Logic of discovery, abduction and types of explanatory models

Whereas Salmon constructs his typology of explanations in the dialogue with Hempel, presenting explanation not as a kind of argument, but as a unity of relevant considerations, others focus attention on otherwise disadvantages of the deductive-nomological model for scientific explanations. For example, the problem of explanatory asymmetries, which submits in comparison of two possible line of explanation with opposite directions (Salmon dedicates some place of his article to this theme, but not overwhelmingly). Deductively valid arguments relying on the same laws are represented in both cases, but the context of explanation sets the only correct direction of explanation. This direction is specified by causality, as in the example of Sylvain Bromberger, we can explain the length of the shadow of the flagpole deducing from the knowledge of the height of the flagpole, the angle made by flagpole and sun, the laws describing the rectilinear propagation of light. But we cannot make an opposite explanation. The presence of the flagpole can explain the presence of the shadow, but not vice versa. The other type of asymmetry, non-causal, is included in the laws of mathematics and other formal disciplines - is a characteristic of theoretical explanation. Theoretical explanation is not reduced to causal because of specific ontological status of abstract objects, such as mathematical objects, which exist not in space-time continuum, not causally, and special human capacities giving access to mathematical entities - mathematical intuition (the position of Platonists) or principles of language. The other, not so controversial for its ontological status, capacity of mind to think not-causally, but analogically in not-scientific explanations sometimes transmits to scientific explanations. So, there can be symmetrical (analogical) and asymmetrical (non-causal, theoretical and causal) explanations.

There are works revising ideas of Hempel. For example, A. Reutlinger proposes monist counterfactual theory of scientific explanation (CTE) that polemizes with so-called “liberal consensus” in philosophy of science established since the mid-2000s with proclaimed dualism of causal and noncausal scientific explanations irreducible to each other or any other covering principle. Hempel was the starting point of working out monistic intuition for Reutlinger. But problems concerning covering laws (indifference between relevant and irrelevant factors for explanation, nonreflecting explanatory asymmetry in the case of noncausal explanations) are underlined and presented in reconstruction.

Reconstruction of Hempel's monist approach

1. Common monist account:

-nomic expectability

Types of explanations:

1.1. Causal explanations

The explanation of phenomena P by identifying the type/token level causes of P.

The explanandum is predicted

-on the basis of laws of succession (causal covering laws)

-and with initial conditions

1.2. Noncausal explanations

The explanandum is predicted

-on the basis of laws of coexistence (noncausal covering laws)

-and with initial conditions

Then Reutlinger adduces his scheme of explanation:

Explanation structure:

statement E - the explanandum phenomenon

generalizations G1, …, Gm - the explanans

auxiliary statements S1, …, Sn - initial/ boundary conditions specifying the state of the explanandum system involved in the explanans

The classification which is proposed is following (expanding and precizing consensus-scheme):

1. Common monist account:

-the notion of counterfactual dependence (determination) of the explanandum on the explanans: what-if-things-had-been-different-question or

“if S1, …, Sn, had been different than they actually are (in at least one way deemed possible in the light of the generalizations), then E or the conditional probability of E would have been different as well”

1.1. Causal explanations

Explanations that contain asymmetry (including time asymmetry), metaphysical contingency, distinctness of the relata.

1.1.1 Detailed mechanistic explanations typical for natural and social sciences

1.1.2 Higher-level/macro causal explanations typical for less detailed cases

1.2. Noncausal explanations

1.2.1. “Purely” mathematical

-graph-theoretic

-topological

-geometric

-statistical explanations

1.2.2. Physical

-symmetry principles

-conservation laws

-kinematics

-renormalization group theory (RG)

-dimensional analysis

-intertheoretic relations

The most important distinction between the scheme of contemporary scholar and the Hempel's scheme is in its justification. Reutlinger suggests criteria of validity of explanations, explaining relationship between explanandum and explanans:

1. “Veridicality condition:

G1, …, Gm, S1, …, Sn, and E are (approximately) true or, at least, well confirmed.

2. Implication condition:

G1, …, Gm, S1, …, Sn logically entail E or a conditional probability P(E|S1, …, Sn) - where the conditional probability need not be `high' in contrast to Hempel's covering-law account.

3. Dependency condition:

G1, …, Gm support at least one counterfactual of the form: had S1, …, Sn

been different than they actually are (in at least one way deemed possible in the light of the generalizations), then E or the conditional probability of E would have been different as well.”

Thus, Reutlinger solves the problem of subsumption under covering laws.

The common pattern of classifications of explanations is to distinguish explanations including causes, explanations of formal sciences and somehow present explanations of human behaviour. Other principle of division is questions asked.

Anyway, it does not look very inspiring in case there is a need for some basis for creating explanatory hypotheses. Explanatory types presented earlier have creative potential of peculiar type. They provide understanding of connections which will bond entities in ideal schemes of hypotheses. Hempel's scheme has direct reference to covering laws. Some kinds of abduction have this goal of subsumption. Explanation schemes that are based on quasi-metaphysical scheme of sciences dividing ontology on the world of natural causes, the world of intentions and the quasi-platonic world of formal entities, properties and rules compose different representations and trigger different search strategies depending on target-world relations. Counter-factual determination as rationale for division of explanatory types is entertaining in the sense of mental architecture with necessary and sufficient conditions, asymmetrical relations, and temporal correlations. Notwithstanding all these explanatory types can help in understanding and representation of some phenomena, but not in discovery.

The objective is to find explanatory types that will be able to create new entities, new laws which can be not only statistical or causal.

Peirce proffers to find the source of analogy namely some set of known phenomena which can become the base of hypotheses to explain surprising phenomena. Peirce writes: “Nothing unknown can ever become known except through its analogy with other things known. Therefore, do not attempt to explain phenomena isolated and disconnected with common experience.” (MS L75:286) “By its very definition abduction leads to a hypothesis which is entirely foreign to the data….” (MS L75:286) My hypothesis is that combination of abduction studies and thematic analysis of Holton will be an example of productive synthesis which will facilitate creation of hypotheses.

The thematic analysis of G. Holton is a great example of search for types of explanatory models (also called thematic hypotheses/thematic propositions/ fundamental preconceptions/ explanatory devices). Holton distinguished thematic propositions from thematic concepts such as force and inertia, methodological themas such as search for constancies, for min and max meanings (extrema) and conditions of non-realization/ impotency (“ It is impossible that. . . ” ). For its part thematic hypotheses can be derived from methodological themas and can contain thematic concepts. They themselves can produce fruitful analogons (mathematical or physical) which will correlate convincingly the puzzling element in the observation (Peircean “surprising fact” we can say) with the consequences of the postulated structure of the analogon.

According to Holton, the thematic structure of scientific research may be interpreted as the structure of choice for a scientist which is independent from analytic and empirical content of the work and which represent features of constancy and continuity, sustainable structure of science. Themata (themes) can be named the third type of meaningful propositions in scientific discourse after phenomenic (propositions concerning empirical matters of fact) and analytic (propositions concerning logic and mathematics) propositions.

After analyzing the texts of Holton, it is possible to make a list (incomplete) of themes:

Table 10

immutability	evolution	catastrophic changes
1	atomism/ discreteness (cells, genes, discrete quantum)/discontinuity	continuity/imponderable fluids/field/wave
2	simplicity (+necessity)	complexity (+contingency)
3	analysis	synthesis
4	symmetry	asymmetry (redundancy or lack of universality) /chirality (property of asymmetry)/ incongruity
5	efficacy of mathematics (or the view of universe as a as a mathematical harmony/ the universe as a central theological order/ static, homocentric, hierarchically ordered, harmoniously arranged cosmos	efficacy of materialistic or mechanistic models (or universe as a purely mechanistic, causal)
6	projection [externalization]	retrojection [internalization]
7	complementarity	separation of methods giving different results
8	evolution	devolution
9	constancy (“ steady-state”/ invariance)	change (evolution/ transformation)
10	experience	symbolic formalism
11	reductionism	holism
12	finiteness	infinity
13	principle of potency/ the active vis/ Kraft/ energy/ force	inertia
14	once-created, real universe / the once-given real world which God supervises from the center of His temple	ontologically agnostic universe
15	use of mechanisms	teleological or anthropomorphic modes of approach
16	hierarchical structure	unity
17	the behavior of things is the consequence of their geometrical shapes	inherently probabilistic behavior
18	a harmoniously ordered world/	disorderliness/ derangement/ fission
19	hierarchy	disintegration
20	classically causal law	statistical law

It should be noted that Holton himself pointed out that his thematic analysis is close to the search for universal structure in linguistics and cultural anthropology, cataloguing of themata. Holton stresses that as distinctive features of themas, they have long history (at least from Antiquity), they are culturally universal and there is a limited number of them (“a total of fewer than 50 couples or triads”). In contrast with structuralist approach the historian of science writes about gradual development of themes, even supplementation of them (complementarity and chirality).

Themata have great influence on the thought: “Science has always had also a mythopoeic function-- that is, it generates an important part of our symbolic vocabulary and provides some of the metaphysical bases and philosophical orientations of our ideology.” So, science can create metaphysical lattice of reality, provide some basic principles of the reality. The metaphysical understanding of perfect being inspired by mathematical intuitions as in Antiquity of course has a special effect on the ideology differing from the dominant theme of necessary evolution for survival, influencing modern economics with its fear of stagnation.

Holton writes that in contradistinction to any natural language, for example Japanese that could be learnt by any child, the language of simple contemporary science (Holton's example is introductory quantum physics) can be inaccessible even to most talented students. Holton is following the idea of Mead that knowledge gap of different social groups form esoteric groups, separating intellectual elites. The real progress of knowledge is realized in the groups of young scientists who mainly communicate to each other (horizontal communication between peers) in highly specialized languages. There is a danger that in the long term there will be a decrease of culture without vertical transmissions, divergencies between generations will be too radical to overcome. Anyway, the accent is on the vocabulary. One of possible solutions, it can be assumed, is thematic analysis as provisional universal grammar that could be grasped not by any child but by any high school student.

Holton understands the limitations of his methodology from the point of view of classical (predominantly positivistic) philosophy of science: themes cannot be verified or falsified, because themata cannot be derived from observation and analytic ratiocination.

Remarkably, Holton focused on search for sustainable themes in scientific imagination, while the prominent direction of research was problem-solving approach, which accounts cognitive activity and especially scientific work for goal-attaining activity of puzzle solutions. This famous image of scientific puzzles of normal science, created by T. Kuhn, was challenged by attention given not to problems-puzzles, but themes by Holton. In the foreword to the Russian translation of his book, Holton writes that the character of discovery of Copernicus is distinctively understood by reading his work “On the Revolutions of the Heavenly Spheres”: “There is nothing like the pedantic folding of puzzle pieces”. Presumably Kuhn would agree with such an assessment of paradigmatic discovery, but the point is that Holton referred to the other aspect of context of discovery, which was missed by researchers of artificial intelligence in 1970s-1980s and which solves the great methodological problem. As it was mentioned before, AI-works on creating programs capable of making re-discoveries has serious disadvantage, connected with its inductive base. Those programs of the end of the 20th-century did not discovery new theories and laws because they were not universal; their heuristics were closely related to the data. As for the Holton's approach, it gives a broad perspective on applications as a resource of explanation heuristics which can motivate to make a quasi-structuralist cognitive map. Other notable feature of the Holton's approach is its openness to the wide cultural background which can influence on choice and development of themata including ideas of philosophy, literature, sociology of science, psychology, aesthetics.

Moreover, Holton characterizes the evolution of scientific thought. Holton writes: “Our interests and tools change, but not in a linear, inevitable way. For example, the historic development from organismic science to a mechanistic and then to the mathematical style could have taken place in the opposite direction.” Even though he specified schemata that are now discredited, he rejects a kind of inner dynamic of thought, logic of changes. It is a controversial thesis. But are there any empirical examples that science really developed in the opposite direction (with the proviso that actually themas are not verified)?

Holton himself says about the relevance of his work: actually, he did not mark as the target time of the usage of thematic analysis crisis times, but time “when a science is young”, when science does not have proper hierarchy of hypotheses. It can be explained by the thesis that in the childhood of the science the path from facts to general hypotheses is faster than in advanced, well-developed fields of knowledge. It is worth noting that Holton examines the dynamics of themes, he has not a relatively speaking Hegelian intuition of thesis, antithesis and synthesis, but the change of theme/antitheme moving from level to level. “The simple harmonies, the simple symmetries, have been found out. How much more satisfying it will be if we can discern harmonies and simplicities directly, through a more highly trained vision, in complex, apparently broken, and deranged configurations! It may be that we are beginning to train new sensibilities which will set a new style.”

Table 11

* Atrophied and/or now discredited themata
1) macrocosmic-microcosmic correspondence 2) inherent principles* 3) teleological drives 4) action at a distance 5) space filling media 6) organismic interpretations 7) hidden mechanisms 8) absolutes of time, space 9) simultaneity

As for explicit connections between abduction theory of Peirce and Holton's themata analysis, ur-hypotheses of Peirce can be mentioned. Tychism refers to chance/determinism opposition, closely related to themes causal/statistical laws; synechism has direct correspondence with discreetness/continuity themata. Peirce could have added to the list such themata as mutual coherence/parallel evolution (as in hypothesis of coordinated harmonious development of nature and mind), homogeneity / heterogeneity (as in game of twenty questions dedicated to the explication of nature of light).

Consequently, basic principles of understanding of the world somehow presented in human mind are the best basis for explanatory hypotheses.

There are a lot of suggestions from the specialists of different disciplines to enhance the typology of explanation appealing to the newest research. Rethought relations of determination (e.g. supervenience, intermingling of bottom-up genetic and top-down epigenetic causation), conversion, structures (e.g. perspectivism in conversion of structures, categorical notion pf mathematical structures in univalent foundations) in such areas of research as philosophy of consciousness, biology, anthropology, linguistics inspire insights about the developing nature of fundamental themata. One of the possible further directions of work is to elaborate the scheme of upgraded list of Holton themata using some instruments of formalization (modal logic) and sophisticated general notions of sciences.

Conclusions

This work was dedicated to the problems of so-called heuristic appraisal in methodology. In contrast to epistemic appraisal with its focus on context of discovery, heuristic appraisal proposes tools for perspective inquiries, appraisal of possible effectiveness of some future investigations.

It was shown that logic of discovery is realized as complex inquiry combining methods of Artificial Intelligence, cognitive psychology, logic and philosophy of science. The main breakthroughs are acceptance of weak form of psychologism and problem-solving approach. Contribution of psychology is representation schemes analysis, admittance of creative potential of tools from context of justification and hypothesis-driven research. AI developed by using distinction of qualitative and quantitative research, pattern recognition using experience of guided by principles perceptual judgments of specialists, extracting “problem isomorphs”.

As for reasoning schemes which will help to create new ideas, the game of twenty questions suggested by Peirce is guiding principle. Creative types of abductions will be the schemes of inference in accordance with the goal of abduction. The result will be certain type of connection. These types are preliminary sketch formed from standard classifications of explanations. The questions will be devoted to properties - themes of compared objects.

The unit of goal of abduction as generating an explanatory hypothesis, type of connection, data handling algorithms and heuristics plus questions specified regarding concrete property will form type of explanatory model.

Table 12

Goal of abduction	Properties asked	Data handling	Type of connection
-create a concept/a law/a theory -unify under category/general concept -subsume under rule -find hidden causes -find analogy …	Finite/infinite Symmetrical/asymmetrical Ordered/disordered Integrated/disintegrated Homogenous/heterogeneous …	Use of search heuristics and algorithms of AI and representational heuristics of cognitive psychology	-causal -statistical -structural -conventional -normative -formal -functional -categorical -analogical (surface or intrinsic similarity)

Explanatory models form abductive-explanatory inner grammar which promotes hypotheses, because they have necessary constructive element required for making an unlimited number of statements with a limited number of elements and rules. Themata may play a role of parts of speech; types of connections can be rules of sequence of tenses, rules of grammatical agreement in this analogy; goal of abduction - as intention of communication.

Prospectively, strict rules should be elaborated for types of explanatory model. But it is a question of formalization.

For the goal of creating philosophical explanations the list of themata may be broaden. For example, by themata which Jean Paul Gustave Ricњur noticed in “The conflict of interpretations”, namely teleology (phenomenology of image of Hegel) and psychoanalysis (archaeology); structuralism (can be named atemporal, but with serious reservations and universally-directed) and hermeneutics (with its accents on historical and unique, concrete).

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