Paul Thagard

Philosophy Department

University of Waterloo

Waterloo, Ontario, N2L 3G1


This paper explores the ethical relevance of a precise new characterization of coherence as maximization of satisfaction of positive and negative constraints. A coherence problem can be stated by specifying a set of elements to be accepted or rejected along with sets of positive and negative constraints that incline pairs of elements to be accepted together or rejected together. Computationally tractable and psychologically plausible algorithms are available for determining the acceptance and rejection of elements in a way that reliably approximates coherence maximization. This paper shows how justification of ethical principles and particular judgments can be accomplished by taking into account deductive, explanatory, analogical, and deliberative coherence.

In Toronto in 1995, Paul Bernardo was convicted of the prolonged sexual torture and murder of two young women. Since Canadian law does not admit capital punishment, he was sentenced to life in prison. Some people who had long argued the immorality of capital punishment felt strongly inclined to judge that execution would nevertheless be appropriate for Bernardo's extraordinarily heinous crimes. How should such people overcome the incoherence in their ethical views?

Many ethical theorists have taken coherence to be central to the justification of judgments of right and wrong. For example, Rawls writes "A conception of justice cannot be deduced from self-evident premises or conditions on principles; instead, its justification is a matter of the mutual support of many considerations, of everything fitting together into one coherent view." (Rawls, 1971, p.21; cf. Rawls 1996, pp. 26, 53, etc.) Unfortunately, ethical theory has remained vague about the nature of coherence and about how ethical principles and judgments can be evaluated with respect to coherence. The term "wide reflective equilibrium" is used to describe a state in which a thinker has achieved a mutually coherent set of ethical principles, particular moral judgments, and background beliefs. But how people do and should reach reflective equilibrium has remained poorly specified.

This paper explores the ethical relevance of a precise new characterization of coherence as maximization of satisfaction of positive and negative constraints (Thagard and Verbeurgt, in press). A coherence problem can be stated by specifying a set of elements to be accepted or rejected along with sets of positive and negative constraints that incline pairs of elements to be accepted together or rejected together. Computationally tractable and psychologically plausible algorithms are available for determining the acceptance and rejection of elements in a way that reliably approximates coherence maximization. This paper shows how justification of ethical principles (such as that capital punishment is wrong) and particular judgments (such as that Paul Bernardo should be executed) can be accomplished by taking into account a wide range of coherence considerations.

The next section of this paper briefly describes the new characterization of coherence as multiple constraint satisfaction. To show that ethical decision is a coherence problem, it is necessary to define the elements, the positive constraints, and the negative constraints that operate in ethical thinking. Ethical conclusions require a complex interplay of four different kinds of coherence: deductive, explanatory, deliberative, and analogical. Each of these kinds of coherence involves different kinds of elements and constraints that contribute to an overall conclusion of what ethical principles and judgments to accept. Reflective equilibrium requires integrated assessment of deductive coherence (fit between principles and judgments), explanatory coherence (fit of principles and judgments with empirical hypotheses), deliberative coherence (fit of judgments with goals), and analogical coherence (fit of judgments with other judgments in similar cases).

I take naturalism to be the view that philosophy overlaps substantially with the empirical sciences, so that philosophical theories and analyses should be closed tied to the results of scientific research. In particular, cognitive naturalism holds that many philosophical issues are intimately connected with the cognitive sciences, including psychology, linguistics, neuroscience, and artificial intelligence. Cognitive naturalism is currently the dominant position in the philosophy of mind, and it has made substantial contributions to epistemology and the philosophy of science. By applying a psychological/computational theory of coherence, this paper demonstrates the relevance of cognitive naturalism to ethics.



Assessment of ethical coherence requires balancing the mutual support and mutual incompatibility of an interconnected set of principles, judgments, and other beliefs. When two of these mutually support each other, their coherence will tend to make them be accepted or rejected together. If an ethical principle entails a particular judgment, we will tend (in the absence of any other considerations) to accept them together or reject them together. For example, the principle that capital punishment is wrong entails the judgment that Paul Bernardo should not be executed. There is thus a positive constraint between the principle and judgment that establishes a tendency either (1) to accept both that capital punishment is wrong and that Paul Bernardo should not be executed or (2) to reject both. Negative constraints between two elements are such that if we accept one of the elements, we will tend to reject the other. For example, if we believe that capital punishment is right under certain circumstances, we will reject the belief that capital punishment is wrong. Ethical discussion is very difficult, because particular judgments often invoke conflicting principles so that it is not possible to satisfy all the constraints that arise. I may desire that Paul Bernardo be executed for the sake of the victim’s families, for deterrence of similar offenders, or because he deserved it; but I somehow have to reconcile these reasons with my principle that unnecessary killing is wrong. All these reasons establish positive constraints on my thinking that cannot be simultaneously satisfied along with the negative constraints that enforce some degree of consistency in my judgments. To reach reflective equilibrium, I need somehow to figure out how best to accept some beliefs and reject others in a way that maximizes satisfaction of the constraints.

Coherence can be understood in terms of maximal satisfaction of multiple constraints, in a manner informally summarized as follows (Thagard and Verbeurgt, in press):

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, facilitation, 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 which is 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 the appendix. Connectionist (neural network) models that are commonly used in cognitive science provide a powerful way of 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. The appendix outlines in more detail how connectionist networks can be used to compute solutions to coherence problems.

Ethical judgment is a complex coherence problem, because it involves at least four different kinds of coherence: deductive, explanatory, deliberative, and analogical. To show the relevance of each of these to ethics, we need to describe the kinds of elements and constraints that are relevant to reaching ethical conclusions.

For deductive coherence as applied to ethics, the elements are propositions, including both general principles and particular moral judgments. The main positive constraint is established by the relation of entailment: if one proposition entails another, then there is a positive constraint between them which will tend to make them either accepted together or rejected together. Such a constraint operates quite differently from logical inference, where from p -> q and p we can infer q by Modus Ponens, and from p -> q and not-q we can infer not-p by Modus Tollens. Coherence judgments do not have the kind of step-by-step linear reasoning found in formal logic. Rather, they require fitting everything together using constraints that are typically soft rather than hard. A soft constraint produces a tendency to accept two positively constrained elements together, but this constraint can be overruled if overall coherence maximization suggests that one of the elements be accepted and the other rejected.

In ethics, positive constraints arise because principles entail judgments, as when the principle that capital punishment is wrong entails that Paul Bernardo should not be executed. Alternatively, the principle that capital punishment is justified for heinous murders implies that Bernardo should be executed. Negative constraints arise because of contradictions between propositions, for example between the two principles just stated and between the two judgments just stated. Figure 1 shows a simple constraint network that shows the relations among these four propositions.



Obviously, the constraint network shown in figure 1 does not offer a solution to the coherence problem, since there are two equally coherent solutions: accepting that capital punishment is wrong and that Paul Bernardo should not be executed while rejecting that capital punishment is justified and that he should be executed, or vice versa. Figure 1 should be expanded to include higher-level principles such as that killing people is wrong which entails that capital punishment is wrong, as well as additional judgments about particular cases of capital punishment. Evaluation of ethical coherence based solely on fit of principles and judgements will generally be open to the standard objection to coherence theories that incompatible sets of propositions can be equally coherent. We will see, however, that broadening ethical coherence to incorporate judgments of explanatory and deliberative coherence can help to overcome this problem by introducing empirical information.

Deductive coherence is important outside ethics also, for example in axiom selection in mathematics. Rarely are axioms selected because they are self-evident. Rather, axioms are selected because they entail the desired theorems, which are in turn accepted because they follow from the axioms. Mathematicians do not proceed from axioms to theorems, nor backwards from desired theorems to axioms, but rather attempt to come up with deductively coherent packages of axioms and theorems (Russell 1973, p. 279). Similarly, ethical principles are not self-evident, but must be selected on the basis of deductive coherence with particular judgments, taking into account additional kinds of coherence.

For much reasoning in science and ordinary life, the most important kind of coherence is explanatory rather than deductive. Scientists accept a theory if it provides the best explanation of the evidence, where "best explanation" is evaluated based on an overall coherence judgment (Thagard 1989, 1992). For explanatory coherence, the elements are propositions, including hypotheses and evidence statements. If one proposition explains another, then there is a positive constraint between them, as when a hypothesis explains a piece of evidence. Contradictory hypotheses, and ones that compete to explain the same data, have negative constraints between them. For example, Darwin's theory of natural selection consists of propositions that have positive constraints with biological evidence and negative constraints with creationist hypotheses; maximizing coherence requires accepting Darwin's theory and rejecting creationism.

Ethics requires attention to explanatory coherence whenever (as frequently occurs) ethical decisions depend in part on evaluation of empirical hypotheses. Particular judgments such as that Paul Bernardo should be punished depend on factual claims such as that he actually committed the crimes of which he was accused. General principles such as adoption of capital punishment can also be closely tied to factual claims: one common argument for capital punishment is that it is desirable as a deterrent to future crimes, which depends on the empirical hypothesis that having capital punishment as a possible punishment reduces crimes of certain sorts. Evaluation of this hypothesis depends on a very complex evaluation of evidence, such as comparison of countries or states with and without the death penalty. The hypothesis that capital punishment is a deterrent must mesh with a variety of sociological and psychological evidence if it is to be put to ethical use.

How can deductive and explanatory coherence interconnect? The principle that preventing serious crimes is good, and the empirical hypothesis that capital punishment helps to prevent crimes, together entail that capital punishment is good. These three propositions form a mutually constraining package shown in figure 2. Unlike a pure deductive principle or moral judgment, however, the empirical hypothesis is subject to a kind of coherence in which empirical evidence is given priority. Priority does not mean that the results of observations must be accepted, only that there is a soft constraint that tends to make them accepted. Now we begin to see how coherence judgments might discriminate objectively between competing sets of principles and judgments: whenever the entailment relation between principles and judgements depends on empirical hypotheses, the coherence of the ethical judgments can be affected by the explanatory coherence of the empirical hypotheses. An opponent of capital punishment might argue that killing innocent people is wrong, and that capital punishment sometimes kills innocent people, so that capital punishment is wrong. This entailment depends on the empirical hypothesis that sometimes innocent people are executed in countries and states that have capital punishment. People who are convinced on the basis of explanatory coherence that the empirical hypothesis that capital punishment leads to execution of innocent people, and convinced on the basis of explanatory coherence that the hypothesis that capital punishment serves as a deterrent is false, will tend to find more coherent the conclusion that capital punishment is wrong.

Figure 2. Deductive coherence depending on an empirical hypothesis.

Thus evaluation of ethical principles requires considerations of explanatory coherence as well as deductive coherence, striving for wide rather than narrow reflective equilibrium. But deductive and explanatory coherence are quite similar, in that both involve propositional elements with positive and negative constraints that can be maximized. The interpenetration of deductive and explanatory coherence gives us some hope that ethical deliberation can be affected substantially by empirical evidence. Adding deliberative coherence shows another way of broadening ethical coherence.

Standard decision theory says that rationality consists in maximizing the satisfaction of preferences or utilities, but says nothing about why people have their preferences and utilities. In contrast, Thagard and Millgram (1995; Millgram and Thagard, 1996) have developed a coherence theory of decision making that involves the evaluation of personal goals as well as actions that potentially accomplish those goals. According to this theory, the elements in deliberative coherence are actions and goals, and the primary positive constraint is facilitation: if an action facilitates a goal, then there is a positive constraint between them. For example, the action of executing Paul Bernardo (or the action of life imprisonment) will facilitate the goal that Paul Bernardo not murder again. Negative constraints arise because some actions are incompatible, since, for example, we cannot both execute Bernardo and imprison him for 50 years. Just as explanatory coherence gives some priority to propositions that state empirical evidence, so deliberative coherence gives some priority to intrinsic goals, ones that an agent has for basic biological or social reasons rather than because they facilitate other higher goals. But just as empirical evidence can be overridden for reasons of explanatory coherence, intrinsic goals can also be revised and overridden for reasons of deliberative coherence, which evaluates intrinsic goals (final ends) as well as instrumental goals and actions.

Deliberative coherence is relevant to ethical decisions that take into account the consequences of actions. Someone might argue that executing Bernardo will be much cheaper than imprisoning him under special security for life; thus execution facilitates the goal of saving Canadian taxpayers money. The deterrence-based argument for capital punishment can be reframed as a matter of deliberative coherence: the action of executing murderers facilitates (it is claimed) the goal of preventing murders. Putting it in this way makes it clear how deliberative coherence depends in part on explanatory coherence. The judgment that an action facilitates a goal depends on a causal judgment about the relation between the action and the goal, where the plausibility of the causal judgment is a matter of explanatory coherence.

In individual decision making, an agent may maximize coherence of actions and goals for the agent alone. Ethical decisions, however, require us to consider what is objectively good, not just for the agent, but also for other people involved. Something is non-morally good for an agent if and only if it would satisfy an objective interest of the agent (Railton, 1986). Normatively, actions should be chosen on the basis of the extent to which they facilitate the objective interests (goals) of all concerned. Thus in deciding whether to execute Paul Bernardo, we take into account the interests of the victims' families, Bernardo himself, and anyone else affected.

Whereas deductive coherence involves a quasi-Kantian concern with general moral principles, deliberative coherence involves a consequentialist concern with goals of those affected by ethical decisions. From the point of view of a coherence theory of ethics, the Kantian and consequentialist positions need not be seen as radically conflicting. Rather, each identifies one kind of coherence that goes into an overall judgment of right and wrong. In everyday debates on ethical issues, people often swing between questions of principle and questions of practical effects. Seeing ethical coherence as involving both deductive and deliberative coherence shows why this can be so.

Questions of objective interests are closely tied with empirical hypotheses about the wants/interests mechanism of human beings. Evidence from biology, psychology, sociology, and anthropology will be needed to evaluate hypotheses concerning what kinds of actions contribute to the interests of human beings. Thus deliberative coherence is intimately tied with explanatory-coherence evaluation of hypotheses about the nature of humans and their societies. Deliberative coherence does not reduce to explanatory coherence, but depends on it in very useful ways that allow for the possibility of revising views about what is good for people and thereby revising decisions about what to do. For example, the families of Paul Bernardo's victims may naturally want to see him killed, but whether execution would bring some relief from their grief is an empirical question. Without psychological evidence about the effects of executions in similar cases, we do not have grounds for saying whether execution is really in the objective interests of the victims' families.

Not all ethical argument considers general principles (deductive coherence) or consequences (deliberative coherence). Often, people argue for moral principles and judgments analogically, supporting a conclusion in one case by comparing it to a similar case whose moral status is more obvious. The morality of capital punishment is similarly subject to analogical dispute: is execution of a murderer comparable to killing a defenseless victim, or is it somehow similar to acts of self-defense? Applying an analogy to an ethical issue requires transferring a moral judgment from an accepted case to a contested case: if capital punishment is relevantly similar to killing a defenseless victim, an act that is obviously wrong, then capital punishment can also be judged to be wrong. Assessing relevant similarity requires establishing correspondences between the source analog about which an ethical judgment has already been made and the target analog to which the ethical judgment is to be applied.

Establishing correspondences between a source and a target analog can be viewed as a coherence problem involving several different kinds of constraints. (Holyoak and Thagard, 1995). The elements are hypotheses about what features of the analogs correspond to each other. Here is a simple representation of two analogs:

source target

holds (abductor, victim) holds (state, prisoner)

kill (abductor, victim) execute (state, prisoner)

wrong (killing)

To perform an analogical mapping between these analogs, we need to create elements for mapping hypotheses such as that kill in the source analog corresponds to execute in the target analog and that victim in the source analog corresponds to prisoner in the target. In the multiconstraint theory of analogy, positive constraints are based on semantic and visual similarity, with people tending to map semantically similar predicates such as kill and execute. Additional positive constraints are based on syntactic structure: if holds in the source is mapped to holds in the target, then the corresponding arguments will also be mapped: abductor to state and victim to prisoner. Structure also provides negative constraints based on a preference for one-to-one mappings; accepting the mapping hypothesis that abductor corresponds to prisoner will tend to lead to rejection of the mapping hypothesis that abductor corresponds to victim. Finally, an additional set of positive constraints arises from the purpose of the analogy, what it is designed to accomplish. In ethical deliberations, the purpose of the analogy is to transfer the ethical judgment about the source over to the ethical judgment about the target.

Analogical arguments are rarely convincing on their own, but they can contribute to the overall coherence of a view. Darwin, for example, used an analogy between artificial and natural selection as one of the ingredients in his case for the explanatory coherence of his theory of evolution. Similarly, analogy can help to establish the deductive and deliberative coherence of an ethical conclusion. A defender of capital punishment might argue that just as it may be legitimate to kill an attacker such as Bernardo in self-defense, so it may be legitimate for society to defend itself against murderous psychopaths like Bernardo by executing them. The argument involves both deductive coherence with fit between principles and judgments (killing in self-defense is right/victims killing Bernardo would have been justified) and analogical coherence (comparison between killing be self-defense and execution). Of course, a critic of capital punishment will attempt to undermine this analogy and employ different ones to suggest the applicability of different principles.

From the perspective of the multicoherence theory of ethics proposed here, reaching ethical conclusions turns out to be a complex psychological process. Normatively, people can proceed as follows in establishing ethical principles and judgments:

1. Identify deductive elements (principles, judgments) and positive and negative constraints among them.

2. Identify deliberative elements (actions, goals) and positive and negative constraints among them.

3. Identify explanatory elements (hypotheses, evidence) and positive and negative constraints among them.

4. Identify constraints linking the explanatory elements with the deductive and deliberative elements.

5. Identify analogical elements (mapping hypotheses) and positive and negative constraints among them.

6. Identify constraints linking the analogical elements with the deductive, deliberative, and explanatory elements.

7. Finally, use algorithms to maximize coherence by accepting some elements and rejecting others in the way that approximately maximizes satisfaction of the positive and negative constraints.

While this procedure is normatively appealing, it is probably too much to expect of people given their psychological resources. At the level of consciousness, working memory is far too limited to simultaneously entertain all the different elements that go into such a complex coherence judgment. Perhaps simultaneous maximization of all the constraints goes on automatically at the unconscious level, just as the brain makes sense of complex visual inputs to produce a coherent interpretation of a scene. More likely, though, the mind must proceed more sporadically, alternating between focusing on one kind of coherence and another. Instead of systematically identifying different kinds of constraints, people focus for a while on a particular kind of coherence such as the deductive fit between principles and judgments, then shift to other kinds of coherence such as deliberative. Within each focus, the mind reaches a tentative coherence conclusion based on the elements and constraints currently active, producing evaluations of elements that can then feed into coherence calculations involving different elements and constraints. This sporadic, unsystematic way of reaching ethical conclusions is obviously subject to the main weakness in any imperfect maximization procedure: instead of reaching a global maximum which achieves the highest possible extent of constraint satisfaction, people may get stuck in a local maximum which, although better than immediately available alternatives, is still inferior to other ways of maximizing constraint satisfaction. One charitable way of explaining the incessant controversies in ethics is by noting the complexity of ethical coherence and conjecturing that disputants have simply fallen into different local maxima.

Those who find inconsistencies in their ethical views, such as the people mentioned at the beginning of this paper who believe both that capital punishment is wrong and that Paul Bernardo should be executed, can at least attempt to implement the seven-step procedure stated at the beginning of this section. The result should be to bring to bear a wide complex of principles, judgments, actions, goals, hypotheses, evidence, and mapping hypotheses in a way that may suggest how to either abandon the principle that capital punishment is wrong or to reject the judgment that Paul Bernardo should be executed. In either case, coherence with a large number of other considerations will be what determines ethical belief change.

It is important to note that the process by which people reach ethical conclusions is often social: "We press each other toward coherence, and these pressures help nudge us toward consensus." (Gibbard, 1990, p. 204). From the perspective of the individual, it may seem rather arbitrary what elements (concepts, propositions, analogs, etc.) make up the coherence network, but the arbitrariness is much diminished in a social context in which people with different ethical judgments introduce competing elements to be integrated into each other's coherence networks. We do not have to worry about there being an unlimited number of trivial elements that are minor variants of each other (as in the "grue" predicates in confirmation theory), so long as ethical coherence is viewed as taking place in human minds in real social contexts.

The complexity of ethical coherence is further illustrated by debates concerning the morality of abortion, which contain a variety of deductive, explanatory, deliberative, and analogical considerations. Deductive arguments are used by both sides of the issue. Defenders of abortion argue that the illegitimacy of the state's banning abortion follows from a right to privacy, whereas critics of abortion claim that its immorality follows deductively from the principle that murder is wrong. Of course, neither of these deductive arguments is convincing to the other side, since they depend on the legitimacy of the principle stated and on the acceptability of additional premises required to make the argument sound, for example that abortion is murder.

Other arguments invoked in the abortion case point to issues of deliberative coherence, for example on the pro side that prohibiting abortion will lead to injuries of numerous women undergoing illegal abortions, and on the con side that abortion causes distress both to fetuses and to women who have abortions. As in the issue of capital punishment, deliberative coherence often interacts with explanatory coherence: factual claims such as that making abortion illegal would cause suffering both from illegal abortions and unwanted children need to be empirically evaluated on the basis of how well they fit with theories and observations in psychology and sociology. Explanatory coherence can also interact with deductive coherence, for example when theists infer that abortion is wrong because God forbids it. This deductive argument presupposes that there is a God, a hypothesis that can be evaluated on the basis of its explanatory coherence.

Analogical coherence also enters into judgments about the morality of abortion, since arguments often rely on comparison to practices such as infanticide or hypothetical cases such as being involuntary connected with an adult (Thomson, 1971). Analogy also plays a major role in legal judgments, when abortion is treated as a case that should be settled in ways similar to precedents such as the judgment that prevented states from banning contraception. Whether abortion is deemed as analogous to legally acceptable practices such as contraception or as analogous to proscribed practices such as infanticide depends on a variety of deductive, explanatory, and deliberative considerations. Analogies contribute to the assessment of explanation and plans, at the same time as the assessment of explanations and plans contributes to the evaluation of analogies. There is no circularity here, because all kinds of coherence can be simultaneously computed by global maximization of satisfaction of different kinds of constraints.


This paper has proposed a multicoherence theory of ethical thinking according to which people reach ethical conclusions by approximately maximizing the satisfaction of deductive, explanatory, deliberative, and analogical constraints. The theory is intended to be both descriptive and prescriptive, characterizing how people think ethically when they are thinking at their best. There are at least four reasons why this theory should be adopted as a normative account of how people should reason about right and wrong.

First, the multicoherence theory of ethics can handle the complexity of moral reasoning. This paper has shown the relevance of all four kinds of coherence to the evaluation of capital punishment as right or wrong. It should not be hard to show that other major ethical issues similarly involve a mixture of deductive, explanatory, deliberative, and analogical considerations.

Second, the multicoherence theory is naturalistic in that it is consistent with substantial amounts of evidence showing that the processes of parallel constraint satisfaction are important in human cognition (Holyoak and Spellman, 1993; Thagard (1996, ch. 7). If vision, language understanding, hypothesis evaluation, concept application, and analogy are all coherence processes, it should not be surprising that ethical thinking is also a coherence process. The ethical theory developed in this paper is not, however, naturalistic in the sense of claiming that moral judgments are reducible to scientific facts about the natural world. Judgments about right and wrong are often closely tied with scientific judgments, as we saw with the interconnections among deductive, deliberative, and explanatory coherence. But these connections do not reduce deductive and deliberative coherence to explanatory coherence. Ethical questions are not simply factual questions, but they are sufficiently linked with empirical issues that we can hope that agreement on psychological, biological, and economic issues can contribute to agreement on ethical issues. My multicoherence account of coherence provides a much fuller account of ethical inference than is found in recent naturalistic accounts that emphasize either perception-like neural networks (Churchland, 1995; Flanagan, 1996) or metaphor (Johnson, 1993, 1996; Lakoff, 1996). These accounts capture aspects of conceptual and analogical coherence, but neglect the contributions of deductive and deliberative coherence to ethical judgments.

Third, the coherence view of ethics proposed here avoids the two major problems of foundationalist approaches to ethics and epistemology. The first problem is that, for epistemology as for ethics, no one has ever been able to find a set of foundations that even come close to receiving general assent. The coherentist approach has no need for a priori intuition or contractarian artifice. The second problem is that proposed foundations are rarely substantial enough to support an attractive epistemic or ethical edifice, so that foundationalism degenerates into skepticism. In contrast, the multicoherence theory of ethics, like coherence theories of knowledge, recommends jumping into issues in midstream, revising ethical beliefs as necessary in order to increase overall coherence, without attempting the impossible task of rederiving all ethical principles and judgments from first principles.

Finally, the multicoherence theory proposed here has the advantage over previous coherentist approaches to ethics that it employs a clearly stated and computationally implemented account of what it is to maximize coherence. Explanatory, deliberative, and analogical coherence all have computational models that have been applied to numerous complex real-world cases. Putting these kinds of coherence together with the deductive coherence of ethical principles and judgments is non-trivial. But the sporadic, incremental way in which people generally shift focus among different kinds of coherence can be seen as a rough approximation to a more ideal process of global maximization of constraint satisfaction. We do not always maximize coherence, but sometimes we manage nevertheless to make quite good sense of right and wrong.


We can 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.