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Many-many mappings and world structure.

In discussions of inter-theoretic relations, some philosophers allude to the existence of "many-many" correlations between properties from higher and lower-level theories.(1) David Hull was the first to mention the phenomenon in any straightforward way, citing the case of Mendelian and molecular genetics:

[T]he relation between Mendelian and molecular

predicate terms express prohibitively

complex, many-many relations. Phenomena

characterized by a single Mendelian predicate

term can be produced by several different

types of molecular mechanisms ... Conversely,

the same types of molecular mechanism can

produce phenomena that must be characterized

by different Mendelian predicate terms

(1974, p. 39).

The claim is provocative, given hopes of a clear and empirically well-sustained one-to-one mapping between "gene" and "DNA," which is to say, because it flouts expectations about inter-level identity in the most extreme way. And others now agree that, by their initial taxonomies, there is indeed a many-many relation between classical and molecular genetics (Churchland, 1986, p. 364; Kincaid, 1988, p. 274; Gasper, 1992, pp. 668-9).

Nor is this complex relation confined to liaisons at the genetic level. Concerning computational types vis-a-vis their physical implementations, Stephen Schiffer says:

First, a given physical state-type can, and

invariably will, have indefinitely many functional

roles. This is just to say that a

state-type can stand in numerous distinct

causal (or transitional) relations to numerous

other states or to outputs, inputs, and so

on. Second, two distinct physical state-types

can have the same functional role: a state-type

that figures in the etiology of my behavior

can have a causal property that is also

had by a different state-type that figures in

the etiology of the behavior of a computer

(1987, p. 21; see also Block, 1996, p. 207).

Hence the many-many notion has become a widely accepted albeit largely unspoken dogma. Yet, as will become apparent, passing references to many-many structures and variable functional roles leave an enormous range of questions completely untouched. What is worse, the relation may be obvious in one case but not in another. For example, Jerry Fodor accepts it for computational types via their physical realizations but finds it deeply problematic for broad content properties via their computational implementations. It is an "architectural problem" whereby "many broad contents might correspond to the same computational state (as in Putnam's case of computational Twins with different beliefs); or many computational states might correspond to the same broad content (as in Frege's case, where people who believe that Fa fail to believe Fb, even though a = b)" (1994, p. 22).(2)

A careful analysis would again be helpful, providing a standard against which one could separate out all legitimate many-many mappings from those that can and should be explained away. But more than that, with an eye to both sides of the inter-theoretic relation, and by incorporating the variability of both high and low-level properties, one can arrive at a comprehensive picture of the metaphysical structure of the world.


By itself, a many-many relation is compatible with any number of philosophical systems, even metaphysically extravagant ones. For to say there is a many-many mapping between properties from two separate domains says nothing about the nature of those properties, still less about the mechanism whereby this many-many relation is effected.

To illustrate, consider anomalous dualism, a position according to which there are no lawful connections between the mental and the physical. Each domain is equal and free from determination by the other. A type of mental substance might correlate with any number of physical types, and a type of physical substance might correlate with any number of mental types, and this, because of the unlawful behavior between the two realms. Like games of chance, various combinations are possible.

Or consider dualistic occasionalism, where all causal relations hold within but not between the mental and the physical. This being so, a many-many correlation is again possible, depending entirely upon the kind of inter-theoretic harmony or disharmony that God had ordered in advance.

And there are various possibilities within the same broad philosophical system. On the many-many functionalist picture alluded to by Schiffer, the first-order state types standing in their variable input/internal/output relations might "cause" distinct second-order properties to be instantiated, or serve to "realize" them, or the second-order properties might "supervene" in one of several proprietary senses upon the variable arrays of first-order state types, and so on.

Generally speaking, there is always a question of formal structure, whether the items are related one-to-one, one-many, many-many; nature, whether the items are concrete or abstract, substantial or functional, mental or physical; and mechanism, whether the items are generated by chance, accidental correlation, pre-established harmony, two-way causal interaction, realization, supervenience, and the like. Taken alone, talk of "many-many correlations" is only a gesture towards a philosophical position.

The first desideratum, consequently, is to specify a mechanism which is both plausible on general metaphysical grounds and adequate to generate the many-many structure.


The many-many relationship will be divided into two separable components: the one-many relation of "compositional plasticity" or "multiple realization" whereby a given higher-level property can be realized by any number of distinct lower-level state types (Putnam, 1960, 1967; Fodor, 1974; Boyd, 1980); and the converse many-one relation of "context sensitivity" or "multiple realization complement" whereby a given lower-level property can serve to realize any number of distinct higher-level state types (Richardson, 1979, pp. 550-55; Block and Fodor, 1980, p. 239; Macdonald, 1989, p. 38; Kincaid, 1990, p. 577; Gasper, 1992, p. 668; Endicott, 1994).(3)

Viewing matters in terms of multiple realization and its complement, moreover, commits one to a specific metaphysical mechanism--that of realization. Two features are particularly relevant to generating a many-many structure and will be selected for attention: "levelhood" and "property-to-property determination." (For other suggested details on realization, see Tye, 1995, pp. 41-50.)

First, unlike causation and other determinative relations, realization is a purely inter-level affair.(4) Higher-level properties are realized by the lower-level types, all in keeping with multi-leveled science, and, happily for present purposes, the targeted inter-theoretic many-many mappings. Yet the notion of a "level" has itself been defined in various ways--in terms of mereology, a difference in patterns of nomic generalization, a form of supervenience between adjoining domains, or a combination of these intermixed with a reiterated role/occupant distinction--and disagreements about their overall significance abound (see Wimsatt, 1976, pp. 237-42; Lycan, 1987, pp. 37-8; Crane and Mellor, 1995). So, to avoid controversy, I will leave the notion of levels as an undefined primitive, allowing various criteria to be substituted within the overall analysis.

Second, realization is typically understood so that, when a lower-level G serves to realize a higher-level F, the one must bring about or determine the other. Hence the oft-cited connection "necessarily, if G is instantiated then so is F" (Lepore and Loewer, 1989, p. 179; Tye, 1995, p. 41).(5) Yet such connections must be slightly amended if they are to underlie the many-many relation.

To see the point, and to bring the entire variable structure closer to view, observe that multiple realization correlates a higher-level property F in one-many fashion with a range of base properties G through H such that, minimally, it is possible for an instance of F to be subserved by an instance of G and not H, and on another occasion, H and not G (a particular system could instantiate the adding function (F) by activating a neural cell assembly (G) and not an Intel 80486 microprocessor (H), and vice versa). Thus, given the aforementioned determinative connection, the result is this:

(i) possibly G and F but not H are instantiated,

and, necessarily, if G is instantiated

then so is F; and

(ii) possibly H and F but not G are instantiated,

and, necessarily, if H is

instantiated then so is F.

But, similarly, the converse relation should correlate a lower-level G in one-many fashion with a range of upper-level properties E through F such that, minimally, it is possible for an instance of G to subserve an instance of E and not F, and on another occasion, F and not E (the system which is an Intel 80486 microprocessor (G) might instantiate the adding function (F) and not the tangent function (E), and vice versa).(6) So, given the same determinative connection, one must say:

(iii) possibly G and F but not E are instantiated,

and, necessarily, if G is

instantiated then so is F; and

(iv) possibly G and E but not F are instantiated,

and, necessarily, if G is

instantiated then so is E.

But the two are inconsistent. The possible situation described in the first clause of (iii) contradicts the modally strong conditional in the second clause of (iv), and the possible situation described in the first clause of (iv) contradicts the modally strong conditional in the second clause of (iii).(7) Thus, unless one joins the anomalous dualist and abandons inter-level determination, the proper course is to become sensitive to context and relation (see Richardson, 1979, pp. 540-55; and Kincaid, 1990, pp. 576-83).

That is, a base property G determines a higher-level property F relative to a background condition C, and a change in that condition will permit G to determine some other property E and not F. Henceforth all determinative connections will be relativized to a context, taking the form: "necessarily, given C, then if G is instantiated then so is F."(8)

Given the stratified nature of realization, however, the level of background conditions must be addressed. In particular, Harold Kincaid argues that some contexts may not be specifiable within the resources of a lower-level theory (1990, pp. 577-82; cf. also Baker, 1987, pp. 88-96). So C may not be identical to a set of base-level properties. But I will assume it is at least determined by them, thus remaining true to the overall spirit of realization as a form of property subservience whereby the lower brings about the higher, and respecting broader commitments to the supervenience of one level upon another.

To sum up, then, stratified contextual determination is the mechanism. What remains is the full variable structure.


Armed with previous points about realization, one can understand the many-many relationship in piecemeal fashion, beginning top-down with its one-many structure:

(MR) A set of properties A is subject to multiple

realization/compositional plasticity in

a set of properties B iff A and B constitute

higher and lower levels, respectively, and for

every F in A there exist distinct properties G

and H in B, and distinct B-determined background

conditions C1 and C2 such that:

(i) possibly F and G but not H are exemplified,

and C1 holds, and, necessarily,

given C1, then if G is instantiated then

so is F;

(ii) possibly F and H but not G are exemplified,

and C2 holds, and, necessarily,

given C2, then if H is instantiated then

so is F.(9)

The converse many-one structure of multiple realization complement thus becomes:

(MRC) a set of properties B is subject to

multiple realization complement/context

sensitivity in a set of properties A iff A and B

constitute higher and lower levels, respectively,

and for every property G in B, there

exist distinct properties E and F in A, and

distinct B-determined background conditions

C1 and C2, such that:

(iii) possibly G and F but not E are exemplified,

and C1 holds, and, necessarily,

given C1, then if G is instantiated then

so is F;

(iv) possibly G and E but not F are exemplified,

and C2 holds, and, necessarily,

given C2, then if G is instantiated then so

is E (cf. other variations in Endicott, 1994).

Hence, combining (MR) and (MRC), and eliminating redundancy created by the first clause of each, the full many-many plasticity can be expressed thus:

(MMR) A many-many realization exists between

sets of properties A and B iff A and B

constitute higher and lower-level properties,

respectively, and there are distinct properties

E and F in A, and distinct properties G

and H in B, and distinct B-determined background

conditions C1, C2, and C3, such that:

(i) possibly F and G but not H or E are

exemplified, and C1 holds, and, necessarily,

given C1, then if G is instantiated

then so is F;

(ii) possibly F and H but not G or E are

exemplified, and C2 holds, and, necessarily,

given C2, then if H is instantiated

then so is F;

(iii) possibly G and E but not F or H are

exemplified, and C3 holds, and, necessarily,

given C3, then if G is instantiated

then so is E.(10)

Clauses (i) and (ii) jointly describe F's compositional plasticity, according to which F has alternate realizations in G and H. And clauses (i) and (iii) jointly describe G's context sensitivity, according to which G has alternate realized properties in E and F. Viewed graphically, and with background conditions left understood, (MMR) depicts the following metaphysical structure:


It is evident, too, that (MMR) represents but a cross-section of property relations, being silent on whether E has plasticity like F, or H has plasticity like G (graphically, whether the structure continues in the direction to the sides). Moreover, (MMR) represents "the world at a layer," being silent on whether the A properties have plasticity with respect to any (still) higher-level properties, or the B properties have plasticity with respect to any (still) lower-level properties (graphically, whether the structure continues above and below).

Indeed, if there is a metaphysically lowest level, then one set of B properties is guaranteed to lack multiple realization.(11) Contrawise, if there is a metaphysically highest level, then one set of A properties is guaranteed to lack its complement.(12) So the properties gleaned from the lowest and highest levels--the "alpha" and "omega" properties--would provide the upper and lower limits upon the world's structure. Pictorially, the result is a lattice structure:


Of course this is not to say that all properties will conform to this picture. There are simpler inter-theoretic structures, for instance, the one-to-one relations charted by the chemical table of elements.(13) As a result, the world is likely to exhibit a number of distinct patterns, its fabric stitched together in a variety of ways. Even so, previous examples will show that the many-many notion applies to a wide range of cases--the liaison between computation and neurophysiology, biofunction and molecular genetics, or form and substance quite generally.

To illustrate (MMR) with a specific case, and using the framework of a two-factor theory of content (McGinn, 1982; Block, 1986), let F be the propositional attitude type "believing H20 is near," which by clauses (i) and (ii) and familiar functionalist considerations will have alternate realizations in distinct neurophysical properties G and H that, relative to contexts where H20 is present, determine the belief in question. Then consider that same neurophysical G which by clauses (i) and (iii) and familiar contextual considerations will have alternate realized properties in F and a distinct type of propositional attitude E, say, "believing that XYZ is present"--the former when G is embedded within a wider social, causal, historical, or evolutionary context involving H20, the latter when G is embedded within a social, causal, historical, or evolutionary context involving XYZ. Such is the many-many relationship.


(MMR) expresses the variability of a many-many relation in the simplest manner, from F to just two properties G and H, and from G to just two properties E and F, representing what must exist, minimally, for the relation to hold between adjacent domains. But there may be other realized and realizing properties, and reflecting on their number and range leads to the notion of a property's degree of plasticity.

Starting with a higher-level type's degree of multiple realization/compositional plasticity, this can be measured along two parameters. As a purely quantitative measure, where B is the set of properties that serve to realize F, and B* the set of properties that serve to realize F*, then F has more compositional plasticity than F* if and only if B is larger than B*, having more members. Hence "being a planet" has more than "being jade," since the former has indefinitely many substance types that may subserve it (all of the minerals, for example) whereas the latter has only two (jadeite and nephrite). One can say, furthermore, that a property F has maximal compositional plasticity if and only if the set B of properties that serve to realize F has infinitely many members.

Yet the degree of plasticity can be measured along a different line, metaphorically, not by its breadth but by its depth. For in the most theoretically interesting cases, F's variability is a deep fact which does not mask any underlying commonality; no one-to-one correlation hidden beneath the one-many structure. Accordingly, where B is the set of all properties that subserve F, then F has deep compositional plasticity if and only if the conditions specified in (MR) or (MMR) hold, and there is no property K in the set B such that, necessarily, F is instantiated if and only if K is instantiated. Functional properties, for example, appear to enjoy both maximal and deep compositional plasticity with respect to the entire range of physical properties.(14)

Similarly, then, concerning a lower-level type's degree of multiple realization complement/context sensitivity; where A is the set of properties that G serves to realize, and A* the set of properties that G* serves to realize, G has more context sensitivity than G* if and only if A is larger than A* ("being carbon-based," for example, has more of this plasticity than "being cordate"). Also, G has maximal context sensitivity if and only if the set A which G subserves has infinitely many members. Finally, for matters of depth, and to rule out any hidden one-to-one structure, where A is the set of all properties that G subserves, then G has deep context sensitivity if and only if the conditions specified in (MRC) or (MMR) hold, and there is no property K in the set A such that, necessarily, G is instantiated if and only if K is instantiated.


The many-many relationship not only subsumes a wide range of inter-theoretic phenomena, it also brings into better focus a number of issues in the philosophy of mind, science, and metaphysics. To wit, the materialist type-identity theory requires that all properties be identical to physical properties, which in turn requires necessary coextension, a lawful one-to-one correlation. Now it is a familiar lesson that multiple realization prevents such correlations. With the many-many plasticity, however, this lesson becomes more pronounced. For the correlation must fail not simply because F can occur without G, but also because G can occur without F. The avenue to identity is blocked from both sides.

Moreover, there can be no identity between F and any other lower-level property if F happens to enjoy deep compositional plasticity, as that notion was earlier defined; and there can be no identity between G and any other higher-level property if G happens to enjoy deep context sensitivity, as that notion was earlier defined. Accordingly, where A is the set of all properties that G serves to realize, and B is the set of all properties that serve to realize F, then deep many-many realization holds for properties F and G if and only if those properties satisfy (MMR), and there is no other property K in the sets A or B such that, necessarily, F or G is instantiated if and only if K is instantiated.

In a word, and quite contrary to the materialist type identity theory, there will be no underlying commonality for F, and no overarching similarity for G, meaning no lawful coextension in any region throughout property space.

Not surprisingly, then, the many-many relation also has dire consequences for any model of scientific reduction which seeks a form of ontological unification by incorporating that same materialist type-identity theory, including the classical account whereby one reduces a theory [T.sub.R] by deducing it from a basic theory [T.sub.B] via biconditional bridge laws that, on the most favored interpretation, express or support property identities (Sklar, 1967; Causey, 1977).


Beyond the denial of identity, many-many realization delivers a more dramatic result by frustrating weaker inter-theoretic constraints on reduction. For example, Kenneth Schaffner maintains that a theory [T.sub.R] approximately reduces to a basic theory [T.sub.B] in virtue of a "strong positive analogy" between [T.sub.R] and a corrected version [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that is strictly deducible from [T.sub.B] (1967). Similarly, Paul Churchland describes the corrected [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as an "equipotent image" of the original [T.sub.R] in ideal cases (1979, p. 83; 1985, p. 10). And Clifford Hooker justifies the claim that [T.sub.R] reduces to [T.sub.B] by "the analogue relation" between [T.sub.R] and the actual target of deduction, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1981, p. 49).

Now this analogical relation from image to original theory must be positive in a fairly significant sense, otherwise the claim about reduction would be trivially true (everything resembles everything in some respect or in some measure). Thus, one plausible requirement is that the reduced and corrected theories have terms whose denoted properties are at least roughly isomorphic in their causal and explanatory roles. Or, since the corrected theory can be deduced from the basic reducing theory, it is plausible to require this from the reducing theory. As Churchland says:

is that it have the resources to conjure up a

set of properties whose nomological powers/

roles/features are systematic analogues of the

powers/roles/features of the set of properties

postulated by the old theory (1985, p. 13).(15)

But consider any two properties F and G that are drawn from many-many related theories. Given multiple realization, the causal role of F goes well beyond that of G by entering into relations with other properties in situations where G is absent; and given its converse, the causal role of G goes well beyond that of F by entering into relations with other properties in situations where F is absent. Worse still, if F and G have maximal and deep plasticity, then each has connections to infinitely many other properties in situations where there are no underlying or no overarching similarities at all.

So it is not the case that "the `causal powers' of F-ness (as outlined in the laws of [T.sub.R]) are a subset of the `causal powers' of G-ness (as outlined in the laws of [T.sub.B])" (Churchland, 1985, p. 11, with a slight change in the variables). Nor are the causal powers of G-ness a subset of the causal powers of F-ness. Rather, the many-many notion guarantees a slight intersection, with a possibly infinite number of other causal powers being entirely disjoint.

And neither is this problem exclusive to accounts based upon analogical relations. Compare the claim that [T.sub.R] reduces to a basic [T.sub.B] only when the latter is "informationally redundant" with respect to the former (Foss, 1995, p. 412). Yet, given a many-many mapping, the terms of the lower-level theory are in nowise redundant, but carry information about properties that occur in a possibly infinite number of situations not covered by the terms of the higher-level theory, and vice versa.

Of course other accounts of reduction are possible.(16) Still, the problem seems perfectly general. Ontologically retentive strategies must hope for a modicum of resemblance between reduced and reducing theories, or some measure of isomorphism between their terms, causal roles, and any information they may carry. With many-many realization, however, that hope is lost--fragmented into a thousand pieces of crisscrossing property relations.


While philosophers like Hull claimed the many-many structure presents a further problem for reduction, beyond what is created by multiple realization, they did not focus on the heightened degree of cross-classification that accompanies the additional converse, maximal, and deep plasticities examined here. Instead, the debate centered on whether many-many mappings create a form of inter-theoretic "indeterminacy" or "ambiguity" which blocks the minimal Nagelian condition of "connectability" between reduced and reducing theories (Hull, 1974, pp. 37, 39; Richardson, 1979, p. 551; Kincaid, 1990, p. 577).

But, given the present analysis, this particular worry appears misplaced.(17) For laws of the form, "necessarily, given C, if G is instantiated then so is F," forge the disputed connection between G and a specific, unambiguous F relative to a designated contextual feature C, thereby securing inter-theoretic determinacy (without this, one would be hard-pressed to avoid anomalous dualism). What has not been secured by previous definitions is a form of determinacy that arises solely from the base properties.(18)

Specifically, while the determining conditions C and G are either base-level properties or (if C is high level) determined by them, the law itself, "necessarily, given C, if G is instantiated then so is F," may not be explainable within the resources of a lower-level theory. A parallel holds for supervenience, where, famously, a physical explanation of supervenience bridge laws is lacking (see Levine, 1983, for the case of qualia; and more generally Blackburn, 1985, p. 145; Schiffer, 1987, pp. 153-54; and Horgan, 1993a, pp. 577-82). One must be prepared for "brutely inter-theoretic" determination.


In light of the foregoing, (MMR) and its extensions offer a new nonreductive position whereby many-many correlations fit comfortably with the contextual determination of higher-level properties. To underscore this novelty, I will finish by comparing another idea currently popular among nonreductivists--the doctrine of supervenience.

"Supervenience" is a term of art designating a family of relations tied together by the general idea that indiscernibility with respect to lower-level properties should entail a corresponding indiscernibility with respect to higher-level properties. Variations on this basic theme are then explained largely by differences regarding the items in their mode of determination, be they individuals, regions, or entire worlds (see Horgan, 1993a).

So consider basic or generalized supervenience, according to which there can be no difference in a set of properties A without a difference in another set of properties B (Lewis, 1986, p. 14; Kim, 1990, pp. 131-40; McLaughlin, 1995, pp. 17-24). As such, (MMR) entails supervenience.(19) The crucial difference lies in the opposite direction.

That is to say, generalized supervenience fails to imply (MMR) because it is compatible with the denial of all property plasticity. Thus, it is compatible with the materialist type-identity theory whereby all higher-level properties are mapped one-to-one with lower-level physical properties, since, under the assumption of identity, indiscernibility with respect to B is ipso facto indiscernibility with respect to A.

It goes without saying, also, that more specific forms of supervenience will not imply (MMR) either. Whether individual or global, the doctrine is still true under the assumption of type identity--the individuals/worlds will be indiscernible in their A and B properties--an assumption that conflicts with many-many plasticity.(20)

This fundamental difference in metaphysical commitment should come as no surprise. For, at bottom, supervenience concerns similarities. Items indiscernible in their base properties must be indiscernible in all other respects, meaning the same lower-level types must determine the same higher-level types. Many-many realization, on the other hand, expresses real differences--the same lower-level types determining different higher-level types; and the same higher-level types being determined by different lower-level types.(21)

Hence supervenience and the present many-many relation belong to distinct but conceptually related families. The one is "determination proper." The other is "property plasticity," where that determination is parceled out among specific properties standing in their variable relations.(22)


(1.) Discussions in the literature are framed in terms of a heterogeneous lot--"predicates," "properties," "types," and "kinds." I will speak of "properties," and for literary relief, "types" (though the former are abstract generals and the latter abstract particulars).

(2.) Hence Fodor says "Twin cases imply many-one mappings from intentional states to computational implementations," and "Frege cases imply many-one mappings from computational implementations to intentional states" (1994, p. 23).

(3.) This converse relation goes by many names. With an eye to Hull's claims, Richardson (1979) calls it "indeterminacy of function"; Gasper (1992) calls it "multiple supervenience"; and Endicott heretofore (1994) "constructival plasticity." As for the terms chosen here, the virtue of "context sensitivity" is its familiarity, and "multiple realization complement" its evident (albeit converse) connection to multiple realization.

(4.) Putnam (1960) introduced the realization idiom into contemporary philosophy by the first-order and second-order properties of Turing machines. But the distinction between orders may or may not correspond to the traditional scientific demarcation of levels (Kim, 1997, pp. 290-91). Specifically, the very same object can instantiate both first- and second-order properties. Hence, if one individuates levels by the mereology of particular objects, there will be different property orders obtaining at the same mereological level. Still, Putnam intended that talk of realization capture inter-level relations, e.g., the psychophysical case of pain and brain states.

(5.) Kim suggests something stronger, i.e., that realization presupposes species-specific biconditional laws (Kim, 1981, p. 180; 1992, p. 315). But such are inappropriate where the many-many mapping applies (see sec. V and fn. 16).

(6.) The exclusionary sense is important. E.g., though Gasper describes a phenomenon he calls "multiple supervenience" as the converse of multiple realizability, he seems to have in mind the coinstantiation of distinct higher-level properties. He says:

The same molecular structure, for instance, underlies a multiplicity

of higher-level properties in a piece of metal: its electrical

conductivity, thermal conductivity, ductility, metallic luster,

opacity and so on ... Similarly, the same molecular process underlies

a multiplicity of properties at the cytological level. To take a

simple example, the chromosomal properties of being less than 1 cm

in length, being less than 0.9 cm in length, being less than 0.8 cm

in length, and so on, may all supervene on the same molecular base

(1992, p. 668).

Yet, since multiple realization is not the mere coinstantiation of base properties G and H, then the true converse must require that F and not E be exemplified, and then E but not F. So one would have to say of the piece of metal that its molecular structure could realize, say, electrical conductivity and not metallic luster, and on another occasion, metallic luster and not electrical conductivity; and one would have to say of the chromosome that its molecular structure could realize being less than 1 cm in length and not less than 0.9 cm in length, and on another occasion, being less than 0.9 cm in length and not less than 1 cm in length (the latter is logically impossible, which only dramatizes the difference between high-level coinstantiation and genuine alternate realized properties).

(7.) They are contradictory assuming the same type of modality throughout (e.g., that the possibilities are not merely "logical" while the necessities are "physical"), and making a quite natural assumption about the accessibility relation between the relevant possible worlds. Of course one could alter the modalities so that the pertinent possibilities and conditionals do not hold at the same world. Yet that would hardly be satisfactory, seeing that the resulting concept would be inapplicable when the possibilities described in clauses (iii) and (iv) are both actualized. Hence the suggested contextualization in the text. (For more detail, see Endicott, 1994, pp. 60-71.)

(8.) Two points about the contextual feature, and one about the implicit quantifiers. First, and obviously, the context C must be a possible one, otherwise the conditional will be vacuously satisfied. Second, C must not be sufficient for F, otherwise G will play no significant role. Third, one should not assume the token identity of F and G instances, since there might be mereology between nonidentical objects (Horgan and Tye, 1985; Baker, 1997), and since there might be nonlocal determination (Post, 1995). So, one way to fully and formally and neutrally state the conditional is this: "for any objects or regions x and y, possibly Cx and Cy, and, necessarily, if Cx and Cy, then if Gx then Fy," where, e.g., x may be a highly gerrymandered object/region in which y is embedded, and where it is false that "necessarily, if Cy then Fy." But I will continue to use the less cumbersome "given C, if G is instantiated then so is F" in the text.

(9.) Note that the background conditions needn't be distinct and are actually unnecessary when (MR) is taken in isolation (consider a chip-for-neuron replacement scenario where having a microprocessor H and having a neural cell assembly G may individually give rise to F in the same brain, i.e., in the same context). So the conditions are only added for conformity with context sensitivity and the full many-many relation which follows. For a simpler definition shorn of this complication, see Endicott (1996). For an even simpler definition, see Yablo (1992, p. 255), though the one-many relation is more graphically displayed by (MR), and though the definition is satisfied even if no G necessitates F when F is multiply realized (realization being cashed out as a nondeterminative relation). Hence the virtue of making it an explicit condition on realization that G and its alternative H bear some determinative relation to F. Otherwise one must take multiple realization in conjunction with a thesis like supervenience, as in Yablo's paper (pp. 254-6).

(10.) Since the background conditions are unnecessary when multiple realization is taken in isolation from its converse (fn. 9), then, strictly speaking, C2 needn't be a distinct condition. But C1 and C3 must be distinct, otherwise the inconsistency discussed in sec. II will arise. Thus, and alternatively, one can say within the definiens: "there are background conditions C1, C2, and C3, such that C1 [not equal to] C3," and then proceed as before.

(11.) Perhaps some set of sub-atomic properties true of the most basic entities will qualify, having context sensitivity but not compositional plasticity. If so, then Leibniz was correct (Monadology, sec. 2 and 3), and there is a most basic level whose objects are the proper exemplars for the alpha properties.

(12.) Perhaps a property like "being the entire universe" would qualify, i.e., it would have compositional plasticity since it could have been realized by distinct types of substances (say, if the big bang were to have generated an alternate system of fundamental elements which bonded together in quite different ways), and it would lack context sensitivity since there would be no other larger object which its instances could serve to construct.

(13.) In fact, many-many realization may hold in a way different than what is described by (MMR). For the definition focuses on the plasticities of F and G with respect to each other. Yet one could choose a set of properties s which satisfies (MR) and a set of properties t which satisfies (MRC) where none of the properties in s bear any determinative relation to those in t, and vice versa. The sets would be completely disjoint, a "one-many, many-one" relation rather than the unified "many-many" relation of (MMR).

(14.) Of course, whether there is deep compositional plasticity depends upon the absence of necessary coextensions created by logico-mathematical operations, e.g., disjunctions of realizing properties (more precisely, disjunctions of realizing properties conjoined with their contexts/ base-level determiners of those contexts, since a disjunction of uncontextualized properties [G or H ...] will be instantiated in situations where F is absent, given the context sensitivity of each disjunct). In any case, the standard Putnam-Fodor argument against materialist type-type identity theories is best interpreted as the claim that psychofunctional properties have maximally deep multiple realization, lacking necessary and sufficient conditions within their possibly infinite physical realization bases. For another extension of plasticity which focuses on the variability within domains, e.g., species and individuals, see Endicott (1993) and Horgan's "strong multiple realization" (1993b, pp. 307-8; and forthcoming, sec. 2).

(15.) Two points. First, Churchland speaks of properties having causal powers rather than being identical to such powers. For present purposes, however, the difference is of no consequence. Second, although the model requires that [T.sub.R] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be analogous, Churchland speaks of the isomorphism between [T.sub.R] and [T.sub.B] because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is deduced from [T.sub.B], and because (according to Churchland and Hooker) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is part of [T.sub.B], developed out of its conceptual resources. See Churchland (1979, pp. 82-4). For criticism of this and other aspects of the Churchland-Hooker model, see Endicott (1998).

(16.) Cf. Kim's account in (1989) and (1992). On this view, a corrected domain-specific psychology [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] need not be a "equipotent image" or even a "rough analogue" of the original cross-species psychology [T.sub.R], given their different scope and explanatory power. But entire theories aside, the domain-specific types paired for reduction must themselves be analogous, e.g., "human pain" ([F.sup.*]) and the appropriate "human neurophysiology" ([G.sup.*]). And if so, then the problems pointed out in the text will apply. E.g., alter the background conditions and [G.sup.*] will serve to realize [E.sup.*] and not [F.sup.*] (rewire the brain and what was once human pain is now human pleasure). For other problems, see Endicott (1993).

(17.) I should add that Kincaid runs a different concern through the same pages, viz., that high-level background conditions will frustrate reduction (1990, pp. 577-82). I think he is right. But whether background conditions are specifiable in low-level terms, as the reductionist should want, is an issue that can only be settled on a case-by-case basis. It has nothing to do with the many-many relation per se. It is a problem about particular laws and their limiting assumptions.

(18.) Other reductive demands are not secured by virtue of the same one-way conditionals either, e.g., that one could derive the theory in which F is embedded from the theory in which G is embedded (Kitcher, 1980; see also Kim, 1996, pp. 212-16), or, indeed, that there could be any ontologically retention, given the radical cross-classification noted in sec.V. Parenthetically, one can accommodate any indeterminacy that might arise from quantum interactions by taking the conditionals in question to express the statistical probability of G being followed by F.

(19.) Assuming, of course, the clause that all background conditions are determined by the B properties, otherwise one could envision a violation of supervenience--simply hold property G in B constant, with variations on C (which now, ex hypothesi, are not determined by differences in B) explaining why F is instantiated on one occasion and not another--a difference in A without a difference in B. (MMR) will, however, fail to imply more specific forms of supervenience without suitable provisos. Strong supervenience, e.g., typically expresses individualism or local determination, and it leaves the background conditions unstated (Kim, 1984, p. 65). What (MMR) entails, then, is strong, contextualized, locally-neutral supervenience, aping Kim's definition: "when property F in A is instantiated, there exists a property G in B and a context C such that G in C is instantiated, and, necessarily, if C holds, then if G is instantiated then so is F."

(20.) Supervenience is also compatible with the possibility that F and G are not identical, though F can be subserved only by G (context sensitivity but no compositional plasticity), or G can only subserve F (compositional plasticity but no context sensitivity).

(21.) The remark about "the same lower-level types determining different higher-level types" needs to be interpreted carefully. Granted, one can say "the same G determines E or F or ..., but relative to different contexts," as I have done here. On the other hand, the conditional "if C1 then if G then F" is logically equivalent to "if C1 and G then F," and the conjunctive property "C1 and G" that determines F is not the same conjunctive property "C2 and G" that determines some other E (Endicott, 1994, pp. 68-71). What remains the same in each case is the "core" property G, not the "total" G-plus-context (see Shoemaker, 1982, pp. 96-7). Perhaps the solution is to say that "determines" is ambiguous. Let "-- [determines.sub.c] ..." take core properties as values, and "-- [determines.sub.t] ..." take total context properties. The above remark is then true on the first reading, but not always on the second. E.g., the total property of having C-fibers-fire-plus-entire-physical-history-of-the-organism could probably serve to realize only pain, not pleasure. There might even be a general metaphysical principle in the offing, viz., the wider the property, the less variability it has to enter into diverse determinative relations.

(22.) My thanks to Charles Carr, Thomas Grimes, Terence Horgan, Jaegwon Kim, and John Post for helpful comments on an earlier draft of this paper.


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Title Annotation:relations between the physical and the theoretical
Author:Endicott, Ronald P.
Publication:American Philosophical Quarterly
Date:Jul 1, 1998
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