1. Introduction

The purpose of this paper is to examine a novel notion of causality, namely, fuzzy set-theoretic causality. It is a conceptual product of the tradition of research that has introduced fuzzy sets in connection to fuzzy logic and has inspired fuzzy technology. Over the last decade, a number of causal models of the same type have appeared in the scientific literature and have circulated widely from one scientific discipline to another. However, they have received no philosophical attention. In this paper, I will discuss the value and limitations of this type of model, and will read into it several dimensions of philosophical significance.

The broadest dimension of philosophical significance I bring to bear is that of the relation between science and philosophy. Causality is a notion common to scientific and philosophical discussion. Fuzzy-set causality is a notion introduced by scientists and, thus, an example of a strong interaction between both. The point of my discussion is not, however, to show how philosophy should dictate or justify science. Rather, it should promote the idea that a common ground for discussion may be both relevant and helpful to both enterprises. The fuzzy-set approach is also relevant to general issues in set theory and logic. But this is of less interest for the purpose of this paper. Fuzzy-set causality also illustrates an aspect of mathematical science that I have discuss in detail elsewhere: mathematical formalisms can be endowed with conceptual contents, in the form of ontological and epistemic categorization schema—we may say ‘a priori’—beyond the purely formal and mathematical structure, and aside from any specific theory in which they are applied. That interpretation often plays a mediating and guiding role in their application. As a consequence, such formalisms, including fuzzy-set causality, can circulate and extend their domain of application beyond one single theory and scientific discipline. In this regard, fuzzy-set causality is an example of dynamical unification in the sciences. For the specific purpose of this paper, I am interested in the difference that this approach makes to empirical research and how this relates to empirical method. But some of my considerations have much broader relevance. The models I discuss below show the relevance of fuzzy-set notions in scientific representation. The central question is: Can the formal apparatus that fuzzy set theory brings to scientific representation provide an adequate account of causal relations or a criterion for detecting them?

2. What Is a Fuzzy Set?

Lotfi Zadeh introduced the notion of fuzzy sets to model vague predicates and informal reasoning in the form of fuzzy logic (Zadeh Reference Zadeh1965). Let X be a classical set of objects, denoted by x, called the universe. Let C be a classical subset of X and $m(x) $ be a function from X to the pair of values {0, 1} such that if $x\in C$ , $m(x) =1$ , and if $x\not\in C$ , $m(x) =0$ . In classical sets, then, elements of the universe either belong to a set or they don't. The set is characterized by a sharp boundary and is identical with its members. By contrast, a fuzzy set, F, is a subset characterized by the set of pairs

where $m(x) $ represents the degree of membership with possible values ranging over the real interval [0, 1]. In each context of application of the predicate F, the fuzzy set F will be normalized if there exists one x, such that $m(x) =1$ .

There is no sharp boundary. The set is characterized by a membership function. In this way, fuzziness formalizes a kind of deterministic uncertainty and ambiguity. For instance, if F represents the vague predicate TALL, the fuzzy set of tall individuals will include very tall individuals, with m closer to 1, and short individuals, with m closer to 0. As an interpretation of fuzzy logic, the membership function designates a degree of truth valuation for statements of the form ‘x is F’. This provides the extensional semantics for multivalent logic.Footnote ¹

For fuzzy sets, one can introduce an algebra of set-theoretic operations. The most basic ones, introduced by Zadeh, are:

• • union: $m(F\cup G) (x) =\mathrm{max}\,(mF(x) ,\,mG(x)) $

• • intersection: $m(F\cap G) (x) =\mathrm{min}\,(mF(x) ,\,mG(x)) $

• • complement: −F is the set of x such that $m^{\prime }(x) =1-mF(x) $ ,

where $mF(x) $ is the degree of membership in F (but $m^{\prime }(x) $ is the degree of membership in −F) and −F is the complement of F. Unlike for classical sets, for fuzzy sets $F\cup -F\neq \Omega $ , where Ω is the universe, and $F\cap -F\neq \emptyset $ . And finally,

• • subset: $F\subset G\,\mathrm{iff}\,\,mF(x) \leq mG(x) $ , for all x.

If $\sum mF(x) =M(F) $ is the size of F and $\sum mG(x) =M(G) $ , the size of G, then $F\subset G\,\mathrm{iff}\,\,M(F) \leq M(G) $ . These operations form the basis for the models of causality I present and discuss in the next sections.

3. Fuzzy-Set Empiricism and Local Subjective and Enhanced Coordination of Theory and Data

The philosophical problem of empiricism is the determination of the coordination of theory and data. Duhem and Neurath raised the issue of the indeterminacy in the coordination of precise technical terms of theory and vague concepts of perception, viz., practical facts and protocol sentences, respectively.Footnote ²

Charles Ragin and other proponents of fuzzy-set causality note that in many scientific contexts the data is precise—GDP, income, etc.—whereas what counts as the ‘theory’ is now formulated in vague terms: dangerous neighborhoods, financially secure households, rich countries, social stability, religious fundamentalism, poverty. Fuzzy sets defined for precise data or individuals in a population operationalize those theoretical categories. Membership functions provide a sort of bridge principle in the form of a scale ranging over the meaningful interval of variation.

The membership function introduces two paradoxically inseparable aspects of fuzzy empiricism. On the one hand, the locality and partial subjectivity of the membership function is an irreducible aspect of fuzzy empiricism. Zadeh himself and other researchers have made clear that the assignment of membership values—and the corresponding truth-values of propositions—is subjective and local (Zadeh Reference Zadeh1975; Dubois and Prade Reference Dubois and Prade1980; Haack Reference Haack1996; Ragin Reference Ragin2000). It is at least partly subjective in all cases because at least one assignment is subjective, namely, to a standard individual, relative to which other individuals get their membership degree. It is subjective because at least that assignment is not based on any intersubjective, formal or material method of assignment that would ensure consensus. It might, however, not be arbitrary or alien to consensus. Instead, especially in the scientific case, it might be guided by background knowledge or assumptions and motivated by certain values. It is based on expert judgment. Consensus among experts can be adopted as a minimal form of objectivity (Helgason and Jobes Reference Helgason and Jobe2003). It is also local, because the assignment of membership degrees varies from context to context (small Chihuahuas are addressed differently from small St. Bernards). Here scientists appeal to expert judgment and consensus. This is, however, a pervasive aspect of scientific practice. On the other hand, the acceptance of degrees of membership enhances the pool of empirical data beyond the clear or prototypical cases, viz., with $m=1$ . Note that the degree of membership in a set provides the degree of truth-value of propositions relating a member of the set and the property the set represents. This fact introduces a notion of approximation.

The most eloquent proponent on epistemological and methodological issues involving fuzzy-set-theoretic relations is Ragin. In the context of sociology, he criticizes correlational studies for focusing on patterns of co-variation in which the independent variables that explain the outcome are always independent and represent pre-constituted homogeneous populations. Heterogeneity in populations stands only for the possibility of error; and complexity in the form of diverse co-relevant and possibly interacting conditions only plays the role of confounding the analysis. And if one is interested in establishing necessary or sufficient conditions (or sets thereof), perfect correlations fail to provide evidence (Ragin Reference Ragin2000).

Interestingly enough, Ragin's epistemic values differ from the considerations behind Helgason and Jobe's model in medicine (see below). They see in fuzzy-set models of causality the opportunity to bypass the population-oriented basis of statistical models and think causally and quantitatively about one single patient and his context.

Set theoretic relations are then of clear relevance and importance in the job of representing and theorizing phenomena in contemporary science, from engineering to physiology to social sciences. The same virtues extend to relations on non-empirical levels of theories or models. It is a separate, but not independent, question whether that relevance can extend to a defensible causal interpretation of fuzzy-set-theoretic relations.

4. Fuzzy Set-Theoretic Models of Causality

4.1. Kosko's Models

The first two models are due to Bart Kosko (Reference Kosko1986, Reference Kosko1990). They inspired the subsequent ones. In the first one, Kosko introduces a definition of causality between conditions represented by fuzzy sets. Given conditions (concepts) C _i, C _j such that$C_{i,j}=(Q_{i,j}\cup -Q_{i,j}) \subseteq M_{i,j}$, where Q _i,j and −Q _i,j are fuzzy quantity sets (e.g., stability) and their respective complements (e.g., instability), and M _i,j are modifiers (e.g., social):

C _i causally increases C _j iff $(Q_{i}\cap M_{i}) \subset (Q_{j}\cap M_{j}) $ and $(-Q_{i}\cap M_{i}) \subset (-Q_{j}\cap M_{j}) $ .

C _i causally decreases C _j iff $(Q_{i}\cap M_{i}) \subset (-Q_{j}\cap M_{j}) $ and $(-Q_{i}\cap M_{i}) \subset (Q_{j}\cap M_{j}) $ .

With these definitions Kosko generalized R. Axelrod's 1970s Cognitive Maps in political science to Fuzzy Cognitive Maps that represent causal reasoning as calculus of causal influence. The calculus is expressed in terms the algebra of adjacency matrices C _ij, where map nodes C _ij represent the sign of causal influence between C _i and C _j.

In general, the causal relation A causes B is defined in terms of subsethood, or set inclusion, $A\subset B$ . The subsethood relation clearly models and generalizes the bivalent logical implication in the form of truth-functional material conditional, $A\supset B$ .

In the second model, Kosko introduces the notion of degree of subsethood formulated more generally, in terms of inclusion in the power set. It is meant as a fuzzy generalization of Zadeh's subsethood relation. The power set of B, $\Pi (B) $ , is set of all subsets of B. If $A\subset B$ , $A\subset \Pi (B) $ . He motivates a definition of degree of subsethood, $S(A,\,B) $ , as

which he identifies with $S(A,\,B) =M(A\cap B) / M(A) $ . M is now the size of the set.

$S(A,\,B) $ represents the fuzzy causality relation between A and B. A and B now stand for properties or conditions involving all sorts of physical systems. Kosko frames the causal model in a geometric formalism: Sets {x _n} representing different predicates can be represented, instead of as functions (Zadeh), as points in an n-dimensional hypercube representing the power set, each dimension representing the degree of membership of an element of a set with a unit distance representing the range of possible membership values. He believes that the association of the general set-theoretic results with geometric properties of the hypercube provides the causal models and other results with both an intuitive presentation and a more general foundation.

4.3. Helgason and Jobe's Model

The fourth model I want to consider is due to Cathy Helgason and Thomas Jobe (Helgason and Jobe Reference Helgason and Jobe2003). It is meant as an alternative to population-statistics-based medicine. The latter, they object, assumes an artificial separation among variables. Their goal is to provide a perception-based representation of the physiological interaction of elements in an individual patient, and to view the evolution over time with given initial conditions.

They adopt Kosko's hypercube formulation with sets as points. They take relevant causal knowledge to be knowledge of the evolution in the hypercube of the causal nexus of interacting conditions of an individual patient, from the initial set of conditions, A, to a later set, B, after a change with or without intervention. This is represented by a trajectory along the corresponding points in the hypercube of relevant variables.

In this model, a rich conceptual scaffolding for causal knowledge is constructed in terms of quantities they call Formal Causal Ground and Clinical Causal Effect, which rest on notions of necessary and sufficient causal conditions, viz., necessary and sufficient causal ground. These are built, in turn, on the notions of the fuzzy subsethood relation, S, which is built, in turn, upon the fuzzy measure M and fuzzy intersection relation:

Necessary Causal Ground (NCG) = $[ M(A\cap B) / M(A) ] ^{2}=S(A,\, B) ^{2}$ .

Sufficient Causal Ground (SCG) = $[ M(A\cap B) / M(B) ] ^{2}=S(A,\, B) ^{2}$ .

A represents the initial set of conditions—internal and contextual—of the patient that are considered relevant. B represents the subsequent set of conditions, following therapeutic intervention. The measure M represents, in their interpretation, the degree of ‘connectedness’ of ‘co-located’ conditions of the patient. Into these set-theoretic relations, Helgason and Jobe read an intricate and multilayered causal story with specific medical significance. NCG represents the degree to which the (causal) antecedent, A, is present—as suggested by the extensional set-theoretic relation of intersection—in the (causal) consequent, B, the putative effect. The model is clearly a set-theoretic interpretation of logical implication; as the authors put it, if A is a necessary condition for B, then there is no B without A. Translated by the authors into more dynamical and modal terms, NCG represents the degree of causal necessity of A for the occurrence of the effect, B; as indicated by the expression in terms of the ratio of the intersection to the measure $M(A) $ . SCG, by contrast, measures the degree of presence of B in the presence of A, as a set-theoretic interpretation of logical implication; if A is a sufficient condition for B, there is no A without B, where the causal emphasis on B is expressed by the ratio of the intersection to $M(B) $ . Notice that the different ratios mark out an alleged causal asymmetry between ‘no B without A’ and ‘no A without B’.

From these quantities they define Formal Causal Ground as

FGC is the joint or simultaneous degree of necessity and sufficiency of the set of initial conditions, in their causal reading of the formalism, the degree to which A causes B. More specifically, the smaller the degree of causal relevance of A to B, the smaller the ‘causal flow’ of A into B, and the larger the ‘causal flow’ of extraneous influences on B.

The Clinical Causal Effect is

CCE is the measure of clinical intervention. In Helgason and Jobe's creative causal reading, FCG and CCE represent causal flows. FCG represents an “involutionary causal flow from A in B”—or “involutional cause of initial conditions”—and CCE represents an “evolutionary flow of B out of A”—or “evolutional cause of the final state” (Helgason and Jobe Reference Helgason and Jobe2003, 85, 87). Finally, out of FCG and CCE, they define Clinical Causal Efficiency, K, as the ratio between these two causal flows,

The name is, however, misleading. They interpret its inverse, $1/ K$ , instead, as “the relative dosage needed of the drug or process needed to move one unit of CCE and [it] tells us how many FCGs or initial conditions are overcome by one unit of the drug” (88). In other words, K is a measure of the efficiency of one unit of drug. As such, the causal efficacy is the measure of the power to overcome the resistance of, and thus is inversely proportional to, the ‘connectedness’ or ‘interactive power’ of the interacting conditions defining the initial state or causal nexus of the patient. $1/ K$ is the dosage needed to counteract that power.

One of the clinical studies they present illustrates the application of their fuzzy-set causal formalism to the case of strokes, produced by platelet aggregation in blood. Different kinds of aggregations are induced by three agonists: adenosine diphosphate (ADP), epinephrine (Epi) and collagen (Coll). They are interested in studying the degree of aggregation inhibition induced by aspirin alone and by aspirin in combination with clopidogrel. The fuzzy hypercube in which the state of the set of conditions of the patient is represented as a point has three dimensions, each representing fuzzy scales—between 0 and 1—of degrees of each relevant condition determined by expert consensus: ADP-, EPI-, and COLL-platelet aggregation. In one patient, with aspirin, A = {ADP, Epi, Coll} = {0.2, 0.4, 0.8}, and after clopidogrel is added, B = {ADP, Epi, Coll}= {0.2. 0.8, 0.8}. Applying the rules of the fuzzy causal calculus, $M(A) =1.4$ and $M(B) =1.8$ , so

Thus, NCG = 1 and SCG = 0.6049, and the patient's FCG = 0.6049, which is the degree to which A causes B (or “the degree to which the initial inhibition of platelet aggregation involutes that of the addition of clopidogrel into it” (93)). The patient's CCE, “the degree to which the inhibition of platelet aggregation by clopidogrel evolves that by aspirin alone,” is the clinical causal effect of the addition of clopidogrel, and CCE = 0.8819. Then, $K=1.457$ , and the recommended effective relative dosage per unit of change $1/ K=0.69$ . These values vary for different individuals.

5. Conceptual and Methodological Criticism of Fuzzy-Set Causal Models

The point of conceptual criticism is the examination of limitations in these models so that they are developed and applied intelligently and valuably, and they don't become mere ad hoc labels. For this purpose, one should keep in mind a distinction between two kinds readings of the causal accounts: Conceptual, or semantic, or ontological, viz., they explain what causality is, and epistemological, or methodological, viz., they offer methods for causal inferences, for hunting causes for the purpose of prediction, intervention, or explanation. I take the distinction to be independent of the distinction between semantic and epistemic interpretations of vagueness in terms or concepts. Scientific practice fails to adjudicate conclusively between such positions, on whether vague descriptions supervene on determinate facts, or on whether determinacy fails in the application of predicates (McGee and Mclaughlin Reference McGee and McLaughlin2004; Williamson Reference Williamson1998, Reference Williamson2004). Recall that the basis for fuzzy set membership assignments are made by expert consensus based partly on background knowledge. More broadly, the combination of semantic and methodological considerations is the fertile common ground for the useful exchange between science and philosophy. In no way is my position that philosophy should dictate or justify science.

The four models presented above represent causality as linking logical implication and empirical data; although the empirical status of the latter is largely irrelevant to many of the criticisms, which are much more general. The logical implication is the one modeled set-theoretically by subsethood relations. The logical formulation of causal relations in terms of set theory has important limitations as an account of causation and as models of causal reasoning.

(1) Under a set-theoretic interpretation (not necessarily fuzzy), the material conditional or implication between A and B, $A\supset B$ , A is a subset of B, $A\subset B$ . A is a sufficient condition for B; and equivalently, B is a necessary condition for A. However, this conditional analysis, in set-theoretical form, fails to capture the causal asymmetry between cause and effect, where the cause includes the possibility of necessary or sufficient causal conditions. I have shown above that a reflection of this predicament is Helgason and Jobe's counterintuitive dual-flow causal metaphysics.

(2) Material implication fails to distinguish causal relations from conceptual relations. Concepts can be defined extensionally or given extensional expression: subsethood relations apply both to conceptual relations and to causal relations. Subsethood alone is not sufficient to isolate causal relations. For instance, being a mammal might be sufficient for having four limbs and yet the former doesn't cause the latter. Also, the population of cordates—animals with a heart—is almost in perfect overlap with the population of renates—animals with kidneys—yet one condition being necessary and sufficient for the other doesn't indicate any asymmetric causal influence.

(3) Being a necessary cause is different from being the only actual cause or the only possible cause. Subsethood and material implication fail to tell the difference.

(4) As Elizabeth Anscombe famously pointed out, causes need not be either necessary or sufficient. In one sense of this assertion, of the three models, two fail to make provisions for partial causes or ‘INUS’ conditions in complex cases: causes can be true causes while being neither necessary nor sufficient, but instead ‘insufficient but necessary parts of sufficient but unnecessary causal complexes’. Example: A such that [ $(A\,\& \,B) $ or $(C\,\& \,D) $ ] causes E. Ragin states that “this type of causation may be the most common form of social causation” (Reference Ragin2000, 93). His account recommends that one analyze all conditions linked to an outcome for necessity first, and then analyze the complex of conditions of which they are part for sufficiency. Members of the sufficient complex that are not themselves necessary are INUS conditions.

(5) In the indeterministic sense of Anscombe's assertion, fuzzy set-theoretic relations need to be generalized to accommodate indeterminism. Here it is important to consider the compatibility between the assumptions behind the fuzzy-set models and the assumptions—for instance, in the constitution of populations—behind the introduction of probabilistic generalizations.

Additional problems arise from the relation of fuzzy sets to probabilities. Kosko argues that degree of subsethood $S(A,\,B) $ is set-theoretically equivalent to conditional probability:

If so, the FC model based on S is as vulnerable to all the known objections to causality defined in terms of high $P(B\vert A) $ (Salmon Reference Salmon1984; Cartwright Reference Cartwright1989; Pearl Reference Pearl2000). The Helgason-Jobe model presumes to replace probability-based statistical knowledge as inadequate and irrelevant to the medical goal at hand. Yet, required prior knowledge might come from statistical models, in the preselection of relevant causal conditions for the transformation of variables into fuzzy membership functions. The set-theoretic structure of the model generates or maps onto a formal probabilistic structure. Does that mean that the model can relate to probabilities or express them but only under a propensity interpretation about one individual? If so, this is inconsistent with the statistical or population interpretation of the probabilistic information required for constructing the fuzzy membership assignments. The same regress problem potentially affects Ragin's introduction of statistical benchmarks. (Clearly, from the interpretive point of view, the risk does not extend to the introduction of probabilities to capture random factors.)

(6) There is a lack of syntactic distinction between stable mechanisms and spurious circumstantial conditions: ‘A implies B’ implies that ‘not-A implies not-B’; for instance, ‘smoking implies cancer’ implies ‘not-cancer implies not-smoking’. However, while the former might have causal implications, the latter lacks the same kind of causal support. Kosko's Reference Kosko1986 model, above, introduces the additional logical implications in terms of causal increase and decrease precisely to eliminate such possibilities. The other models don't.

(7) The empirical truth-functional material implication is true even if A is false enough: $t(A\,\mathrm{implies}\,\,B) =\mathrm{min}\,\{ 1,\,1-t(A) +t(B) \} $ . But only stronger, non-truth-functional, forms of necessity or implication can make the implication false enough in that case.

In stronger, non-truth-functional forms of implication, the truth of A and B is insufficient to make the logical implication true. Causality is one such form. Empirical regularities or contiguous sequences are not sufficient for a stronger form of implication or necessity. This is a problem for conceptual accounts; the Humean reductionist project of understanding causation in terms of factual regularities fails to capture the element of necessity that distinguishes causal action. Of course, Humeans will disagree here. The problem for epistemological accounts is the limitation it implies on evidential relations. In the strongest case they aren't conclusive. The same problem extends to improved fuzzy cognitive maps, in epistemological and ontological versions that include time lags, based on sequences or histories (Chaib-Draa and Desharnais Reference Chaib-Draa and Deshairness1998; Helgason and Jobe Reference Helgason and Jobe2003). The statistical version of this problem is that correlation is not causation. Cartwright and others have argued that probabilistic models of causality fail to be reductive because in order to avoid Simpson-paradox-type difficulties, they need to be conditionalized upon relevant causal factors. But this requires prior causal—not just empirical—knowledge. Therefore, in the strongest formulation, probabilistic models approximately measure but do not define causation (Cartwright Reference Cartwright1983, Reference Cartwright1989). At both the epistemological and ontological level, provisions must be made to isolate the relevant causal candidates. All three models fail in this regard.

(8) In cases of anti-correlations or conserved quantities, knowledge of the value of a property of one system is necessary and sufficient to know the value of the other. Is that causality? The controversy surrounding the case of EPR correlations in quantum mechanics illustrates this point. The issue would be best addressed in the context of a fuzzy-set representation of quantum mechanics (following limited examples by Pykacz, Da Costa, Krause, and others) and its corresponding fuzzy-logic interpretation.

(9) The problem of underdetermination of sufficient causes (similar to the problem of logical underdetermination of theories by empirical data) is pervasive: for any sufficient cause C, $C\,\& \,X$ is sufficient as well. At both at the epistemological and ontological level, provisions must be made to isolate the relevant causal candidates. All three models fail in this regard.

(10) The problem of overdetermination also plagues fuzzy-set models: if at least two causes are sufficient, neither will prove necessary. There is the additional problem of spurious causation: if either shooter, A or B, can kill a prisoner D, and C acts by ordering A to shoot, in the measured relation either C or B is sufficient for the death of D. If both A and B fail to act (despite C's order to A) but D is dead nonetheless, then the measured relation still entails that C's action (giving orders) caused D to die. The problem is a limitation also in probabilistic models of causality, conceptually and epistemologically.

(11) As stated, the logical models fail to distinguish causal connection from common cause. Provisions might be introduced.

(12) The three models are vulnerable to the problem of reversibility: `increase of A implies increase of B' implies that `decrease of A causes decrease of B', which is not always the case.

(13) The three models obscure research into the intervening mechanisms that might actually explain the subset relation—in lieu of ‘laws of nature’. Attention to the intervening causes would involve a change in fuzzy cognitive maps and the associated adjacency matrices. Intervening mechanisms and shielding constraints—such as a pipe or a blood vessel—might be worth taking into account as they may present degrees of failure (leaks) and may be required in order to understand and intervene (prevention).

7. Conclusion: Broader Philosophical Significance: A Priori Unification and Interaction

The previous discussion points to semantic insufficiency and methodological weakness in the formulation and application of the fuzzy-set formalism. To previous considerations one might add the value of models of causality that represent an alternative to conditional analysis. The discussion also raises two meta-questions: Are fuzzy necessary and sufficient conditions necessary and sufficient (even in the fuzzy sense) for causality? Is the notion of causality an example of a higher-order fuzzy category?Footnote ³ In other words, should we evaluate the models using the strict sharp-boundary criteria that analytic philosophy applies to the evaluation of concepts? The discussion sheds doubt on such suggestions. Recall that the issue is not whether set-theoretic relations are relevant to representing or theorizing about phenomena, but whether the formal apparatus—the superstructure—they import can adequately capture causal relations. An alternative approach to accommodating the usefulness of fuzzy sets would be to apply other models of causality to conditions or variables represented by fuzzy categories without the difficulties and weaknesses introduced by the set-theoretic models of logical implication. The next step would be to add aspects of the alternative models to the set-theoretic model in those domains, no matter how idealized, in which the latter seems relevant.

The different causal models may be related as conceptual accounts or as cooperating or competing local epistemological models. The first task in order to address the suggestions above is to figure out in detail where and how successfully fuzzy and other models apply; and then to know how they can relate to one another. Pluralism about causal criteria may prove productive, both in offering the opportunity of considering alternatives and of considering their joint cooperation. But, as Catherine Elgin has said, the possibility of many rights doesn't imply that there no wrongs. Finally, attention to fuzzy-set models of causality offers interesting lessons aside from issues of limits of application; causal models in general, and fuzzy-set models of causality in particular, have doubtless philosophical significance in all the aspects of actual science they illustrate: from the complexity of empiricism, to the horizontal and dynamical unity of different sciences, to the vagaries of the interaction between scientists and philosophers.

Models of fuzzy-set causality, unlike many conceptions of causation, have been articulated and put forward by scientists, not philosophers. It is a conceptual kind of contribution that mediates the application of an abstract mathematical formalism to the empirical description of the world. It is also a contribution to an area of conceptual research now engaged by philosophers. This is no surprise. Science has a rich conceptual dimension that mediates and guides the mathematical description of the world. In addition, throughout their history, science and philosophy have long been engaged in exchanging notions and developing in response to each other's practices and deliverances, each other's devices and desires. Until the seventeenth century, any distinctions between scientific and philosophical disciplines are typically internal to the subject matter and mode of inquiry of philosophy. Subsequent developments in each discipline appear both to engage in interaction and to articulate and promote external distinction (Friedman Reference Friedman2001). What can be said about the philosophical significance of Einstein's relativity theories can be matched by the intricate relation of Gestalt psychology to the history of twentieth-century scientific philosophy and philosophy of science (see Cat, Reference Catforthcoming). And philosophy of science is still considered a branch or subdiscipline of philosophy.

The specific topic of causality is one in the changing overlap and interaction of science and philosophy. Even for Kant the synthetic a priori status characterizes both scientific and causal judgments. Causal claims and causal reasoning pervade scientific knowledge and practice. Philosophy has since its recognizable early moments addressed questions regarding the nature of causality and the reliability of methods to yield and ascertain causal claims. In philosophy of science, attention has been paid to scientific claims and to causal assumptions that make intelligible and possible much scientific practice.

What is relevant from this point of view to the case of fuzzy-set models of causality is that they have been both issued and applied by scientists themselves. This fact obliterates the distinction between science and philosophy in terms of a strict division of labor.

The standard picture of how scientific terms get their meaning is that in each theory an abstract symbolic calculus is interpreted relative to a particular kind of phenomena or models. By contrast, I believe that mathematics is often applied with a certain conceptual interpretation, or understanding, over and above any particular theory. These interpretations help us apply within different theories more abstract and general categories or representations such as individual particle, determinism, locality, causal relation, information, etc. For instance, to Maxwell and Kelvin, differential equations constituted a canonical representation of contiguous action, whereas integral laws exposed action at a distance. Similarly, fuzzy-set models of causality mediate the application of fuzzy set theory in a novel formal treatment of phenomena; the latter receive, in turn, a causal interpretation above any specific theory or domain of application.Footnote ⁴ Even more abstractly, the same formalism of fuzzy-sets, as well as others, provides the precondition for a generalization of formalisms such as the differential calculus based on topological manifolds and, indirectly, new tools for empirical science (Cat, forthcoming).

All those assumptions share the property of being ‘relatively a priori’. They are adopted prior to the adoption of the tested hypotheses or the description or the explanation of phenomena. Reichenbach, Carnap, and Friedman would consider some of those more strongly as ‘relativized a priori’: as a priori constitutive conditions of conceptual possibility (Friedman Reference Friedman2001). They are valid for the definition of concepts in specific theories or frameworks. Fuzzy-set models of causality are a priori formal criteria that specify and facilitate the application of causal talk. They constitute a bridge principle that concretizes the notion of causality for application to phenomena represented through fuzzy sets.

Ideals of unification of science abound. Ideals and ideas of fundamental languages, concepts, laws or entities, or ideals of a unique method provide values, standards, or projects that drive and evaluate research. They are not universal values shared among different scientific communities, even within the same discipline, such as physics (Dupré Reference Dupré1993; Galison and Stump Reference Galison and Stump1996; Cat Reference Cat1998).

In the cases considered so far, the introduction of a new formalism such as fuzzy-set theory and of a causal criterion in such terms constitute a framework of unity of disciplines—theories, fields, including experimental methodology—both internally and across boundary divides. Real unification is often the outcome of a partial reduction of elements of different theories and hybrid models and inter-field theories (Sarkar Reference Sarkar2000; Cat Reference Cat1998). Less noted is the actual unification that consists in the circulation of techniques and models within subdisciplines (e.g., renormalization and symmetry-breaking within physics) or among different sciences—fuzzy-set theory, statistical analysis, analogies such as genetic algorithms, cognitive maps, etc. The examples of fuzzy-set theory and fuzzy-set causality are clear ones, based on the latter's nature as interpreted formalism. And in each, they can cooperate or replace other elements, opening up a dimension of both unity and pluralism. Each alternative is relevant, and often, as in the case of bare fuzzy-set causality, it is best served in interaction with others. Such formalisms, I have noted above, have the corresponding property of being more general than any particular theory or model in which they are applied to a domain of phenomena. This makes their circulation from science to science possible. It is an example of another, dynamical unification of the sciences.

Finally, the formalism of fuzzy-set causality establishes a unifying link between scientists and philosophers, and an opportunity for coordination and cooperation.