KNOWLEDGE AND COHERENCE
Paul Thagard, Chris Eliasmith, Paul Rusnock, and Cameron Shelley
Philosophy Department
University of Waterloo
Waterloo, Ontario, N2L 3G1
pthagard@watarts.uwaterloo.ca
1. Introduction
Many contemporary philosophers favor coherence theories of knowledge (Bender 1989, BonJour 1985, Davidson 1986, Harman 1986, Lehrer 1990). But the nature of coherence is usually left vague, with no method provided for determining whether a belief should be accepted or rejected on the basis of its coherence or incoherence with other beliefs. Haack's (1993) explication of coherence relies largely on an analogy between epistemic justification and crossword puzzles. We show in this paper how epistemic coherence can be understood in terms of maximization of constraint satisfaction, in keeping with computational models that have had a substantial impact in cognitive science. A coherence problem can be defined in terms of a set of elements and sets of positive and negative constraints between pairs of those elements. Algorithms are available for computing coherence by determining how to accept and reject elements in a way that satisfies the most constraints. Knowledge involves at least five different kinds of coherence - explanatory, analogical, deductive, perceptual, and conceptual - each requiring different sorts of elements and constraints.
After specifying the notion of coherence as constraint satisfaction in more detail, we show how explanatory coherence subsumes Susan Haack's recent "foundherentist" theory of knowledge. We show how her crossword puzzle analogy for epistemic justification can be interpreted in terms of explanatory coherence, and describe how her use of the analogy can be understood in terms of analogical coherence. We then give an account of deductive coherence, showing how the selection of mathematical axioms can be understood as a constraint satisfaction problem. Moreover, visual interpretation can also be understood in terms of satisfaction of multiple constraints. After a brief account of how conceptual coherence can also be understood in terms of constraint satisfaction, we conclude with a discussion of how our "multicoherence" theory of knowledge avoids many criticisms traditionally made against coherentism.
2. Coherence as Constraint Satisfaction
Although coherence theories have been very popular in contemporary epistemology and ethics, coherence has been very poorly specified compared to deductive logic and probability theory. Coherence can be understood in terms of maximal satisfaction of multiple constraints, in a manner informally summarized as follows (Thagard and Verbeurgt 1998):
1. Elements are representations such as concepts, propositions, parts of images, goals, actions, and so on.
2. Elements can cohere (fit together) or incohere (resist fitting together). Coherence relations include explanation, deduction, similarity, association, and so on. Incoherence relations include inconsistency, incompatibility, and negative association.
3. If two elements cohere, there is a positive constraint between them. If two elements incohere, there is a negative constraint between them.
4. Elements are to be divided into ones that are accepted and ones that are rejected.
5. A positive constraint between two elements can be satisfied either by accepting both of the elements or by rejecting both of the elements.
6. A negative constraint between two elements can be satisfied only by accepting one element and rejecting the other.
7. A coherence problem consists of dividing a set of elements into accepted and rejected sets in a way that satisfies the most constraints.
More precisely, consider a set E of elements which may be propositions or other representations. Two members of E, e1 and e2, may cohere with each other because of some relation between them, or they may resist cohering with each other because of some other relation. We need to understand how to make E into as coherent a whole as possible by taking into account the coherence and incoherence relations that hold between pairs of members of E. To do this, we can partition E into two disjoint subsets, A and R, where A contains the accepted elements of E, and R contains the rejected elements of E. We want to perform this partition in a way that takes into account the local coherence and incoherence relations. For example, if E is a set of propositions and e1 explains e2, we want to ensure that if e1 is accepted into A then so is e2. On the other hand, if e1 is inconsistent with e3, we want to ensure that if e1 is accepted into A, then e3 is rejected into R. The relations of explanation and inconsistency provide constraints on how we decide what can be accepted and rejected. Different constraints can be of different strengths, represented by a number w, the weight of a constraint.
This informal characterization of coherence as maximization of constraint satisfaction can be made mathematically precise, and algorithms are available for computing coherence: see appendix A. Connectionist (neural network) models that are commonly used in cognitive science provide a powerful way of approximately maximizing constraint satisfaction. In such models, each element is represented by a neuron-like unit, positive constraints are represented by excitatory links between units, and negative constraints are represented by inhibitory links between units. Appendix A outlines in more detail how connectionist networks can be used to compute solutions to coherence problems.
Characterizing coherence in terms of constraint satisfaction does not in itself provide a theory of epistemic coherence, for which we need a description of the elements and the constraints relevant to establishing knowledge. Epistemic coherence is a composite of five kinds of coherence, each with its own kinds of elements and constraints. We will first review explanatory coherence, showing how it subsumes Susan Haack's recent "foundherentist" epistemology.
3. Haack's "Foundherentism" and Explanatory Coherence
Susan Haack's (1993) book, Evidence and Inquiry, presents a compelling synthesis of foundationalist and coherentist epistemologies. From coherentism, she incorporates the insights that there are no indubitable truths and that beliefs are justified by the extent to which they fit with other beliefs. From empiricist foundationalism, she incorporates the insights that not all beliefs make an equal contribution to the justification of beliefs and that sense experience deserves a special, if not completely privileged, role. She summarizes her "foundherentist" view with the following two principles (Haack 1993, p. 19):
(FH1) A subject's experience is relevant to the justification of his empirical beliefs, but there need be no privileged class of empirical beliefs justified exclusively by the support of experience, independently of the support of other beliefs;
(FH2) Justification is not exclusively one-directional, but involves pervasive relations of mutual support.
Haack's explication of "pervasive relations of mutual support" relies largely on an analogy with how crossword puzzles are solved by fitting together clues and possible interlocking solutions.
To show that Haack's epistemology can be subsumed within our account of coherence as constraint satisfaction, we will reinterpret her principles in terms of Thagard's theory of explanatory coherence (TEC), and describe how crossword puzzles can be solved as a constraint satisfaction problem by the computational model (ECHO) that instantiates TEC. TEC is informally stated in the following principles (Thagard 1989, 1992a, 1992b):
Principle E1. Symmetry. Explanatory coherence is a symmetric relation, unlike, say, conditional probability.
Principle E2. Explanation. (a) A hypothesis coheres with what it explains, which can either be evidence or another hypothesis; (b) hypotheses that together explain some other proposition cohere with each other; and (c) the more hypotheses it takes to explain something, the lower the degree of coherence.
Principle E3. Analogy. Similar hypotheses that explain similar pieces of evidence cohere.
Principle E4. Data priority. Propositions that describe the results of observations have a degree of acceptability on their own.
Principle E5. Contradiction. Contradictory propositions are incoherent with each other.
Principle E6. Competition. If P and Q both explain a proposition, and if P and Q are not explanatorily connected, then P and Q are incoherent with each other. (P and Q are explanatorily connected if one explains the other or if together they explain something.)
Principle E7. Acceptance. The acceptability of a proposition in a system of propositions depends on its coherence with them.
The last principle, Acceptance, states the fundamental assumption of coherence theories that propositions are accepted on the basis of how well they cohere with other propositions. It corresponds to Haack's principle FH2 that acceptance does not depend on any deduction-like derivation, but on relations of mutual support. Principle E4, Data Priority, makes it clear that TEC is not a pure coherence theory that treats all propositions equally in the assessment of coherence, but, like Haack's principle FH1, gives a certain priority to experience. Like Haack, however, TEC does not treat sense experience as the source of given, indubitable beliefs, but allows the results of observation and experiment to be overridden based on coherence considerations. For this reason, it is preferable to treat TEC as affirming a kind of discriminating coherentism rather than as a hybrid of coherentism and foundationalism (see the discussion of indiscriminateness in section 8).
TEC goes beyond Haack's foundherentism in specifying more fully the nature of the coherence relations. Principle E2, Explanation, describes how coherence arises from explanatory relations: when hypotheses explain a piece of evidence, the hypotheses cohere with the evidence and with each other. These coherence relations establish the positive constraints required for the global assessment of coherence in line with the characterization of coherence in section 2. When a hypothesis explains evidence, this establishes a positive constraint that tends to make them either accepted together or rejected together. Principle E3, Analogy, also establishes positive constraints between similar hypotheses. The negative constraints required for a global assessment of coherence are established by principles E5 and E6, Contradiction and Competition. When two propositions are incoherent with each other because they are contradictory or in explanatory competition, there is a negative constraint between them that will tend to make one of them accepted and the other rejected. Principle E4, Data Priority, can also be interpreted in terms of constraints, by positing a special element EVIDENCE that is always accepted and which has positive constraints with all evidence derived from sense experience. The requirement to satisfy as many constraints as possible will tend to lead to the acceptance of all elements that have positive constraints with EVIDENCE, but their acceptance is not guaranteed. Constraints are soft, in that coherence maximizing will tend to satisfy them, but not all constraints will be satisfied simultaneously.
As appendix A shows, the idea of maximizing constraint satisfaction is sufficiently precise that it can be computed using a variety of algorithms. The theory of explanatory coherence TEC is instantiated by a computer program ECHO that uses input about explanatory relations and contradiction to create a constraint network that performs the acceptance and rejection of propositions on the basis of their coherence relations. We have used ECHO to simulate the solution of Haack's crossword puzzle analogy for foundherentism. Figure 1 is the example that Haack uses to illustrate how foundherentism envisages mutual support. In the crossword puzzle, the clues are analogous to sense experience and provide a basis for filling in the letters. But the clues are vague and do not themselves establish the entries, which must fit with the other entries. Filling in each entry depends not only on the clue for it but also on how the entries fit with each other. In terms of coherence as constraint satisfaction, we can say that there are positive constraints connecting particular letters with each other and with the clues. For example, in 1 across the hypothesis that the first letter is H coheres with the hypotheses that the second letter is I and the third letter is P. Together, these hypotheses provide an explanation of the clue, since "hip" is the start of the cheerful expression "hip hip hooray." Moreover, the hypothesis that I is the second letter of 1 across must cohere with the hypothesis that I is the first letter of 2 down, which, along with other hypotheses about the word for 2 down, provides an answer for the clue for 2 down. These coherence relations are positive constraints that a computation of the maximally coherent interpretation of the crossword puzzle should satisfy. Contradictions can establish incoherence relations: only one letter can fill each square, so if the first letter of 1 across is H, it cannot be another letter.
<INSERT FIGURE 1 ABOUT HERE>
Figure 1. Crossword used to illustrate coherence relations, adapted from S. Haack, Evidence and Inquiry, p. 85..
Appendix B shows in detail how a solution to the crossword can be simulated by the program ECHO, which takes input of the form:
(explain (hypothesis1 hypothesis2 ... ) evidence).
For the crossword puzzle, we can identify each square using a system of letters A-E down the left side and numbers 1-6 along the top, so that location of the first letter of 1 across is A1. Then we can write A1=H to represent the hypothesis that the letter H fills this square. Writing C1A for the clue for 1 across, ECHO can be given the input:
(explain (A1=H A2=I A3=P) C1A)
This input establishes positive constraints among all pairs of the four elements listed, so that the hypothesis that the letters are H, I, and P tend to be accepted or rejected together in company with the clue C1A. Since the clue is given, it is treated as data and therefore the element C1A has a positive constraint with the special EVIDENCE element which is accepted. For real crossword puzzles, explanation is not quite the appropriate relation to describe the connection between entries and clues, but it is appropriate here because Haack is using the crossword puzzle example to illuminate explanatory reasoning.
The crossword puzzle analogy is useful in showing how beliefs can be accepted or rejected on the basis of how well they fit together. But TEC and ECHO go well beyond the analogy, since they demonstrate how coherence can be computed. ECHO not only has been used to simulate the crossword puzzle example, it has been applied to many of the most important cases of theory choice in the history of science, as well as to examples from legal reasoning and everyday life (Eliasmith and Thagard 1997; Nowak and Thagard 1992a, 1992b; Thagard 1989, 1992b, 1998). Moreover, ECHO has provided simulations of the results of a variety of experiments in social and educational psychology, so it meshes with a naturalistic approach to epistemology tied with human cognitive processes (Read and Marcus-Newhall 1993, Schank and Ranney 1992, Byrne 1995). Thus the construal of coherence as constraint satisfaction, through its manifestation in the theory of explanatory coherence and the computational model ECHO, subsumes Haack's foundherentism.
4. Analogical Coherence
Although explanatory coherence is the most important contributor to epistemic justification, it is not the only kind of coherence. In contrast to the central role that the crossword puzzle analogy plays in her presentation of foundherentism, Haack nowhere acknowledges the important contributions of analogies to epistemic justification. TEC's principle E3 allows such a contribution, since it establishes coherence (and hence positive constraints) among analogous hypotheses. This principle was based on the frequent use of analogies by scientists, for example Darwin's use of the analogy between artificial and natural selection in support of his theory of evolution.
Using analogies, as Haack does when she compares epistemic justification to crossword puzzles, requires the ability to map between two analogs, the target problem to be solved and the source that is intended to provide a solution. Mapping between source and target is a difficult computational task, but in recent years a number of computational models have been developed that perform it effectively. Haack's analogy between epistemic justification and crossword puzzles uses the mapping shown in table 1.
Epistemic justification |
Crossword puzzles |
Observations |
Clues |
Explanatory hypotheses |
Words |
Explanatory coherence |
Words fitting with clues and each other |
Table 1. Mapping between epistemic justification and crossword puzzle completion.
Analogical mapping can be understood in terms of coherence and multiple constraint satisfaction, where the elements are hypotheses concerning what maps to what and the main constraints are similarity, structure, and purpose (Holyoak and Thagard 1985). To highlight the similarities and differences with explanatory coherence, we now present principles of analogical coherence:
Principle A1. Symmetry. Analogical coherence is a symmetric relation among mapping hypotheses.
Principle A2. Structure. A mapping hypothesis that connects two propositions, R(a, b) and S(c, d), coheres with mapping hypotheses that connect R with S, a with c, and b with d; and all those mapping hypotheses cohere with each other.
Principle A3. Similarity. Mapping hypotheses that connect elements that are semantically or visually similar have a degree of acceptability on their own.
Principle A4. Purpose. Mapping hypotheses that provide possible contributions to the purpose of the analogy have a degree of acceptability on their own.
Principle A5. Competition. Mapping hypotheses that offer different mappings for the same object or concept are incoherent with each other.
Principle A6. Acceptance. The acceptability of a mapping hypothesis in a system of mapping hypotheses depends on its coherence with them.
In analogical mapping, the coherence elements are hypotheses concerning which objects and concepts correspond to each other. Initially, mapping favors hypotheses that relate similar objects and concepts (A3). Depending on whether analogs are represented verbally or visually, the relevant kind of similarity is either semantic or visual. For example, when Darwin drew an analogy between natural and artificial selection, both analogs had the verbal representations of selection that had similar meaning. In visual analogies, perceptual similarity can suggest possible correspondences, for example when the atom with its electrons circling the nucleus is pictorially compared to the solar system with its planets revolving around the sun. We then get the positive constraint: if two objects or concepts in an analogy are visually or semantically similar to each other, then an analogical mapping that puts them in correspondence with each other should tend to be accepted. This kind of similarity is much more local and direct than the more general overall similarity that is found between two analogs. Another positive constraint is pragmatic, in that we want to encourage mappings that can accomplish the purposes of the analogy such as problem solving or explanation (A4).
Additional positive constraints arise because of the need for structural consistency (A2). In the verbal representations (CIRCLE (ELECTRON NUCLEUS)) and (REVOLVE (PLANET SUN)), maintaining structure (i.e. keeping the mapping as isomorphic as possible) requires that if we map CIRCLE to REVOLVE then we must map ELECTRON to PLANET and NUCLEUS to SUN. The need to maintain structure establishes positive constraints, so that, for example, the hypothesis that CIRCLE corresponds to REVOLVE will tend to be accepted with or rejected with the hypothesis that ELECTRON corresponds to PLANET. Negative constraints occur between hypotheses representing incompatible mappings, for example between the hypothesis that the atom corresponds to the sun and the hypothesis that the atom corresponds to a planet (A5). Principles A2 and A5 together incline but do not require analogical mappings to be isomorphisms. Analogical coherence is a matter of accepting the mapping hypotheses that satisfy the most constraints.
The multiconstraint theory of analogy just sketched has been applied computationally to a great many examples and has provided explanations for numerous psychological phenomena. Additional epistemological importance lies in the fact that the constraint-satisfaction construal of coherence provides a way of unifying explanatory and analogical epistemic issues. Take, for example, the traditional philosophical problem of other minds: you know from your conscious experience that you have a mind, but how are you justified in believing that other people, whose consciousness you have no access to, have minds? One common solution to this problem is analogical inference: other people’s actions are similar to yours, so perhaps they are also similar to you in having minds. Another common solution to the problem of other minds is inference to the best explanation: the hypothesis that other people have minds is a better explanation of their behavior than any other available hypothesis, for example that they are radio-controlled robots. From the perspective of coherence as constraint satisfaction, analogical inference and best-explanation inference are complementary, not alternative justifications, because analogical and explanatory coherence considerations can simultaneously work to justify as acceptable the conclusion that other people have minds. Figure 2 shows how analogy-based positive constraints mesh with explanation-based positive constraints to establish the acceptability of the hypothesis that other people have minds.
<INSERT FIGURE 2 ABOUT HERE
Figure 2. The problem of other minds incorporating both explanatory and analogical coherence. Plus signs indicate positive constraints and minus signs indicate negative constraints. The hypothesis that you have a mind is evaluated both by comparing its explanatory power with other hypotheses that explain your behavior, and by analogy with my explanations of my own behavior.
Thus metaphysics, like science, can employ a combination of explanatory and analogical coherence to defend important conclusions. Mathematical knowledge, however, is more dependent on deductive coherence.
5. Deductive Coherence
For millennia, epistemology has been enthralled by mathematics, taking mathematical knowledge as the purest and soundest type. The Euclidean model of starting with indubitable axioms and deriving equally indubitable theorems has influenced many generations of philosophers. Surprisingly, however, Bertrand Russell, one of the giants of the axiomatic method in the foundations of mathematics, had a different view of the structure of mathematical knowledge. In an essay he presented in 1907, Russell remarked on the apparent absurdity of proceeding from recondite propositions in symbolic logic to the proof of such truisms as 2+2=4. He concluded:
The usual mathematical method of laying down certain premises and proceeding to deduce their consequences, though it is the right method of exposition, does not, except in the more advanced portions, give the order of knowledge. This has been concealed by the fact that the propositions traditionally taken as premises are for the most part very obvious, with the fortunate exception of the axiom of parallels. But when we push the analyses farther, and get to more ultimate premises, the obviousness becomes less, and the analogy with the procedure of other sciences becomes more visible (Russell 1973, p. 282).
Just as science discovers hypotheses from which facts of the senses can be deduced, so mathematics discovers premises (axioms) from which elementary propositions (theorems) such as 2+2=4 can be derived. Unlike the logical axioms that Russell, following Frege, used to derive arithmetic, these theorems are often intuitively obvious. Russell contrasts the a priori obviousness of such mathematical propositions with the lesser obviousness of the senses, but notes that obviousness is a matter of degree and even where there is the highest degree of obviousness, we cannot assume that the propositions are infallible, since they may be abandoned because of conflict with other propositions. Thus for Russell, adoption of a system of mathematical axioms and theorems is much like the scientific process of acceptance of explanatory hypotheses. Let us try to exploit this analogy to develop a theory of deductive coherence.
The elements are mathematical propositions - potential axioms and theorems. The positive and negative constraints can be established by coherence and incoherence relations specified by a set of principles that are adapted from the 7 principles of explanatory coherence in section 3.
Principle D1. Symmetry. Deductive coherence is a symmetric relation among propositions, unlike, say, deducibility.
Principle D2. Deduction (a) An axiom or other proposition coheres with propositions that are deducible from it; (b) propositions that together are used to deduce some other proposition cohere with each other; and (c) the more hypotheses it takes to deduce something, the less the degree of coherence.
Principle D3. Intuitive priority. Propositions that are intuitively obvious have a degree of acceptability on their own. Propositions that are obviously false have a degree of rejectability on their own.
Principle D4. Contradiction. Contradictory propositions are incoherent with each other.
Principle D5. Acceptance. The acceptability of a proposition in a system of propositions depends on its coherence with them.
When a theorem is deduced from an axiom, the axiom and theorem cohere symmetrically with each other, which allows the theorem to confer support on the axiom as well as vice versa, just as an explanatory hypothesis and the evidence it explains confer support on each other (D1, D2). Principle D2, Deduction, is just like the second principle of explanatory coherence, but with the replacement of the coherence-producing relation of explanation by the similarly coherence-producing relation of deduction. These coherence relations are the source of positive constraints: when an axiom and theorem cohere because of the deductive relation between them, there is a positive constraint between them so that they will tend to be accepted together or rejected together. Clause (c) of the principle has the consequence that the weight of the constraint will be reduced if the deduction requires other propositions. Just as scientists prefer simpler theories, other things being equal, Russell looked for simplicity in axiom systems: "Assuming then, that elementary arithmetic is true, we may ask for the fewest and simplest logical principles from which it can be deduced" (Russell 1973, pp. 275-276).
Although some explanations are deductive, not all are, and not all deductions are explanatory (Kitcher and Salmon 1989). So explanatory coherence and deductive coherence cannot be assimilated to each other. The explanatory coherence principle E4, Data priority, discriminated in favor of the results of sensory observations and experiments, but deductive coherence in mathematics requires a different kind of intuitive obviousness. Russell remarks that the obviousness of propositions such as 2+2=4 derives remotely from the empirical obviousness of such observations as 2 sheep + 2 sheep = 4 sheep. Principle D3, Intuitive priority, does not address the source of the intuitiveness of mathematical propositions, but simply takes into account that it exists. Different axioms and theorems will have different degrees of intuitive priority. D3 provides discriminating constraints that encourage the acceptance of intuitively obvious propositions such as 2+2=4. Russell stressed the need to avoid having falsehoods as consequences of axioms, so we have included in D3 a specific mention of intuitively obvious falsehoods being rejected, even though it is redundant: a falsehood can be indirectly rejected because it contradicts an obvious truth. Principle D4, Contradiction, establishes negative constraints that prevent two contradictory propositions from being accepted simultaneously. For mathematics, these should be constraints with very high weights. The contradiction principle is obvious, but it is much less obvious whether there is competition between mathematical axioms in the same way there is between explanatory hypotheses.
Whereas there are ample scientific examples of the role of analogy in enhancing explanatory coherence, cases of an analogical contribution to deductive coherence in mathematics are rarer, so our principles of deductive coherence do not include an analogy principle, although analogy is important in mathematical discovery (Polya 1957). Moreover, analogical considerations can enter indirectly into the choice of mathematical principles, by virtue of isomorphisms between areas of mathematics that allow all the theorems in one area to be translated into theorems in the other, as when geometry is translated into Cartesian algebra.
Russell does not explicitly defend a coherentist justification of axiom systems, but he does remark that "we tend to believe the premises because we can see that their consequences are true, instead of believing the consequences because we know the premises to be true." (Russell 1973, pp. 273-274; there are additional noncoherence considerations such as independence and convenience that contribute to selection of an axiom set.) Philip Kitcher (1983, p. 220) sees the contribution of important axiomatizations by Euclid, Cayley, Zermelo, and Kolmogorov as analogous to the uncontroversial cases in which scientific theories are adopted because of their power to unify. Principle D5, Acceptance, summarizes how axioms can be accepted on the basis of the theorems they yield, while at the same time theorems are accepted on the basis of their derivation from axioms. The propositions to be accepted are just the ones that are most coherent with each other, as shown by finding a partition of propositions into accepted and rejected sets in a way that satisfies the most constraints.
We have discussed deductive coherence in the context of mathematics, but it is also relevant to other domains such as ethics. According to Rawl's notion of reflective equilibrium, ethical principles such as "Killing is wrong" are to be accepted or rejected on the basis of how well they fit with particular ethical judgments such as "Killing Salmon Rushdie is wrong" (Rawls 1971). Ethical coherence is not only deductive coherence, however, since wide reflective equilibrium requires finding the most coherent set of principles and particular judgments in the light of background information, which can introduce considerations of explanatory, analogical, and deliberative coherence (Thagard, in press).
6. Perceptual Coherence
Explanatory and deductive coherence both involve propositional elements, but not all knowledge is verbal. Perceptual knowledge can also be thought of as a coherence problem, in accord with modern views of perception as involving inference and constraint satisfaction (Rock, 1983; Kosslyn, 1994). Vision is not simply a matter of taking sensory inputs and transforming them directly into interpretations that form part of conscious experience, because the sensory inputs are often incomplete or ambiguous. For example, the Necker cube in figure 3 can be seen in two different ways with different front faces. We shall not attempt here anything like a full theory of different kinds of perception, but want to sketch how vision can be understood as a coherence problem similar to but different from the kinds of coherence so far discussed.
<INSERT FIGURE 3 ABOUT HERE>
Figure 3. The Necker cube. The top edge can be seen either as being at the front or at the back of the cube. Try to make it flip back and forth by concentrating on different edges.
Visual perception begins with two-dimensional image arrays on the retina, but the visual interpretations that constitute sensory experience are much more complex than these arrays. How does the brain construct a coherent understanding of sensory inputs? In visual coherence, the elements are non-verbal representations of input images and full-blown visual interpretations, which fit together in accord with the following principles:
Principle V1. Symmetry. Visual coherence is a symmetric relation between a visual interpretation and a sensory input.
Principle V2. Interpretation A visual interpretation coheres with a sensory input if they are connected by perceptual principles such as proximity, similarity, and continuity.
Principle V3. Sensory priority. Sensory inputs are acceptable on their own.
Principle V4. Incompatibility. Incompatible visual interpretations are incoherent with each other.
Principle V5. Acceptance. The acceptability of a visual interpretation depends on its coherence with sensory inputs, other visual interpretations, and background knowledge.
Principle V2, Interpretation, asserts that how an interpretation fits with sensory input is governed by innate perceptual principles such as ones described in the 1930s by Gestalt psychologists (Koffka 1935). According to the principle of proximity, visual parts that are near to each other joint together to form patterns or groupings. Thus an interpretation that joins two visual parts together in a pattern will cohere with sensory input that has the two parts close to each other. According to the Gestalt principle of similarity, visual parts that resemble each other in respect to form size, color, or direction, unite to form a homogeneous group. Hence an interpretation that combines parts in a pattern will cohere with sensory input that has parts that are similar to each other. Other Gestalt principles encourage interpretations that find continuities and closure (lack of gaps) in sensory inputs. The visual system also has built into it assumptions that enable it to use cues such as size constancy, texture gradients, motion parallax, and retinal disparity to provide connections between visual interpretations and sensory inputs (Medin and Ross 1992, ch. 5). These assumptions establish coherence relations between visual interpretations and sensory inputs, and thereby provide positive constraints that tend to make visual interpretations accepted along with the sensory inputs with which they cohere.
Image arrays on the retina are caused by physical processes not subject to cognitive control, so we can take them as given (V3). But even at the retinal level considerable processing begins and many layers of visual processing occur before a person has a perceptual experience. The sensory inputs may be given, but sensory experience certainly is not. Sensory inputs may fit with multiple possible visual interpretations which are incompatible with each other and are therefore incoherent and the source of negative constraints (V4).
Thus the Gestalt principles and other assumptions built into the human visual system establish coherence relations that provide positive constraints linking visual interpretations with sensory input. Negative constraints arise between incompatible visual interpretations, such as the two ways of seeing the Necker cube. Our overall visual experience arises from accepting the visual interpretation that satisfies the most positive and negative constraints. Coherence thus produces our visual knowledge, just as it establishes our explanatory and deductive knowledge.
We cannot attempt here to sketch coherence theories of other kinds of perception - smell, sound, taste, touch. Each would have a different version of principle V2, Interpretation., involving its own kinds of coherence relations based on the innate perceptual system for that modality.
7. Conceptual Coherence
Given the above discussions of explanatory, deductive, analogical, and perceptual coherence, the reader might now be worried about the proliferation of kinds of coherence: just how many are there? We see the need to discuss only one additional kind of coherence - conceptual - that seems important for understanding human knowledge.
Different kinds of coherence are distinguished from each other by the different kinds of elements and constraints they involve. In explanatory coherence, the elements are propositions and the constraints are explanation-related, but in conceptual coherence the elements are concepts and the constraints are derived from positive and negative associations among concepts. Much work has been done in social psychology to examine how people apply stereotypes when forming impressions of other people. For example, you might be told that someone is a woman pilot who likes monster truck rallies. Your concepts of woman, pilot, and monster truck fan may involve a variety of concordant and discordant associations that need to be reconciled as part of the overall impression you form of this person.
Principle C1. Symmetry. Conceptual coherence is a symmetric relation between pairs of concepts.
Principle C2. Association. A concept coheres with another concept if they are positively associated, i.e. if there are objects to which they both apply.
Principle C3. Given concepts. The applicability of a concept to an object may be given perceptually or by some other reliable source.
Principle C4. Negative association. A concept incoheres with another concept if they are negatively associated, i.e. if an object falling under one concept tends not to fall under the other concept.
Principle C5. Acceptance. The applicability of a concept to an object depends on the applicability of other concepts.
The stereotypes that some Americans have of Canadians include associations with other concepts such as polite, law-abiding, beer-drinking, and hockey-playing, where these concepts have different kinds of associations each other. The stereotype that Canadians are polite (a Canadian is someone who says "thank-you" to bank machines) conflicts with the stereotype that hockey players are somewhat crude. If you are told that someone is a Canadian hockey-player, what impression do you form of them? Applying stereotypes in complex situations is a matter of conceptual coherence, where the elements are concepts and the positive and negative constraints are positive and negative associations between concepts (C2, C4). Some concepts cohere with each other (e.g. Canadian and polite), while other concepts resist cohering with each other (e.g. polite and crude). The applicability of some concepts is given, as when you can see that someone is a hockey player or told by a reliable source that he or she is a Canadian (C3). Many psychological phenomena concerning how people apply stereotypes can be explained in terms of conceptual constraint satisfaction (Kunda and Thagard 1995).
There are thus five primary kinds of coherence relevant to assessing knowledge: explanatory, analogical, deductive, perceptual, and conceptual. (A sixth kind, deliberative coherence, is relevant to decision making; see Thagard and Millgram 1995, and Millgram and Thagard 1996). Each kind of coherence involves a set of elements and positive negative constraints, including constraints that discriminate to favor the acceptance or rejection of some of the elements, as summarized in table 2. A major problem for the kind of multifaceted coherence theory of knowledge we have been presenting concerns how these different kinds of coherence relate to each other. To solve this problem, we would need to describe in detail the interactions involved in each of the fifteen different pairs of kinds of coherence. Some of these pairs are straightforward. For example, explanatory and deductive coherence both involve propositional elements and very similar kinds of constraints. In addition, section 4 above showed how explanatory and analogical coherence can interact in the problem of other minds. The relation, however, between propositional elements (explanatory and deductive) on the one hand and visual and conceptual elements is obscure, so that it is not obvious how, for example, a system of explanatory coherence can interface with a system of visual coherence. One possibility is that a deeper representational level, such as the systems of vectors used in neural networks, may provided a common substratum for propositional, perceptual, and conceptual coherence (see Eliasmith and Thagard, forthcoming, for a discussion of vector representations of propositions and their use in analogical mapping).
Note that simplicity plays a role in most kinds of coherence. It is explicit in explanatory and deductive coherence, where an increase in the number of propositions required for an explanation or deduction decreases simplicity; and deliberative coherence is similar. Simplicity is implicit in analogical coherence, which encourages 1-1 mappings. Perhaps simplicity plays a role in perceptual coherence as well.
elements |
positive constraints |
discriminating constraints |
negative constraints | |
explanatory |
hypotheses, evidence |
E2, explanation E3, analogy |
E4, data priority |
E5, contradiction E6, competition |
analogical |
mapping hypotheses |
A2, structure |
A3, similarity A4, purpos |
A5, competition |
deductive |
axioms, theorems |
D2, deduction |
D3, intuitive priority |
D4, contradiction |
visual |
visual interpretations |
V2, interpretation |
V3, sensory priority |
V4, incompati- bility |
conceptual |
concepts |
C2, association |
C3, given concepts |
C4, negative association |
Table 2. Kinds of coherence and their constraints. Names such as "E2" refer to principles stated in the text.
Explanatory, analogical, deductive, visual, and conceptual coherence add up to a comprehensive, computable, naturalistic theory of epistemic coherence. Let us now see how this theory can handle some of the standard objections that have been made to coherentist epistemologies.
8. Objections to Coherence Theories
Vagueness
One common objection to coherence theories is vagueness: in contrast to fully specified theories of deductive and inductive inference, coherence theories have generally been vague about what coherence is and how coherent elements can be selected. Our general characterization of coherence shows how vagueness can be overcome. First, for a particular kind of coherence, it is necessary to specify the nature of the elements, and define the positive and negative constraints that hold between them. This task has been accomplished for the kinds of coherence discussed above. Second, once the elements and constraints have been specified, it is possible to use connectionist algorithms to compute coherence, accepting and rejecting elements in a way that approximately maximizes compliance with the coherence conditions (appendix A). Computing coherence then can be as exact as deduction or probabilistic reasoning, and can avoid the problems of computational intractability that arise with them. Being able to do this computation does not, of course, help with the problem of generating elements and constraints, but it does show how to make a judgment of coherence with the elements and constraints on hand. Arriving at a rich, coherent set of elements - scientific theories, ethical principles, or whatever - is a very complex process that intermingles both (a) assessment of coherence and (b) generation of new elements; the parallel constraint satisfaction algorithm shows only how to do the first of these. Whether a cognitive task can be construed as a coherence problem depends on the extent to which it involves evaluation of how existing elements fit together rather than generation of new elements.
Indiscriminateness
The second objection to coherence theories is indiscriminateness: coherence theories fail to allow that some kinds of information deserve to be treated more seriously than others. For example, in epistemic justification, it has been argued that perceptual beliefs should be taken more seriously in determining general coherence than mere speculation. The abstract characterization of coherence given in section 2 is indiscriminating, in that all elements are treated equally in determinations of coherence.
But all the kinds of coherence discussed above are discriminating in the sense of allowing favored elements of E to be given priority in being chosen for the set of accepted elements A. We can define a discriminating coherence problem as one where members of a subset D of E are favored to be members of A. Favoring them does not guarantee that they will be accepted: if there were such a guarantee, the problem would be foundationalist rather than coherentist, and D would constitute the foundation for all other elements. As Audi (1993) points out, even foundationalists face a coherence problem in trying to decide what beliefs to accept in addition to the foundational ones. Explanatory coherence treats hypothesis evaluation as a discriminating coherence problem, since it gives priority to propositions that describe observational and experimental results. That theory is not foundationalist, since evidential propositions can be rejected if they fail to cohere with the entire set of propositions. Similarly, table 2 makes it clear that the other five kinds of coherence are also discriminating.
Computing a solution to a discriminating coherence problem involves only a small addition to the characterization of coherence given in section 2:
For each element d in the discriminated set D, construct a positive constraint between d and a special element es which is assigned to the set A of accepted elements.
The effect of having a special element that constrains members of the set D is that the favored elements will tend to be accepted, without any guarantee that they will be accepted. For the connectionist algorithm (appendix A), this new discriminating condition is implemented by having an excitatory link between the unit representing d and a special unit that has a fixed, unchanging maximum activation (i.e. 1). The effect of constructing such links to a special unit is that when activation is updated it flows directly from the activated special unit to the units representing the discriminated elements. Hence those units will more strongly tend to end up activated than nondiscriminated ones, and will have a greater effect on which other units get activated. The algorithm does not, however, enforce the activation of units representing discriminated elements, which can be deactivated if they have strong inhibitory links with other activated elements. Thus a coherence computation can be discriminating while remaining coherentist.
We can thus distinguish between three kinds of coherence problems. A pure coherence problem is one that does not favor any elements as potentially worthy of acceptance. A foundational coherence problem selects a set of favored elements for acceptance as self-justified. A discriminating coherence problem favors a set of elements but their acceptance still depends on their coherence with all the other elements. We have shown how coherence algorithms can naturally treat problems as discriminating without being foundational.
Isolation
The isolation objection has been characterized as follows:
This objection states that the coherence of a theory is an inadequate justification of the theory, because by itself it doesn't supply the necessary criteria to distinguish it from illusory but consistent theories. Fairytales may sometimes be coherent as may dreams and hallucinations. Astrology may be as coherent as astronomy, Newtonian physics as coherent as Einsteinian physics (Pojman, 1993, p. 191).
Thus an isolated set of beliefs may be internally coherent but should not be judged to be justified.
Our characterization of coherence provides two ways of overcoming the isolation objection. First, as we just saw, a coherence problem may be discriminating, giving non-absolute priority to empirical evidence or other elements that are known to make a relatively reliable contribution to solution of the kind of problem at hand. The comparative coherence of astronomy and astrology is thus in part a matter of coherence with empirical evidence, of which there is obviously far more for astronomy than astrology. Second, the existence of negative constraints such as inconsistency shows that we cannot treat astronomy and astrology as isolated bodies of beliefs. The explanations of human behavior offered by astrology often conflict with those offered by psychological science. Astrology might be taken to be coherent on its own, but once it offers explanations that compete with psychology and astronomy, it becomes a strong candidate for rejection. The isolation objection may be a problem for underspecified coherence theories that lack discrimination and negative constraints, but it is easily overcome by the constraint satisfaction approach.
Having negative constraints, however, does not guarantee consistency in the accepted set A. The second coherence condition, which encourages dividing negatively constrained elements between A and R, is not rigid, so there may be cases where two negatively constrained elements both end up being accepted. For a correspondence theory of truth, this is a disaster, since two contradictory propositions cannot both be true. It would probably also be unappealing to most advocates of a coherence theory of truth. To overcome the consistency problem, we could revise the second coherence condition by making it rigid: a partition of elements (propositions) into accepted and rejected sets must be such that if ei and ej are inconsistent, then if ei is in A then ej must be in R. We do not want, however, to defend a coherence theory of truth, since there are good reasons for preferring a correspondence theory based on scientific realism (Thagard, 1988, ch. 8).
For a coherence theory of epistemic justification, inconsistency in the set A of accepted propositions is also problematic, but we can leave open the possibility that coherence is temporarily maximized by adopting an inconsistent set of beliefs. One way of dealing with the lottery and proofreading paradoxes is simply by being inconsistent, believing that a lottery is fair while believing of each ticket that it will not win, or believing that a paper must have a typographical error in it somewhere while believing of each sentence that it is flawless. A more interesting case is the relation between quantum theory and general relativity, two theories which individually possess enormous explanatory coherence. According to the eminent mathematical physicist Edward Witten, "the basic problem in modern physics is that these two pillars are incompatible. If you try to combine gravity with quantum mechanics, you find that you get nonsense from a mathematical point of view. You write down formulae which ought to be quantum gravitational formulae and you get all kinds of infinities" (Davies and Brown, 1988, p. 90). Quantum theory and general relativity may be incompatible, but it would be folly given their independent evidential support to suppose that one must be rejected. Another inconsistency in current astrophysics derives from measurements that suggest that the stars are older than the universe. But astrophysics carries on, just as mathematics did when Russell discovered that Frege's axioms for arithmetic lead to contradictions.
From the perspective of formal logic, contradictions are disastrous, since from any proposition and its negation any formula can be derived: from p to p or q by addition, then from not-p to q by disjunctive syllogism. Logicians who have wanted to deal with inconsistencies have been forced to resort to relevance or paraconsistent logics. But from the perspective of a coherence theory of inference, there is no need for any special logic. It may turn out that at a particular time that coherence is maximized by accepting a set A that is inconsistent, but other coherence-based inferences need not be unduly influenced by the inconsistency, whose effects may be relatively isolated in the network of elements.
Related to the isolation objection is concern with the epistemic goal of achieving truth. A set of beliefs may be maximally coherent, but what guarantee is there that the maximally coherent system will be true? Assuming a correspondence theory of truth and the consistency of the world, a contradictory set of propositions cannot all be true. But no one ever suggested that coherentist methods guarantee the avoidance of falsehood. All that we can expect of epistemic coherence is that it is generally reliable in accepting the true and rejecting the false. Scientific thinking based on explanatory and analogical coherence has produced theories with substantial technological application, intersubjective agreement, and cumulativity. Our visual systems are subject to occasional illusions, but these are rare compared with the great preponderance of visual interpretations that enable us successfully to interact with the world. Not surprisingly, there is no foundational justification of coherentism, only the coherentist justification that coherentist principles fit well with what we believe and what we do. Temporary tolerance of contradictions may be a useful strategy in accomplishing the long term aim of accepting many true propositions and few false ones. Hence there is no incompatibility between our account of epistemic coherence and a correspondence theory of truth.
Conservatism
Coherence theories of justification may seem unduly conservative in that they require new elements to fit into an existing coherent structure. This charge is legitimate against serial coherence algorithms that determine for each new element whether accepting it increases coherence or not. The connectionist algorithm in Appendix A, on the other hand, allows a new element to enter into the full-blown computation of coherence maximization. If units have already settled into a stable activation, it will be difficult for a new element with no activation to dislodge the accepted ones, so the network will exhibit a modest conservatism. But if new elements are sufficiently coherent with other elements, they can dislodge previously accepted ones. Connectionist networks can be used to model dramatic shifts in explanatory coherence that take place in scientific revolutions (Thagard, 1992).
Circularity
Another standard objection to coherence theories is that they are circular, licensing the inference of p from q and then of q from p. The theories of coherence and the coherence algorithms presented here make it clear that coherence-based inference are very different from those familiar from deductive logic where propositions are derived from other propositions in linear fashion. The characterization of coherence and the algorithms for computing it (appendix A) involve a global, parallel, but effective means of assessing a whole set of elements simultaneously on the basis of their mutual dependencies. Inference can be seen to be holistic in a way that is non-mystical, computationally effective, and psychologically and neurologically plausible (pairs of real neurons do not excite each other symmetrically, but neuronal groups can). Deductive circular reasoning is inherently defective, but the foundational view that conceives of knowledge as building deductively on indubitable axioms is not even supportable in mathematics, as we saw in section 5. Inference based on coherence judgments is not circular in the way in the way feared by logicians, since it effectively calculates how a whole set of elements fit together, without linear inference of p from q and then of q from p.
9. Conclusion
This paper has described epistemic coherence in terms of five contributory kinds of coherence: explanatory, analogical, deductive, visual, and conceptual. By analogy to previously presented principles of explanatory coherence, it has generated new principles to capture existing theories of analogical and conceptual coherence, and it has developed new theories of deductive and visual coherence. All of these kinds of coherence can be construed in terms of constraint satisfaction and computed using connectionist and other algorithms. We showed that Haack's "foundherentist" epistemology can be subsumed within the more precise framework offered here, and that many of the standard philosophical objections to coherentism can be answered within this framework.
Further work remains to be done on the "multicoherence" theory of knowledge offered here. The most pressing is the "intercoherence" problem that requires finding additional connections among the elements and constraints involved in the different kinds of coherence. In the tradition of naturalistic epistemology, the intercoherence problem is also a problem in cognitive science, to explain how people make sense of their world using a variety of kinds of representations and computations. We have made only a small contribution to that problem here, but have shown how satisfaction of various constraints involving multiple different elements provide a novel and comprehensive account of epistemic coherence.
Acknowledgements. We are grateful to Elijah Millgram for comments on an earlier draft, and to the Natural Science and Engineering Research Council of Canada for funding.
Appendix A: Formal Details
Thagard and Verbeurgt (1998) define a coherence problem as follows. Let E be a finite set of elements {ei} and C be a set of constraints on E understood as a set {(ei, ej)} of pairs of elements of E. C divides into C+, the positive constraints on E, and C-, the negative constraints on E. With each constraint is associated a number w, which is the weight (strength) of the constraint. The problem is to partition E into two sets, A and R, in a way that maximizes compliance with the following two coherence conditions:
1. if (ei, ej) is in C+, then ei is in A if and only if ej is in A.
2. if (ei, ej) is in C-, then ei is in A if and only if ej is in R.
Let W be the weight of the partition, that is, the sum of the weights of the satisfied constraints. The coherence problem is then to partition E into A and R in a way that maximizes W. Maximizing coherence is a difficult computational problem: Verbeurgt has proved that it belongs to a class of problems generally considered to be computationally intractable, so that no algorithms are available that are both efficient and guaranteed correct. Nevertheless, good approximation algorithms are available, in particular connectionist algorithms from which the above characterization of coherence was originally abstracted.
Here is how to translate a coherence problem into a problem that can be solved in a connectionist network:
1. For every element ei of E, construct a unit ui which is a node in a network of units U. These units are very roughly analogous to neurons in the brain.
2. For every positive constraint in C+ on elements ei and ej, construct an excitatory link between the corresponding units ui and uj.
3. For every negative constraint in C- on elements ei and ej, construct an inhibitory link between the corresponding units ui and uj.
4. Assign each unit ui an equal initial activation (say .01), then update the activation of all the units in parallel. The updated activation of a unit is calculated on the basis of its current activation, the weights on links to other units, and the activation the units to which it is linked. A number of equations are available for specifying how this updating is done (McClelland and Rumelhart 1989). Typically, activation is constrained to remain between a minimum (e.g. -1) and a maximum (e.g. 1).
5. Continue the updating of activation until all units have settled - achieved unchanging activation values. If a unit ui has final activation above a specified threshold (e.g. 0), then the element ei represented by ui is deemed to be accepted. Otherwise, ei is rejected.
We then get a partition of elements of E into accepted and rejected by virtue of the network U settling in such a way that some units are activated and others rejected. Intuitively, this solution is a natural one for coherence problems. Just as we want two coherent elements to be accepted or rejected together, so two units connected by an excitatory link will be activated or deactivated together. Just as we want two incoherent elements to be such that one is accepted and the other is rejected, so two units connected by an inhibitory link will tend to suppress each other’s activation with one activated and the other deactivated. A solution that enforces the two conditions on maximizing coherence is provided by the parallel update algorithm that adjusts the activation of all units at once based on their links and previous activation values. Certain units (e.g. ones representing evidence in an explanatory coherence calculation) can be given priority by linking them positively with a special unit whose activation is kept at 1. Such units will strongly tend to be accepted, but may be rejected if other coherence considerations overwhelm their priority.
Appendix B: ECHO simulation of Haack’s crossword puzzle
Here is a formulation of Haack’s crossword puzzle example as presented in figure 1 above. The propositions representing clues and possible solutions are taken from Haack's sample crossword puzzle, and we have added alternate solutions to produce a problem that is solved by the computer program ECHO. In the possible solutions given below, A1=H means that the entry for square in the top right is H. 2A1=J is a second possible entry for that square.
Hypotheses - possible solutions
(proposition 'A1=H)
(proposition 'A2=I)
(proposition 'A3=P)
(proposition '2A1=J)
(proposition '2A2=O)
(proposition '2A3=Y)
(proposition 'B2=R)
(proposition 'B3=U)
(proposition 'B4=B)
(proposition 'B5=Y)
(proposition '2B2=S)
(proposition '2B3=T)
(proposition '2B4=A)
(proposition '2B5=R)
(proposition 'C1=R)
(proposition 'C2=A)
(proposition 'C3=T)
(proposition '2C1=C)
(proposition 'C5=A)
(proposition 'C6=N)
(proposition '2C6=X)
(proposition 'D1=E)
(proposition 'D2=T)
(proposition '2D1=I)
(proposition 'D4=O)
(proposition 'D5=R)
(proposition '2D4=N)
(proposition '2D5=O)
(proposition 'E2=E)
(proposition 'E3=R)
(proposition 'E4=O)
(proposition 'E5=D)
(proposition 'E6=E)
(proposition '2E4=A)
(proposition '2E5=S)
(proposition 'A2=I)
(proposition 'B2=R)
(proposition 'C2=A)
(proposition 'D2=T)
(proposition 'E2=E)
(proposition '2A2=F)
(proposition '2B2=E)
(proposition '2C2=I)
(proposition '2D2=N)
(proposition '2E2=S)
(proposition 'A3=P)
(proposition 'B3=U)
(proposition 'C3=T)
(proposition '2A3=T)
(proposition '2B3=R)
(proposition '2C3=Y)
(proposition 'B5=Y)
(proposition 'C5=A)
(proposition 'D5=R)
(proposition 'E5=D)
(proposition '2B5=F)
(proposition '2C5=O)
(proposition '2D5=O)
(proposition '2E5=T)
(proposition 'C1=R)
(proposition 'D1=E)
(proposition '2C1=O)
(proposition '2D1=F)
(proposition 'D4=O)
(proposition 'E4=O)
(proposition '2E4=A)
Clues, providing discriminating positive constraints:
(proposition 'C1A "A cheerful start (3).")
(proposition 'C4A "She's a jewel (4).")
(proposition 'C6A "No it's Polonius (3).")
(proposition 'C7A "An article (2).")
(proposition 'C8A "A visitor from outside fills this space (2).")
(proposition 'C9A "What's the alternative (2).")
(proposition 'C10A "Dick Turpin did this to York; it wore 'im out (5).")
(proposition 'C2D "Angry Irish rebels (5).")
(proposition 'C3D "Have a shot at an Olympic event (3).")
(proposition 'C5D "A measure of one's back garden (4).")
(proposition 'C6D "What's this all about (2).")
(proposition 'C9D "The printer hasn't got my number (2).")
Explanations providing solutions to clues, and producing positive constraints:
(explain '(A1=H A2=I A3=P) 'C1A)
(explain '(B2=R B3=U B4=B B5=Y) 'C4A)
(explain '(C1=R C2=A C3=T) 'C6A)
(explain '(C5=A C6=N) 'C7A)
(explain '(D1=E D2=T) 'C8A)
(explain '(D4=O D5=R) 'C9A)
(explain '(E2=E E3=R E4=O E5=D E6=E) 'C10A)
(explain '(A2=I B2=R C2=A D2=T E2=E) 'C2D)
(explain '(A3=P B3=U C3=T) 'C3D)
(explain '(B5=Y C5=A D5=R E5=D) 'C5D)
(explain '(C1=R D1=E) 'C6D)
(explain '(D4=O E4=O) 'C9D)
Other explanations providing alternative solutions and additional positive constraints:
(explain '(2A1=J 2A2=O 2A3=Y) 'C1A)
(explain '(2B2=S 2B3=T 2B4=A 2B5=R) 'C4A)
(explain '(2C1=C C2=A C3=T) 'C6A)
(explain '(C5=A 2C6=X) 'C7A)
(explain '(2D1=I D2=T) 'C8A)
(explain '(2D4=N 2D5=O) 'C9A)
(explain '(E2=E E3=R 2E4=A 2E5=S E6=E) 'C10A)
(explain '(2A2=F 2B2=E 2C2=I 2D2=N 2E2=S) 'C2D)
(explain '(2A3=T 2B3=R 2C3=Y) 'C3D)
(explain '(2B5=F 2C5=O 2D5=O 2E5=T) 'C5D)
(explain '(2C1=O 2D1=F) 'C6D)
(explain '(D4=O 2E4=A) 'C9D)
Contradictory solutions for a letter's square, providing negative constraints:
(contradict '2A2=O '2A2=F)
(contradict '2A3=Y '2A3=T)
(contradict '2B2=S '2B2=E)
(contradict '2B3=T '2B3=R)
(contradict '2B5=R '2B5=F)
(contradict '2C1=C '2C1=O)
(contradict 'C2=A '2C2=I)
(contradict 'C3=T '2C3=Y)
(contradict 'C5=A '2C5=O)
(contradict '2D1=I '2D1=F)
(contradict 'D2=T '2D2=N)
(contradict '2D5=F '2D5=O)
(contradict 'E2=E '2E2=S)
(contradict '2E5=D '2E5=T)
Additional contradictions saying that two different letters can't be in the same space, providing additional negative constraints:
(contradict '2A2=O 'A2=I)
(contradict '2C1=C 'C1=R)
(contradict '2C6=X 'C6=N)
(contradict '2D1=I 'D1=E)
More of the same, but not necessary for correct answer:
(contradict '2A1=J 'A1=H)
(contradict '2A3=Y 'A3=P)
(contradict '2B2=S 'B2=R)
(contradict '2B3=T 'B3=U)
(contradict '2B4=A 'B4=B)
(contradict '2B5=R 'B5=Y)
(contradict '2D4=N 'D4=O)
(contradict '2D5=O 'D5=R)
(contradict '2E4=A 'E4=O)
(contradict '2E5=S 'E5=D)
(contradict '2A2=F 'A2=I)
(contradict '2B2=E 'B2=R)
(contradict '2C2=I 'C2=A)
(contradict '2D2=N 'D2=T)
(contradict '2E2=S 'E2=E)
(contradict '2A3=T 'A3=P)
(contradict '2B3=R 'B3=U)
(contradict '2C3=Y 'C3=T)
(contradict '2B5=F 'B5=Y)
(contradict '2C5=O 'C5=A)
(contradict '2D5=O 'D5=R)
(contradict '2E5=T 'E5=D)
(contradict '2C1=O 'C1=R)
(contradict '2D1=F 'D1=E)
(contradict '2E4=A 'E4=O)
When this input is given to ECHO, it creates a constraint network and uses a connectionist algorithm to select the entries shown in figure 1.
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