Abstract
This paper sets out some new skeleton mathematically-couched models for dealing with supervenience in some, if not all, its many guises. Our models are based around a naïve invocation of a ‘topology’ induced on object sets by property sets. We have two aims: one is to provide an overview of supervenience with enough rigour and detail to act as a self-contained introduction to the subject; and the other is to set out our new approach—but without getting too bogged down in excessive mathematical, or indeed philosophical, niceties—so that it can be used as a foundation for further work. Throughout we have scattered examples and comments to help clarify the points we make. The early sections use our topological approach to recast what appears elsewhere in the literature. And in the latter sections of the paper, we introduce some new ideas and techniques which help in exploring the general structures underpinning supervenience.
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Notes
In this paper, we do not investigate the modes of dependence that might accompany a supervenience relationship.
[A] and [B] are added here.
Davidson’s unpacking is more subtle than we imply here. We simplify considerably as we are using his definitions solely as a segue into a general notion of supervenience.
Davidson restricts his mental states to dispositional attitudes. See Davidson (2001, pp. 209–210).
We shall take it here that any association between physical and mental states is taken at specified times. This can possibly be taken to contradict empirical evidence, for instance Libet et al. (2000), Libet (2006) and Klemm (2010), but this does not affect the central thrust of the arguments presented.
We ignore the possibility that by ‘alter’ Davidson intends to inject some time based aspects into his idea of supervenience. The methods we develop in Sect. 8 can be used to modify our static model of supervenience into one that can accommodate alterations in time. As such we couch our model in terms that take [B] as the contrapositive of [A].
We take the stance that we can talk of mental states supervening on brain states as a convenient expository tool here. We do not claim that this is what Davidson holds to or that it is the only way of cashing out supervenience relationships between the mental and the physical.
The use of ‘thinking’ is a convenient oversimplification here.
Earlier, when we considered Janet and John’s brains we considered them at a number of differing times. As such whenever we referred to John’s brain or Janet’s brain they would be taken as a separate object at each separate time; that is, we used the term ‘brain’ where we should, to be scrupulous, have used the term ‘John’s brain at time \(t_1\)’ or ‘Janet’s brain at time \(t_7\)’ etc.
For the moments, we shall only consider in our initial property sets those properties that are realised by at least one object in the specified object set. We make one exception here: for technical reasons we shall assume throughout that the empty property, \(P_\emptyset \), is a member of every property set, where \(P_\emptyset (O) =0 \,\, \forall O \), where ‘\( \forall \)’ ranges over all [possible] objects. Later, when we come to consider Boolean completion in Sect. 5, there will be properties that are not realised by any objects in the underlying object set and which are distinct from the empty property. We make a few further comments on this at the end of Sect. 10.
Strictly we ought to relate each property with a map from objects to either 0 or 1 and then use these maps as representatives of the properties—telling us whether an object possesses a given property or not. However, for ease, we allow some blurring of the idea of ‘property’ throughout this paper.
Two sets are disjoint if they have no elements in common.
A set of subsets is exhaustive if each element in question belongs to at least one of the subsets.
We say this as there are complications from some interpretations of quantum theory. In some interpretations, the possession of a property is not guaranteed by the object alone and is dependent of other aspects of the set up. We set issues of supervenience and quantum properties aside in this paper. For a brief discussion of properties in quantum theory, see Isham (1995).
For completeness, then, we shall introduce a further definition that we could call upon when distinction between the property sets is explicitly needed. We shall call a specification a triple consisting of a set of objects, a set of properties and the induced pre-topology. This will be denoted as: \(S_{i;j} \equiv \lbrace \mathbb {O}_{i};{\varPi }_{j};\varvec{T}_{i;j}\rbrace \). This allows us to make finer distinctions than the pre-topology definition alone. For instance, as we mentioned, the property F and the property not-F each induce the same pre-topology on a given object set so using pre-topologies alone does not allow us to take account of the differences between F and not-F. The specification, however, does. Two specifications are equal if they have both the same set of objects and the same set of properties. Most of the time, this sort of distinction will be clear and talk of pre-topologies alone will suffice so we shall have no explicit need for specifications in what follows.
In fact the example as we have unpacked it gives an example of symmetric supervenience. That is we also have \(\lbrace \mathbb {O}_{pictures}, {\varPi }_{pixel} \rbrace \looparrowright \lbrace \mathbb {O}_{pictures}, {\varPi }_{picture} \rbrace \) as we are assuming each different pixel configuration gives a unique notion of a picture.
We use \( \wedge \) here in place of \(\cap \) which we use below and in the rest of the paper.
Later, in Sect. 10, we set out a more efficient way of achieving the aims of this section.
Kim allows infinite conjunctions and infinite disjunctions for technical reasons and we shall allow countably infinite combinations, even though there may well be problems with this. For instance, when considering the property of being an open set of the real line and allowing infinite intersections. We shall ignore such difficulties here.
Some accounts use logic notation for the combination of properties: using \( \wedge \) for ‘and’ and \( \vee \) for ‘or’ and \( \lnot \) for ‘not’. From here on, we use mostly set theoretic notation—with the exception ‘\(\lnot \)’ for ‘not’—in place of this. We take it that our notation is directly translatable as follows: \( \wedge = \cap \) and \( \vee = \cup \).
There is an additional problem that is apposite to mention here: the use of properties in some interpretative versions of quantum theory. There it is often maintained that the objects under consideration, such as electrons, photons and so forth, do not possess properties prior to measurement: properties are ‘brought into being’ by the act of measurement. Further, the Heisenberg uncertainty principle is often taken to say there is an ontological restriction on what definite properties quantum objects can possess at any one time. So that an electron that possesses a definitive position in space then cannot possess any specified momentum. We shall ignore these issues for the moment and take it that property combinations are allowable under restricted circumstances and not make much of what these circumstances may be.
Recall that we use \(\varvec{T}_{i;a}\looparrowright \varvec{T}_{i;b} \) and such like as a casual way of denoting the supervenience relationship between property sets with respect to some specified object set.
For instance, see Kim (1993, p. 58).
We leave aside issues of finite versus infinite intersections in this paper. Strictly a [mathematical] topology allows countably infinite unions but only countably finite intersections.
We have simplified the notation by not specifying the object set in our notation. More strictly, we ought to write something such as \(Top: \lbrace P, \mathbb {O} \rbrace \longrightarrow \lbrace \mathcal {O} \in \mathbb {O}: P(\mathcal {O}) \rbrace =1 \) but this is cumbersome for the points being made here.
There is a subtlety here. If there are objects in \( \mathbb {O} \) that do not possess any of the properties under consideration, they will ‘disappear’ under the Top map. This is not a major concern because here we are using the Top map to explain why we distinguish between pre-topologies and topologies. And what is more, if we want to ensure all objects fall under the Top map regardless of what property set we start with, then we could use the ideas we set out in Sect. 10 and apply Top to our property set’s counterpart set.
Following the ideas set out in Sect. 10, we can see that if we were to use \( Top(\overline{{\varPi }}_{\mathbb {O}} )\) then we can easily build up a mathematical topology by using it to construct \( Top(\uplus (\overline{{\varPi }}_{\mathbb {O}}))\).
Many of these are collected in Kim (1993).
In this paper we tend to stick fairly closely to just a few flavours of supervenience. There are many more to be found in the literature, many of which are touched upon in McLaughlin and Bennett (2011).
The quote here is from page 80. The definition on page 65 has slightly different wording.
Kim notes that the quote comes from an unpublished manuscript. The manuscript is entitled, “Why Try to Bake an Intentional Cake With Physical Yeast and Flour?”.
For a related discussion, see Sider (1999).
We ignore issues with trans-world notions of time.
We could define other maps between the object sets that are not bijective and work out how they fit with supervenience, but it is not clear what immediate philosophical advantage this would furnish. We do not do so here as such non-bijective mappings are not generally found in the literature.
It is useful to note that, as we are using Boolean completed sets, our maps are closely related to continuous maps between the corresponding mathematical topologies. It is not clear what advantages would be gained by extending maps to those that are not ‘continuous’ or how we might reconcile non-continuous maps with supervenience in a philosophically interesting manner. We do not pursue this here.
For simplicity we restrict our cases here to those within a single universe/world and look only at the basic notion of supervenience as outlined in Sect. 2. Additionally we assume that all properties under consideration are realised by at least one object in the object set. We make a further comment on this last assumption at the end of Sect. 10.
Recall, all property sets include the empty property, \( P_\emptyset \). For simplicity we do not explicitly include \( P_\emptyset \) here.
Recall also that for a pre-topology we can have objects that sit outside the property sets, but for a topology we cannot. Further, allowing objects to outrun properties is generally not a problem unless our property set consists of all [possible] properties. However, it is not clear if it makes sense to countenance objects that cannot possess any properties, whereas the Boolean operations on property sets suggest we ought to be aware of the issue of empty properties.
This might have some implications for the way we understand purely logical forms of supervenience which we put aside here.
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Robson, D. Topological supervenience. Synthese 193, 2865–2897 (2016). https://doi.org/10.1007/s11229-015-0891-1
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DOI: https://doi.org/10.1007/s11229-015-0891-1