turn, consists of three categories: speculative grammar (which is the general theory of the nature of signs and their meanings), critic (which is usually considered as "formal logic"), and methodeutic (the logic of scientific method). As with each of his classifications, a particular division is always dependent upon all of those which precede it. Peirce further claimed that there is no room for the psychological in critical logic:
[N]o psychological doctrine can be admitted into critical logic. The true doctrine is deduced mathematically from the categories. The justification of abduction follows from it; and from this in turn follow the rules of abduction. 
So, with the above in mind, where does the concept of abduction as musement fit in? Remember, musement is a sort of speculation that arises during Pure Playthe activity that "has no rules, except this very law of liberty. It bloweth where it listeth. It has no purpose, unless recreation ." Can we have a norm for a form of logic which depends upon mathematics and ethics, but "has no rules, except this very law of liberty?" Or, is the abduction of musement of another sort than the abduction of critical logic? In his memoir of methodeutic, Peirce wrote:
I here consider precisely what methodeutic is. I show that it is here permissible to resort to certain methods not admissible in [speculative grammar] or in critic. Primarily, methodeutic is nothing but heuretic and concerns abduction alone 
So, we now are back to the two parts of our abduction/retroduction dilemma, for we have two claims for abduction:
Thus in one sense, Peirces notion of abduction is the aesthetic activity of musement. In another sense, abduction is a type of inference which is mathematically deduced from the categories and cannot be psychological. And, in yet another sense, abduction is a method of reasoning that comprises the entirety of the branch of logic (methodeuic), which is "nothing but heuretic." Peirces use of the term "heuretic" here is a noun form which means the art of discovery or invention ." The more familiar form of this word is the adjective, "heuristic." A heuristic device is a tool (often an analytical tool in the form of a diagram, model, analogy, or metaphor) which helps to show how the qualities and relations of qualities are to be sought for a particular purpose. Thus if methodeutic logic is "nothing but" the art of discovery or invention, then it is likely comprised of the qualities, and patterns of relations among qualities, by which this "art of discovery" should be undergone.
So, let us take up the latter assertion first. By saying that methodeutic logic is nothing but heuretic, Peirce is saying that his methodeutic is nothing but a device for demonstrating how the qualities and relations of qualities of (or by) "abduction alone" are to be sought in "the art of discovery or invention." He also tells us that methodeutic logic may resort "to certain methods not admissible in" critical logic. From this we might infer that "psychological factors" (such as "surprise," "value," and other modes dependent upon sense or affect) might be employable as heuristic devices for explaining abduction, while not actually belonging to abduction, to mathematics, or to any of the sciences of discovery. We might also suspect that the deliberate form of recursive analysis (the interplay of abduction, deduction, and induction), which we are referring to as "retroduction," might provide the qualities and relations of qualities by which the individual processes within it (abduction, deduction, and induction) can be tested. Additionally, we might also reasonably suspect that, when taken as a whole, Peirces methodeutic, which is ("nothing but heuretic and concerns abduction alone") might provide the method by which the recursive analytical process of "retroduction" can itself be tested.
If any or all of the above is so, it would not mean that the logic of methodeutic (or of abduction as musing, guessing, or responding to surprise) cannot be represented mathematically. It would only mean that, in addition to mathematical representations, other means "not admissible in critical logic" are available to this branch of logic for representing qualities and the relations of qualities of the normative sciencesincluding aesthetics, ethics, and logic. For, such psychological attitudes as "musement," "guessing," or "surprise," would be heuristic devices for representing the qualities of abduction. And, in the same way, certain "practical" applications of Peirces methodeutic (or of any of the components of that method) could also be considered as heuristic devices representing the qualities and relations of qualities of retroduction.
Abduction, then, would not actually BE musement, or BE the reaction to a surprising fact and an ensuing hunch. Rather, abduction would be "the qualities and relations among qualities" of abduction which can be ascertained to some degree by extracting these from heuristic representations of these qualities. Such representations could provide forms and models from which to construct and test hypotheses concerning both abduction as musement and retroduction as methodeutic. For methodeutic, as the logic of scientific method, norms the logical methods which the construction and testing of hypotheses should occur. For Peirce, "testing" (not mathematics) provides the sole logical proof of any question concerning Real objects[32,33]." Thus, the "less strict" rules of methodeutic logic allows us to develop a variety of ways to demonstrate and evaluate the development, explication, and testing of hypotheses(including any hypothesis about the method for developing hypotheses). Peirce wrote:
[S]ince the whole business of heuretic, so far as its theory goes, falls under methodeutic, there is no kind of argumentation that methodeutic can pass over without notice. Nor is methodeutic confined to the consideration of arguments. On the contrary, its special subjects have always been understood to be the definition and division of terms. The formation of systems of propositions, although it has been neglected, should also evidently be included in methodeutic. In its method, methodeutic is less strict than critic. 
For methodeutic to "concern abduction alone" and still be the logic of scientific reasoning, the term "abduction" (when Peirce says that "methodeutic concerns abduction alone") should really be termed "retroduction." Since it is the logic of scientific method, methodeutic logic cannot possibly refer only to the noticing of an anomaly and the forming of a hunch. Thus, methodeutic concerned with "abduction alone" really means that it is concerned with "retroduction alone"for Peirces methodeutic is the form of deliberate, recursive analysis which involves the interplay of abduction, deduction, and induction for the development, explication, and (at the least) preliminary testing of hypotheses. As such, "retroduction" as methodeutic is Peirces method of "right reasoning." Right reasoning, according to Peirce, can only occur by individuals adhering to the norms of logic that is informed by ethics and guided by the "science of the ideal" when engaged in scientific inquiry.
Now, let us examine Peirces suggestion that abduction is the aesthetic activity of musing as this contrasts with to his assertions that:
That, for Peirce, the normative sciences of aesthetics, ethics, and logic are interdependent (as are each of the three types of logical reasoning) is an important concept to keep in mind when considering the "notion of abduction." Such interdependence is an especially important concept when reconciling abduction as musement with the recursive analytical method of retroduction. For, as you will see, the key to the "normative" nature of Peirces notion of abduction resides in the category of aesthetics.
Although normative, Peirce placed aesthetics in the position of firstness, overarching (informing) both ethics and logic. Thus, the category of aesthetics as a "normative" science is (like abduction as musing) not subject to ethical norms, but rather to the norming of normsthe exploration and discovery of that which is the "ideal" (and which should therefore, as the ideal, inform both ethics and logic). For Peirce, the "the morally [ethically] good appears as a particular species of the esthetically good ," and logic is a "species under the genus" of ethical conduct. He wrote:
I regard Logic as the Ethics of the Intellectthat is, in the sense in which Ethics is the science of the method of bringing Self-control to bear to gain our Satisfactions . As to what one ought to desire, it is what he will desire if he sufficiently considers it, and that will be able to make his life beautiful, admirable. Now the science of the Admirable is true Esthetics. Thus, the Freedom of the Will, such as it is, is a one-sided affair, it is freedom to become Beautiful . There is no Freedom to be or do anything else. Nor is there any freedom to do right if one has neglected the proper discipline .
For Peirce, aesthetic denotes a qualitative statewithin which one has the "freedom of the will" (as a one-sided affair) only to strive for the ideal (that which will make ones life beautiful and admirable). The aesthetic is a state of potentiality from which we respond to and select among qualities and values, based upon the relations among the qualities themselves for achieving the aesthetic ideal--rather than upon their value in the production of outcomes. Thus the aesthetic provides the "norm" (as the "admirable" ideal) for other norms and, as such, is the only "norm" that can be obeyed at the same time as obeying the "very law of liberty" (or "freedom of the will"). From its position of "firstness," the category of aesthetics (science of the ideal) provides the normative force (as feeling, energy, value, purpose, being) from which the very "law of liberty" should be obeyedsuch is its "norm." And, since abduction is the method by which qualities as potentialities are noticed, related, and formed into meaningful relationships, abduction must at least begin with the aesthetic normfor it is the method by which the aesthetic ideal is expressed. In other words, abductive reasoning (as a properly performed form of logic) must at some point have an intimate relationship with the aesthetic. 
Of course, even if we accept this aesthetic-abduction connection, the second aspect to the abduction/retroduction dilemma, which we discussed earlier, arises from Peirces statement that methodeutic (which is "nothing but heuretic") "concerns abduction alone." The category of methodeutic is supposed to provide the norms for scientific investigationnorms which must include the norms for other forms of inference (deduction and induction) as well as for abduction. Indeed, after laying out his "Neglected Argument for the Reality of God" (an argument which includes explanations of the roles of deduction and induction as well as "musement" as abduction), Peirce declares that this argument comprises the "First Stage of scientific inquiry." Thus (according to Neglected Argument at least), it is not abduction as musement, but rather the recursive interplay among abduction, deduction, and induction that comprises the "first stage of a scientific inquiry," which is, of course, the development of a hypothesis. Therefore, all three inferencing methods interact during the engendering of a hypothesisnot abduction alone. So, when Peirce tells us that the normative logic of methodeutic is "concerned with abduction alone," it is likely that the concept he means is the recursive analytical process that we have been referring to as "retroduction." And when he tells us that abduction is a logical method for responding to a "surprising fact," for developing a "hunch," and for "musing" he is referring to the aesthetic norm for seeking the beautiful and the admirable (which, for Peirce, was the ultimate aim and end of Truth).
WHY THE TWO CONCEPTS SHOULD BE SEPARATED
Also, by taking advantage of the broader range of subjects and tools available to the unique analytical method of Peirces methodeutic it is possible to begin building a clearer understanding of what abduction as a mathematical concept (and as an aesthetic method, also dependent upon mathematical description) might be. By distinguishing between these two concepts within Peirces methodeutic (defining one as abduction and the other as retroduction), it becomes possible to observe and identify patterns of actions during the actual performance of each of the distinct inferencing processes (abduction, deduction, and induction). These patterns become especially clear as each one plays out in actuallyas in the performance of an open-ended task. Because the looser "rules" of the methodeutic branch of logic allow for a variety of ways to demonstrate these specific and overarching reasoning methods, it is possible to use methodeutic logic to construct models for testing the patterns of action that make up each of the three types of inferences. 
Another good reason for separating these two concepts (abduction from retroduction) is that retroduction cannot be considered without also considering the relational patterns of (and among) all three types as inferencesas well as the relational aspects of the context for which it is being applied. The broader concept of "retroduction" as methodeutic renders it much more difficult to define in terms of its qualities and their relations, since there are so many more components to consider. However, until an operational definition of abduction (as the pattern of actions distinct from deduction and induction) is available, we cannot hope to develop an operational definition for retroduction. The pattern of actions of abductive reasoning must be known and well-tested before we can effectively identify the qualities and relations among qualities of the overarching pattern of retroductive reasoning. In other words, we must know how to "abduce" before we can possibly hope to learn how to "retroduce."
HOW IT WOULD WORK
Then, the reasoner returns to abduction and repeats that cycle as necessary. In this sense, the term "retroduction" would be reserved as a definition for the entire abductivedeductiveinductive cycle of Peirces methodeutic, saving the term "abduction" to mean a distinct type of inference that is separate and distinct from either deduction or induction.
If we were to define our terms in this
way, the term "retroduction" would denote the following recursive cycle
that goes into the building up of knowledge:
From the above perspective, then, abduction (as opposed to retroduction) is an aesthetically dependent method of logic for addressing surprising (or anomalous) facts and exploring the qualities of these. Once a surprising fact is formed into a hunch, deduction and induction interact recursively with abduction to engender a hypothesis. Once a hypothesis is formed, deduction is the method by which that idea is explicated and readied for testing. Induction is the reasoning method by which the idea is tested, evaluated, and eventually secured. So, within the activity of hypothesis construction, both deduction and induction are needed at times. Thus, the word "retroduction" can stand as an inclusive term (overarching the three inference methods of abduction, deduction, and induction) for Peirces methodeutic, defining roles of all three of the processes recursively applied for the discovery and construction of worthy hypotheses (conditional purposes).
Without an operational definition of abductive reasoning, we will never be able to develop an operational definition of retroduction (since abduction is necessary for performing a retroduction, as are deduction and induction). Constructing an operational definition of retroduction will be much more complex than constructing one for abduction. However, without a clear understanding of the abductive reasoning process of abduction and all that it entails, we will not be able to attempt an adequate operational definition of retroduction.
The greatest potential benefits from having such definitions lie in the field of education. Once we remove abduction and retroduction from the realm of vagueness and can clearly delineate these processes and their applications for all to see, we can begin to develop effective ways of teaching others how to effectively apply these reasoning processes to all sorts of situations. The greatest yet unrealized benefit of Peirces work to humankind lies in its potential for improving the ability of those at all levels of intellect and in all walks of life to reason more effectively. Before Peirces methods of right reasoning can be taught, however, abductive reasoning must have an effective operational definitionone that can be demonstrated and generally understood, so that all of the reasoning methods can be effectively taught and, more importantly, mastered.
Therefore, the proposal that we should use the term "abduction" for the reasoning method by which conditional purposes (hypotheses) are constructed and "retroduction" as the overarching method by which theories are engendered (by the interplay of abduction, deduction, and induction) should not be taken lightly. By clearly defining the process of abduction and placing it (along with deduction and induction) under the overarching method of "retroductive" reasoning, we are creating the possibility that all of these reasoning methods can be taught, learned, and, hopefully, mastered. Then, once these processes are mastered, individuals can learn to effectively perform and apply each of these methods appropriately for the construction, explication, and evaluation of all sorts of conditional purposes (including hypotheses)thus mastering the skills of Peirces concept of "right reasoning."