Logic of discovery, abduction and types of explanatory models

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Introduction

It has long been recognized that logic of discovery is impossible. Logic of discovery is used “in the sense that it might reveal the logical mechanisms of the invention of hypotheses, that is to say provide a method for finding hypotheses.” It is now widely accepted that “No one, in fact, seems ever to have seriously believed in such a utopia”.

This work is dedicated to the first attempt to collect the image of appearing before our eyes new utopia. That is why the structure of the work is enough simple. Firstly, achievements in the field of logic of discovery in its technical sense are presented. Secondly, the prominent theory of creative type of reasoning is introduced in its dynamic from the inference of the truth of the minor premise of a Barbara syllogism to the field of diversified abduction logics. Thirdly, classifications of explanations which are most prevalent in psychology and philosophy of science are adduced in order to compare them with real source of creative hypotheses.

The aim of this paper is to find some kinds of reasoning schemes, types of explanatory models, which will help to create new ideas as a first step to realization of utopia. The hope is based on quasi-structuralist or quasi-transcendentalist (but actually Peirce's one) idea that there is some correspondence of human mind to laws of nature. The finite list of themata as source of explanatory hypotheses of highest levels and presence of three elementary types of reasoning (deduction, induction and abduction) may be a key to hypothesis that there is a kind of abductive-explanatory inner grammar which promotes hypotheses. “What would the consequences for psychologism be (in view, especially, of Chomsky's claim that certain grammatical structures are innate) of the conjecture that logical form can be identified with grammatical form?”

The hypothesis of the study is that creative types of explanatory models can be inner structures of human mind capable of producing hypotheses “ex nihilo” as abduction, or guessing instinct, is capable of creating explanatory hypotheses.

So, the tasks of this work are

1) to find out how logic of discovery is realized now,

2) to unleash the creative potential of abduction,

3) to construct some example of synthetic explanatory schemes.

The relevance of this utopian quest may be not so fantastic actually. At its best, the goal of creation of generator of explanations in computational theory of discoveries may be realized to help in proposing relevant ideas in crisis situations. But more realistically may look retrospective analysis of the dynamic of disciplines from historical-philosophical position. Hegelian it may sound, but big data of millions of little and more serious discoveries may show that there is an internal logic of development of sciences from one kind of explanation types to other. In that case predictions may be possible.

1. Logic of discovery: fruitful dialogue of disciplines

logic abduction deduction probabilistic

Logic of discovery is possible in the new framework of so-called “balanced philosophy of science”, which includes methods of history of science and means of formal reconstruction of discoveries by the tools of logics.

There are a plenty of approaches to the formalization of logic of discovery. Since debates of the first half of the 20th century in the field of philosophy of science inspired by famous demarcation of context of discovery and context of justification elaborated by Vienna Circle, Hans Reichenbach, Karl Popper critical number of evidences of possibility of logic of discovery was attained in different disciplines. New fields of study appeared guided by ideas of devising new programs of research, reasoning as problem-solving (Langley et al.), computational philosophy of science (Thagard). The crucial factor of success was freedom from the bias of psychologism in logics concerning dangers of relativistic influence on scientific results of “individual psychology of thinking and the sociology of thought”. New scientific work dedicated to the problems of logic of discovery was focused on rational aspects of discovery realized by both strict algorithms and flexible strategies embodied in heuristics. It did not neglect the role of insight and intuition in scientific creativity, but gave more deserved attention to strategic rules governing scientific inquiry, investigating new modes of reasoning, patterns in scientific research. Logic in these cases is understood broadly: research includes working with inductive inferences, deductive logic, but often authors write that the term “logic of discovery” is used as opposition to guess/irrational and unpredicted aspects of inquiry such as chance-factor/psychology of creativity [in scientific discovery]. The real accent is on heuristics, data-driven strategies of optimization of search with probability higher than probability of lucky guesses. Thus, logic is understood as following some rational criteria, scientific procedures of gaining knowledge and rules.

This chapter will be dedicated to the productive dialogue of philosophy (especially epistemology and philosophy of science), psychology (cognitive and gestalt) and Artificial Intelligence (AI) on issues of logic of discovery. It should be mentioned that despite the fact that sometimes appeal to problem field of logic of discovery happened simultaneously, the development of research was parallel. The benefit from mutual influence was realized recently, but there are good results of mutual rectification.

The starting point will be AI, because its approach to logic of discovery is the most counter-intuitive and controversial. Proponents of AI methods to logic of discovery defend thesis that scientific creativity as a type of general human capacity to solve problems may be formalized and simulated by programs. However unfamiliar this thesis may sound, significant achievements have been made in the field of artificial intelligence. So-called Artificial Intelligence hypothesis says that providing implementation of written computer programs that will be able to discover laws or theories, it will be evidence that logic of discovery is possible, that scientific discovery is a rational process. Now these programs really exist. In artificial intelligence researchers program hypothesis, concepts and theories formation, invent algorithms producing explanations, programs that “rediscover” quantitative laws (e. g. Kepler's laws of planetary motion (program BACON)), programs that can learn new heuristics in the working process (EVRISCO), programs that simulate discovering qualitative taxonomies with clustering methods (GLAUBER for laws in chemistry).

It should be noticed that some authors claim that the status of programs such as BACON is lower than was stated. This criticism is concerned novelty aspect: the programs described by the collective of authors with Simon and Langley did not discover new law and furthermore, no new theory was discovered. Generally this kind of programs includes techniques to discover correlations and regularities. The problem is simplified by the fact that there are already a lot of strictly developed inductive methods (Bacon's tables of discovery, Mill's canons of induction etc.). Now it is hard to conceive that programs of artificial intelligence will be able to make discovery which are considered as Discoveries - discovery of new entity/force/capacity, “representations of unobservable levels of reality”, structural descriptions of reality, discoveries unifying theories or making effective transfer of concepts from one field to another, great explanations. Incredulity towards programs making generalizations from data are connected with methodological positions that (a) there are a lot of possible laws for any data set; (b) there are no universally accepted criteria for choice from possible laws the most beautiful, the most simple etc.; (c) generalized laws should be coherent with data and all our understanding of the real world, should explain data; (d) real representation of discovery should contain all the stages from facts to hypotheses, including abductions, analogies, assumptions, models etc. So, the main skepticism towards AI approach is its preference of induction methods and status of discoveries. The feedback of AI specialists was realized in cooperation with psychologists and logicians, so the problem of diversity of methods is somehow solved. Interestingly, this problem is explicitly stated and realized, so it is in the active stage of solution. As for second point of criticism, it is just a prospect of future dialogue of philosophers and AI technicians. But there will be some ways to address new and emerging challenges in this work.

Notwithstanding there are undeniable evidence for AI success in logic of discovery. They are connected with practical results, discovering some target property, making low level generalizations in cases where there are no solutions for a moment. Some programs are already better than humans in diagnosing (for example, ASSISTANT), some make cheap calculations instead of expensive inquiries in laboratories (GOLEM predicts the secondary structure of proteins from their primary structure).

As for psychological contribution to the area of logic of discovery, uncontemplated argument against context division can be mentioned.

Survey in the field of cognitive psychology has shown that logic of discovery can be not just data-driven as in the paradigmatic applications of artificial intelligence, but hypothesis-driven. And hypotheses with explanatory role can be invented in turn thanks to tools. These are tools of justification, which can be analytical (rules of statistics, the normative rule of logical consistency) or physical (such as instrument of measurement, for example clocks, or machine like steam engine), and which suggest new concepts, analogies, theories. Of course tools belong to the context of justification and such a case reflects the artificial character of division between two contexts: there is no true conceptual division and there is no temporal, because items of context of justification can inspire discoveries. So what are the examples of these interactions of two contexts?

First example, the case of A. Quetelet who used as a tools-to-theories heuristic the statistical error law (normal distribution) from astronomy with its aim to find the real position of a star handling observational error to sociology. The analogy is reflected in his understanding of social physics: the concept of the "average man" (l'homme moyen) for the mean values of measured variables that follow a normal distribution and that represent in analogy the true position of a star. The distribution of actual persons stated in analogy for observational errors. Interestingly, L. Boltzmann and J.C. Maxwell were encouraged by social theory of Quetelet and created a model describing the “behaviour” of gas molecules as if it was the behaviour of individuals: easily predictable in the scope of collective and unpredictable at the level of individuals. In some interpretations this transfer of tools is seen as a change of worldview: from deterministic Newtonian world to the world of the Statistics.

Another example has meta-level of awareness, because it relates to the cognitive psychology itself; it is a discovery of cognitive theory. The acceptance of new explanatory model was closely related to the experimental psychological practice in the USA universities in 1940s-1950, time of institualization of inferential statistics in American experimental psychology. The new paradigm was model of representation human cognition as intuitive statistics, representation of mind as a statistician calculating an analysis of variance (ANOVA) to process data and justify hypotheses in conclusion. Community accepted new theory of mind as inductive machine fast in 1960s and started to fruitfully use it. So far, the tool had legitimacy of mathematics and legitimacy of practice in experimental psychology. The scientists were already familiar with the method by that moment, so model adoption rate was really high. Successful mathematical instrument was a great argument for becoming a proponent of new paradigm of a discipline. Moreover, the new theory inspired new type of research due to the fact that it gave for development of psychology new type of facts. New facts appeared in theory of signal detectability (TSD) in the frame of theory of perception. It became clear that the mind "decides" in each perceptual act what is a stimulus and what is noise. So, the new type of data which new theory brings is division of two kinds of errors of mind: false alarms and misses. This type of facts was unreachable within the confines of previous theory of mind. The previous theory conceived mind as a fixed threshold with the twin notions of observer's attitudes and observer's sensitivity. Nevertheless, the honest analysis of research applying new theory has shown inconsistency of theory with existing data. More questions arise provided that historian of science will step back to the origin of the theory. Actually, there was no full consent that the model of scientific search replicated the scheme of process in mind. Undoubted evidences from the history of Science confirmed position of rejecting quantifier of universality in the thesis that “As far as mind is a statistician in his nature, all the research is done inductively”. In astronomy of early nineteenth century scientists would have not behaved as a statistician in mind, they had not revised the data, conversely, they trusted to the theory. But the ANOVA mind trusts data and does not trust theory as a technique for inductive inferences from data to hypotheses. Anyway, there are a lot of implementations of this theory in psychology: speech perception, adaptive memory, pattern recognition. Consequently, there is yet another warning of inappropriate accent on induction in prejudice of other types on inference.

There are examples of productive synthesis and exchange of ideas of psychology, AI and logic. This happens in problem-solving approach. Its psychological direction is very close to human intuition to search the roots of scientific creativity in special organization of human cognition. Nonetheless, the beginning of this approach pertains to Gestalt psychology and mathematics almost simultaneously. The book of mathematician George Polya's “How to Solve It” appeared in the same year as the books of gestalt psychologists Duncker (“On problem-solving”) and Wertheimer (“Productive Thinking” (1945)). Gestalt psychologists came up with model of reproductive and productive thinking by extension of the organizational principles of visual perception to the domain of problem solving. Such division of two modes of thinking was an explanatory hypothesis for interpretation of results in solution of problems that require analysis and insight. The structure of problem situation includes the initial state, the goal state (decision), and the set of possible operators that can help to get from the initial state to the goal state. There was a view that reproductive thinking cannot solve insight-tasks because it just uses resources of memory and habits of thought without genuine understanding. Productive thinking in its turn is capable of restructuring of the problem, constructing a new solution from the scratch, new representation of the problem with the cost of rejecting of previous intellectual schemes. Neo-Gestalt theory shows that reproductive (analytic) thinking can succeed in solution of problems demanding for insight-solutions. So, the schemes of reproductive and productive thinking are presented below.

Table 1. Approaches to problem-solving in Gestalt psychology

As is clear from the table psychologists distinguish special strategies for economy of cognitive resources (heuristics) that people use solving problems. In addition, they differentiate two directions of search in antagonistic pairs typical to different types of thinking: from parts to whole and vice versa, conceptual and data-driven, subsumption to solution and experiments with changing the goal. Psychological investigations of Gestalt psychologists were in decline during World War II, but the problem-solving approach blossomed again in 1970s after the book “Human Problem Solving” (1972) written by Newell and Simon. They transformed thinking-aloud protocols of participants of problem-solving experiments into search-trees in problem space basing their work on information-processing approach. Scientists created the General Problem Solver that used algorithms and heuristics, such as means-ends analysis. The decisive element of Newell and Simon's approach is the neglect of background knowledge. Historical development shows that psychologists focus on representation in problem-solving and scholars from AI field on search strategies. The term of crucial significance in the following research “problem isomorph” was constructed in the framework of AI workings. It designates problems with the same structure or same problem space which stimulate analogous inference. One of the results of this approach is signifying principal otherness of expert appraisal and solutions from the solutions of novices'. First of all, the answer is pattern extracting - meaningful groups (chunks) of information including stored solution plans and/or compiled procedures. Secondly, problem representation in accordance with basic principles of the domain and structural features of entities is the key to solution. As a result, the significance of problem-solving approach for logic of discovery is colossal. It is the main approach to the human scientific activity because it consists of elaborated structures for description of stages of inquiry, it proposes algorithms and heuristics. It has required level of abstractness allowing making conclusions about problems from very different problem spheres (“problem isomorphs”). Problem-solving approach is open to breakthroughs from formal disciplines as well as natural sciences. It provides means to construct nested structures reflexing sequence of information processes, for example pattern recognition based on experts' perceptual heuristics, formal handling and structuring guided by search strategies, terminating in visualizations.

The inspiring example of problem-solving approach is not unique instance of synthesizing projects. There are examples when cognitive psychology and AI are involved in collaboration by means of philosophy. The approach that is in the middle of artificial intelligence and cognitive psychology was promoted by J. Holland, K. Holyoak, R. Nisbett and P. Thagard. In their book problem-solving program “PI” (“processes of induction”) is developed to show the relevance of computational insights to the problems of context of discovery and context of justification. Thagard is famous for his project of computational philosophy of science, synthesizing approaches of artificial intelligence (AI), logic, philosophy of science and cognitive psychology. Thagard proceeds from the position of weak psychologism which postulates prescriptive character of logic for mental processes. It is substantial methodological item, so let us say a few words to clarify the position. Weak psychologism in epistemology includes respectful attitude towards data of empirical psychology, but claims that logic has normative status. Weak psychologism states that logic is applicable to reasoning and has prescriptive character in relation to reasoning. Nevertheless, prescriptiveness is understood in its limited sense that inference should be realized in compliance with rules of logic. Weak psychologism (of Peirce, for example) should be clearly distinguished from strong psychologism with its thesis of descriptive character of logic corresponding reasoning (Kantian position) and anti-psychologism rejecting role of logic in description or control of mental processes (position of Frege). From the point of view of weak psychologism Thagard posits some prerequisites of his approach, such as his thesis that computers are capable of a strong simulation of mental processes thanks to computational theory of mind. Computational theory of mind considers mental processes as computational, makes analogy between mental structures and algorithms, thought and data, programs and mental processes. AI is defined by Thagard as the branch of computer science regarding computer performance of intelligent tasks. According to Thagard the aim of AI in computational philosophy of science is to improve human practice by constituting vocabulary of structures and mechanisms, evaluating coherence of theories, deducing results from hypotheses and predicting results of inductive tests of consequences of hypotheses. Computational philosophy of science shed the light on data structure of scientific rules, concepts, and problem solutions which constitute scientific explanations. Comparing to the different attitudes to the problem of influence of psychology on epistemology, Thagard indicates the resembling debates in the field of AI. He defends the position of Newell and Simon, combing methods of history of science, logic and AI in their problem-solving approach distancing from so-called “neats” camp (McCarthy, Nilsson) of constructors of purely deductive systems and the camp of “scruffies” (Minsky, Schank), proponents of psychological approach. Thus, epistemology and AI acknowledge the importance of dialogue with psychology. Philosophers of science in their turn admit the possibility of huge capacities of computational methods for their disciplines not just from the positivistic point of view that only context of justification can be formalized.

Last but not least example of productive mutual influence is the newest, connected with the progress of technologies. Since first achievements in the domain of programming scientific discoveries, serious changes in approach were realized. Common area of research considering discovery process in a metaphor of form of computation was divided in two paradigms. First initiative, paradigm of data mining is based on great facilities of data sets with open access, but has problem of understandability of results for scientific community, because of crucial difference between data mining notation (decision trees, logical rules, and Bayesian networks) and science and engineering formalisms. Explorers who stick to this direction prefer data-driven search. Second direction of research, computational scientific discovery, is fixed on the goal of making discovery processes (finding laws and theories in scientific observations and rediscoveries from history of science) realized by programs more comprehensible to researchers from different fields. This approach is oriented on interdisciplinary dialogue taking into account of three revolutions in information processing, namely the scientific revolution, the search revolution and the data revolution. Among contributions of scientific revolution (since 16-18th centuries) such ideas can be named as system of evaluation laws and theories through observation, communicable practices of usage of discipline formalisms, notations in each domain of scientific research. The search revolution (since 1950s) bonded with development of computers as instrument for symbols handling in processes involving search through a space of symbolic structures prepared algorithmically based programs of discovery. List-processing language approbated on automate search for logical proofs helped to produce first heuristically-guided search programs. The data revolution (since mid-1990s) provides an access to data storing technologies for all people. Electronic data sets became a new source of investigation to support knowledge discovery. More emphasis on classical procedures in scientific practices allow computer scientists to develop programs realizing computational scientific discoveries in the sense of effective revision allowing to get rid of the costs associated with working from scratch, enhance work with relations of variables, initial conditions, temporal dynamic of research. Also, successful formalisms of results representation facilitated the task of formation of explanations. New programs provide complicated explanations including processes and theoretical variables, not only compiling descriptive summaries. The issue of lopsided development of quantitative models in detriment to qualitative models is also solved by these means. Thus, gradual complication of understanding of the processes engaged in scientific discoveries encouraged modern advances.

Although hypotheses of computational mind and metaphor of mind as an inductive machine proved their usefulness and there is an urgent need for further logic assistance in computer science development for stimulating knowledge discovery processes, there are some problems left unsolved of special philosophical interest.

Theory of computational scientific discovery lacks of general theory of explanation. Cognitive psychology requires cooperative project investigating productive thinking, generation of hypotheses. AI demands for clear specification what a Discovery is.

The results of studies in the history of cognitive psychology show that discovery no less than justification can be associated with logics, mathematics and statistics. Theories and data that are tool-laden are empowered by context of justification and have advantage of explanation of related phenomena. The theory of mind as inductive machine being a paradigm of thinking reflects in understanding science as predominantly inductive inquiry. Better part of human progress in science is conceived as evidence of success of Induction. From the broader perspective of logic of discovery as a trial to predict some theoretic changes and shifts in principal paradigm explanations this situation looks as a great example of prejudices of the era. Gigerenzer in his brilliant reference to the logic of discovery of astronomers of early nineteenth century who trusted more to theories and hypotheses than facts indicated shift in consciousness of researchers. Maybe modern era with its hope for induction is in the same relation to future as deductively oriented projects of logic of discovery of Ramon Llull (Ars Magna) and Leibniz (calculus rationator) towards contemporaneity? The future project of logic of discovery may be based on hypothesis. There are some preconditions for this from the history of philosophy. For example, Peirce thought that only hypotheses starts from facts and induction is only capable of verification of consequences of hypothesis which are produced by deduction. This position will be discussed in more detail later.

2. Insights from Abduction Studies

One of the most perspective approaches to creation of logic of discovery previously not mentioned is concentrated on the development of ideas of American scientist, logician and philosopher Charles Sanders Peirce (1839-1914), especially his project of abduction as rational process of seeking explanation for puzzling phenomena, and also theorematic reasoning and the whole project of unifying logic and science.

Here it is important to make a digression and highlight the position of many researches that the relevance of Peirce's heritage, and in a more general way, his idea of revision of meaning of `logic' is very relevant nowadays. “The present situation resembles the time of pre-Fregean logic, when the boundaries between logic and general methodology were still rather ?uid, as witnessed by the work of Bolzano in his several conceptions of consequence relations, and later by Charles Peirce's investigations into styles of reasoning (deduction, induction, abduction) from a logical perspective. In our view, current post-Fregean logical research is slowly moving back towards this same broader agenda.” The analysis in previous chapter reveals positive effects of AI, philosophy and psychology convergence. The same may be true on a smaller scale: for logic, philosophy of science and methodology convergence.

As for the title of this chapter, there is no established term “abduction studies”, but there is need for it, because of huge proliferation of works on abduction.

Understanding of abduction differs from researcher to researcher. For example, epistemologists often use the term abduction as synonym of IBE (inference to the best explanation), relying on the following textual evidence: “The first starting of a hypothesis and the entertaining of it, whether as a simple interrogation or with any degree of confidence, is an inferential step which I propose to call abduction. This will include a preference for any one hypothesis over others which would equally explain the facts, so long as this preference is not based upon any previous knowledge bearing upon the truth of the hypotheses, nor on any testing of any of the hypotheses, after having admitted them on probation. I call all such inference by the peculiar name, abduction, because its legitimacy depends upon altogether different principles from those of other kinds of inference.” (CP 6.525). Gabbay and Woods write that abductive hypothesis provide a basis for consideration of following actions involving degrees of risk accompanying the strength of H's assumption. Unfortunately, researchers outlined that abductive success is not necessarily connected with originary thinking. Researchers from the field of dynamic epistemic logic tend to analyze abductions not in scientific context but as information processing in multiagent scenarios of belief revision. Abductions are interpreted as processes that caused by epistemic actions that make agent knowing/believing something new, processes of search for explanation and incorporating this new epistemic fact in own system of belief. Within the scope of dialogical logic abduction is studied as “not-conceded-preserving dialogues triggered by a concession problem”. Specialists from the field of Artificial Intelligence sometimes understand abduction as “a form of logical inference which seeks to uncover all possible causes of an observation”, “the process by which we try to answer the question `Why?'.”

For the reasons that the aim of this work is research on creative, not selective potential of abduction and volume limits which preclude correct formalizations, some lines of investigation in abduction area will not be considered here.

Abduction: “the only kind of argument which starts a new idea”

Anyway, there are grounds for s described earlier proliferation in interpretations of abduction. Peirce elaborated his theory of abduction lifelong (from 1865 till 1914). He says that abduction represents “any reasoning of a large class of which the provisional adoption of an explanatory hypothesis is the type.”(CP 4.541)

A brief assessment of general historical tendency of work of Peirce (that is of course very tentative reconstruction) shows gradual shift from understanding of abduction as inference to understanding of abduction as a cluster of procedures, preliminary stage of scientific inquiry, explanatory hypotheses, elaboration and choice of the best explanatory theory making knowledge clear and distinct. Hereafter the focus of the summary will be on creative upside of abduction, the heuristics it provides, strategies enhancing discoveries.

Peirce often writes that the first philosopher who discovered abduction was Aristotle. The Greek philosopher distinguished three types of inference: synagцgй or anagцgй (Deduction), epagцgй (Induction), apagцgй (Abduction) and the mixture of two last, induction and abduction - paradeigma, in strictly Aristotelian, not Platonic, sense (Analogy). Peirce writes that deduction, induction and abduction just three elementary kinds of reasoning and that there are mixed types which are more common (CP 2.774).

Aristotelian example of usage of abduction was connected with ethical and mathematical judgements.

Peirce suggests conversed interpretation of Aristotle (giving translation “comprehension” for epistйmй):

"Let {A} be capable of being taught, {didakton}; {B}, science or comprehension, {epistйmй}; {G}, righteousness, {dikaiosynй}. Now that comprehension is capable of being taught is plain; but that virtue is comprehension is not known. If, however, this is as antecedently likely or more so, than that virtue should be capable of being taught [which, it seems needless to say, everybody knows to be the fact], then there is ground for the abduction; since we are brought by the hypothesis, '{to proeilйphenai}' nearer to a comprehension of virtue being capable of being taught, than we were before.” (7.250)

Peirce formalizes this inference as:

Comprehension can be taught,

Virtue is comprehension;

.·. Virtue can be taught.

(CP 7.251)

Just Aristotelian direction of thought developed by Peirce is associated with understanding of abduction (or retroduction (CP 1.68)/ hypothesis (CP 6.525)/ presumption (CP 2.776)) as a type of inference. Peirce defines inference as "the conscious and controlled adoption of a belief as a consequence of other knowledge" (CP 2.442, 2.144, 5.109) consisting of in the thought that the inferred conclusion is true because in any analogous case an analogous conclusion would be true" (CP 5.130), directed "to find out, from the consideration of matters and things already known, something else that we had not known before" (MS 628:4). Representation of abduction as the inference of the truth of the minor premiss of a syllogism where the major premiss is selected as known already to be true while the conclusion is found to be true. (CP 1.65, 7.250, 8.209). Sometimes Peirce writes concisely: “reasoning from consequent to antecedent” (CP 6.469).

The more known example of abductive inference is presented by Peirce:

DEDUCTION.

Rule.--All the beans from this bag are white.

Case.--These beans are from this bag. .

..Result.--These beans are white.

INDUCTION.

Case.--These beans are from this bag.

Result.--These beans are white.

.·.Rule.--All the beans from this bag are white

HYPOTHESIS.

Rule.--All the beans from this bag are white.

Result.--These beans are white.

.·.Case.--These beans are from this bag.

(CP 2.623)

Peirce advances the interpretation of Aristotle by adding to it the element of puzzlement and amazement, confrontation with something that is in clear opposition to the expectations (CP 2.776). Observer notices something curious and tries to adopt it to some pre-established knowledge.

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.

(CP 5.189)

But anyway, it is presented in the form of syllogism: “possible Explanation …a syllogism exhibiting the surprising fact as necessarily consequent upon the circumstances of its occurrence together with the truth of the credible conjecture, as premisses”. (CP 6.469)

Peirce explains the sense in which he understands the meaning of surprise. Surprise regards belief that can be either formulated or not, active or passive, but disbanded by marvel in pure theory as in mathematics or in nature. (EP 2:287)

Turning to the issue of change in the concept of abduction as it was elaborated by Aristotle, Peirce calls inference, where agents can abduct resemblance in some aspect from the strong resemblance of two objects, hypothetical inference. (CP 2.624) Abduction in this sense differs a bit from analogy, because analogy is inference from the resemblance of many features to resemblance of one feature (CP 1.69).

The more Peirce distances from the Aristotelian definition, the more understanding of abduction expands to the understanding of it as a hypothesis. Peirce also clarifies his usage of the term: “the conclusion of an argument from consequence and consequent to antecedent” (W 2:219). Peirce defines mathematical hypothesis as “an ideal state of things concerning which a question is asked” (NEM 2:10)

In addition to more or less Aristotelian theory of inferences, Peirce elaborates theory of probable inferences dividing probable reasoning into two groups: deductive and ampliative. In turn, ampliative divides in two groups: inductive and hypothetical, inferring from parts to whole and analogy, inferring from particulars to particulars. Peirce explains the sense in which deduction can be understood as probable reasoning: predicted ratio in deduction is approximately verified in a large scale, but can be wrong in a small number of sortitions. In case of ampliative reasoning ratio may be wrong due to small samples, but as soon as sample becomes larger, the ratio will be corrected. In the framework of this theory Peirce calls hypothesis a kind of induction in its extended sense. Peirce implies induction concerning characters or qualities that can be just weighed, not counted as things in induction (in its narrow sense):

FORM V (ii) Hypothesis.

M has, for example, the numerous marks P', P", P'", etc.

S has the proportion r of the marks P', P", P'", etc.:

Hence, probably and approximately, S has an r-likeness to M.

FORM V (i) Induction.

S', S", S'", etc., form a numerous set taken at random from among the M's;

S', S", S'", etc., are found to be--the proportion p of them--P's:

Hence, probably and approximately the same proportion, p, of the M's are P's.

(W 2:220, 268; W 4:416, 419, 420-421)

Another probabilistic interpretation of induction and hypothesis presents them as inverse statistical syllogisms (Form IV, in a footnote). Induction and hypothesis are the inferences from the conclusion and one premise of a statistical syllogism to the other premise, which is the observed fact. Induction is understood as looking for an answer to how-questions and hypothesis as explanatory syllogism answering why-questions (W 4:423-424).

Yet another one interpretation says that induction is inference from one set of facts to others and hypothesis is inference from facts of one type to facts of other type (W 3:336).

Forementioned schemes of inference are the great example of serious claim of Peirce about rigour of abduction: “It must be remembered that abduction, although it is very little hampered by logical rules, nevertheless is logical inference, asserting its conclusion only problematically or conjecturally, it is true, but nevertheless having a perfectly definite logical form.” (CP 5.188) Nevertheless, Peirce writes that abduction is “a form of Argument rather than of Argumentation” (CP 6.469). Argument is defined by Peirce as “any process of thought reasonably tending to produce a definite belief” (EP 2:435) and argumentation as “an argument proceeding upon definitely formulated premisses” (CP 6.456).

Talking about the operation of search for target in the process of subsuming unexpected facts to well-known laws, some remarks should be done. Bringing under the rule is one of the functions of power of judgment in Kantian heritage. “Judgment in general is the ability to think the particular as contained under the universal. If the universal (the rule, principle, law) is given, then judgment, which subsumes the particular under it, is determinative … But if only the particular is given and judgment has to find the universal for it, then this power is merely reflective”. Determinative function of power of judgement is embodied in abductive syllogisms as in the example with beans from bag. (CP 2.623) Reflective function is realized in forming concepts. American philosopher writes that general invariable elements of different perceptual judgements of percepts of specific object are generalized in law of the specific property of concrete object (for example, the law of caninity for general idea of being dog). Complicated mess of predicates attached to one subject which should be substituted by notion which has higher intensity. (EP 2:223, W 3:337) The reason why this process of forming notions can be understood as abductive (and partly inductive) is that for Peirce general concepts are laws. These laws of perceptual judgement are the real witness for constantly working hypothesis of reality as a world of percepts. Also continuity of perception stream is additional argument in favor of existence of reality (PMSW 175-176). According to Peirce, there may be discoveries of universals. It offers an opportunity to map out possible directions and options in the field of artificial intelligence with its problem of pattern recognition. For Peirce, forming general concepts is abductive process because of its explanatory meaning. During abduction of universals people consciously extract invariables from continuous stream of percepts and interpret these elements, bring facts in order in synthesizing act. (EP 2:229, EP 2:287). This insight is in need in the fields of cognitive psychology and AI working on the problem of formalizing experts' knowledge for pattern recognition and working with representations of data.

Hypothetic inference of perceptive abductive judgement:

Table 2

Thereby, it can be inferred that Peirce reconsidered Kantian thought. Necessary objectively existing functions of building relation between particular and universal were saved, but the whole vision of transcendental apparatus is reframed. And the fact that perceptual judgements and abductive inferences can be semiconscious or even not conscious is a good example of this intuition. For Peirce, the focus is not on human powers of making world definite and ordered. The world is already structured and organized, laws are in nature, not in mind. There is something in nature to which our mind corresponds at it is laws of nature and its inferences. Nature, by hypothesis of Peirce, is continuous process of generalizations, of becoming more and more ordered. In the development of nature new, more complicated laws, are realized by the power of originality and abduction in nature. Nature evolutions autonomously, it can make its own deductions, inductions and abductions. (PMSW 175-176) People are only involved participants who have very effective evolutionary developed guessing instinct. Concerning purposes of this work, let us notice that abduction can be realized as creation of new objects to produce valuable explanations, as process of extracting new phenomena from the continuous experience.

Also, Peirce refers to some guiding thread: remarks of specific relations between features of surprising phenomenon, outstanding traits. (CP 2.776) Moreover, philosopher indicates “some conception with which his mind is already stored” (CP 2.776) that will offer some theory to explain curious aspects of new phenomena. Here it can be illuminated that the strictly logical idea of inference is closely connected with philosophical views on the nature of consciousness, evolution of mind and matter, which are presented implicitly in the theory of abduction.

In addition, Peirce reveals the guiding potential of this stored knowledge in guessing. Peirce proposes as analogy between scientist-Nature dialogue-questionary of inquiry and the game of guessing between two gamers. One of the gamer (that can be either group or individual) who fixes upon object (fictitious or real, but known for educated people) and the second gamer who guesses. This game includes twenty questions and two possible answers (yes/no). Peirce mathematically proves that any object in the world can be guessed right in case of sound division of hypothesis into logical components. Notably, Peirce gives the example of theory of light as a scheme of reasonable trajectory of questions. Firstly, notion of light which was given in Antiquity and which encompasses such concepts as the beginning of light (eye), its motion (to the objects, reflection of light from object to eye), the end of movement (eye). Peirce persuades that this hypothesis was quickly refuted. So, there was answer “no” to the question “is the nature of light is so that it is something human-induced, started from human eyes?” Anyway, first question could be focused on the issue of the source of light. Second, the question of consistence or composition of light in terms of homogeneity or heterogeneity of the ray of light along its length in order to get a “yes” or “no” answer. The answer of nature is “no” as it becomes clear thanks to the phenomenon of diffraction. The third question is dedicated to the problem of continuity of qualities of light, or, as Peirce posed it: “Is the ray homogeneous on all sides?” The answer of Nature is again “no” due to the property of electromagnetic waves to change the direction of the electric field (linear polarization, circular or elliptical polarization). (CP 7.220) This game furnishes the other side of Peirce's thought - his strategy of guessing that will astoundingly echoes in the thematic analysis of Holton. Surely, Peirce appeals to the dichotomies of homogeneity/heterogeneity, part/the whole, movement in a close trajectory (from eye to eye) and linear motion, discreteness and continuity.

Engagement of abduction as a necessary stage of scientific inquiry shifts the focus from subsuming role in inference schemes to more creative and explanatory role. Abduction becomes a process of “provisional adoption of a hypothesis” (CP 1.68), “the only kind of argument which starts a new idea” (CP 2.96). Abduction is itself a conjecture allowing facts to suggest a theory with no probative force. (CP 8.209) Abduction starts from colligation of different observed facts about the subject for the hypothesis (CP 5.581) “in one or another of the three Universes, of some surprising phenomenon, some experience which either disappoints an expectation, or breaks in upon some habit of expectation of the inquisiturus”(CP 6.469). The hypothesis should be verified by experiment in induction (CP 1.68) saving the possibility to be refuted by facts which disagree with primary hypothesis. But abduction itself does not grant security (CP 6.470). Importantly the method of verification or refutation should be the one. Hypothesis encourage inquirer to some inclination, not to the assertion of the thesis (Conclusion), but to the question (whether facts, deduced from the premiss-hypothesis, are real). (CP 4.541, CP 6.528) This question promotes further experiment, external observation, whether the hypothesis is right. So, deduction is in need to formulate what are the necessary prediction that must be deduced from the hypothesis, from its “may-be-modus” (CP 5.171), because deduction can work with ideal states of things given in a picture or diagram of relations. Actually, there are two parts of deductive work with hypothesis and second part has two steps. Firstly, Explication, makes hypothesis clear. Second is Demonstrtation, or Deductive Argument, which divides to Corollarial Demonstration (explication of conclusions of hypothesis) and Theorematic Demonstration (working with diagrams) (CP 6.471) Induction in experimental verification shows the actual modus, certainty concerning the real. Thus, the proper sequence of action in scientific inquiry is abduction - deduction - induction (CP 5.171, CP 7.218, CP 8.209). Peirce lays special emphasis on this sequence, explaining that induction can never start with facts, only with hypothesis. And hypothesis in its turn always base on facts. (CP 2.755) Deduction, induction and abduction all together form the “well-rounded system of Formal Logic” (PMSW 167). Peirce claims that cognition of a rule may have habitual, not conscious nature. So, abductions can be produced faster than deductions, but with limitations on validity. Obviously, abduction as well as induction represents the kind of positive synthetic judgement which by definition contains new information in predicate (subject does not contain predicate concept, but the latter is somehow related to the subject concept). Abduction and induction are indirect probable syllogisms, “both lead to the acceptance of a hypothesis because observed facts are such as would necessarily or probably result as consequences of that hypothesis.” (W 4: 421, NEM 4:357, CP 7.218)

Hypothesis. The observed longitudes of Mars are such as they would be if Mars moved in an ellipse