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Having laid down the foundations for a theory of causal significance in chapter 9, we can begin to investigate applications of its principles in order to get a firmer grasp on the concepts and derive some theoretical results. Rosenberg demonstrates applications of the theory of causal significance by using a simple toy physics, where there is only one type of levelzero effective property, charge, which can take on values of either + or . ('Charge' in this context is just a simple levelzero effective property introduced for the sake of this demonstration, and is not to be confused with the word as it is used in physics.)
Diagrams are presented in the text to illustrate instances of charge embedded in various kinds of causal circumstances. In these diagrams, charge is given a spatiotemporal context by being plotted against an axis of space and an axis of time. Also explicitly depicted in the diagrams is the underlying receptive structure connecting instances of charge in time and space, producing a directed graph where receptive connections are the edges and instances of charge are the nodes.
The following is a (crude) representation of what these diagrams look like in the text:
The +s and s are instances of charge located in space and time. The extended lines between them denote receptive connections binding them into causal nexii. These receptive connections create three distinct levelone natural individuals in this diagram. (In the text, boxes are drawn around the individuals to explicitly denote their presence.)
The first individual, at time t1 (call this I_{1}), is an individual with spatial breadth. The two 'o's on either end of the receptive connection represent the 'slots' in the receptivity into which the + and  'fit.' The instances of charge in I_{1} are symmetrically connected, meaning that each receives causal constraint from the other. Notice that this is a causal relation existing across space rather than across time. Causal relationships are thought of as strictly temporal according to our common sense notions of causation, but this need not be the case in the more general theory of causal significance.
The next individual, I_{2}, is located at time slice t2, extending across space similarly to I_{1}. A notable difference is that the rightmost 'o' in the receptive connection has been replaced with an arrowhead. This signifies that there is an asymmetric causal link between the  and +. The + is receiving causal constraint from the , but the  is not receiving causal constraint from the +.
Finally, there is an individual I_{3} that stretches across time, from t2 to t3. I_{3} corresponds to the traditional notion of a temporally extended causal process. Notice that the  at t2 is a member of both I_{2} and I_{3}; individuals can be bound to more than one instance of receptivity.
Rather than reproduce diagrams like the one above for each illustrative example, instead I will use the more concise notation for natural individuals introduced in chapter 9. The general form of this notation for a symmetrically connected natural individual I_{m} is [I_{n1.i}.I_{n1.j}...I_{n1.k}]_{n.m}. The brackets and dots represent I_{m}'s receptivity, which symetrically binds I_{m}'s component individuals, denoted as the 'I_{n1.i}' terms. n is the level of nature at which I_{m} exists, and the i, j, k, and m subscripts are labels used to refer to each distinct individual at each level of nature. For instance, I_{1} from the diagram above is written as [+_{0.1}._{0.2}]_{1.1} using this notation. A generic individual name can be replaced with its value in the notation in order to make its value explicit; for instance, if a charge I_{0.1} is positive, it can be written +_{0.1} to signifiy this fact. If its value is indeterminate, I will sometimes write it as ?_{0.1} in the following to emphasize its indeterminacy.
In the case where component individuals are bound asymmetrically, we use an arrow (=>) in place of the dots, pointing from the constraining individual to the constrained individual. So, I_{2} from the diagram above would be written as [_{0.3} => +_{0.4}]_{1.2}, and similarly I_{3} is written as [_{0.3} => +_{0.5}]_{1.3}. Note that this notation does not differentiate between temporally and spatially extended individuals, and for this reason the examples that will be introduced below will be ambiguous about whether the causal processes involved are spatial or temporal, even though they are explicitly presented as one or the other in the text. However, this ambiguity will not matter for most of the examples' illustrative and theoretical purposes. For those examples where a spatiotemporal context can be helpful for one reason or another, I will explicitly provide one.
The Character of Causal Processes: Tiers of Constraint
First tier: causal laws. There is still some work to be done in constructing our toy world: We must specify a causal law that describes the compatibility relationships that obtain among instances of charge that are bound by a common receptivity. This will be a universal law that operates on each set of instances of charge that share a common levelone receptivity. The causal law determines which joint states for the effective properties in a nexus are permissible and which are impermissible. A simple causal law for this toy world is as follows: Each value of charge, + or , must have an odd number of occurences in any natural individual where that value of charge occurs at all. (Note that the diagram above conforms to this causal law.)
Causal laws, receptive structures, and effective properties combine to realize operations of causal significance. The causal law describes the general relations of compatibility for effective properties; receptivity creates an infrastructure that defines the scope of an effective property's causal significance for other sets of effective properties; and the independently possible values of effective properties occupying that receptive infrastructure constrain each other directly, as described by the causal law. As Rosenberg sums it up:
The process of causal constraint. Let us analyze how these principles apply to specific causal circumstances. Let's start with three instances of charge which are, as of yet, indeterminate. Call these C_{0.1}, C_{0.2}, and C_{0.3}. Given no receptive context, there are eight potential joint states for these charges: {+,+,+}, {+,+,}, {+,,+}, {+,,}, {,+,+}, {,,+}, {,+,}, and {,,}. These are the charges' independently possible joint states. Now suppose that these instances of charge are bound up into the following receptive structures:
I_{1.1}: [?_{0.1}.?_{0.2}]_{1.1}
I_{1.2}: [?_{0.2}.?_{0.3}]_{1.2}
Given a receptive context, causal laws can now operate. The manner in which causal laws operate is by placing constraints on effective properties' (EPs') independently possible joint states. The EPs themselves can only instantiate those values which are consistent with the joint states that are not filtered out by the causal law. In some cases, exclusion of certain joint states entails the exclusion of certain possible values for an EP, in which case the EP becomes more determinate.
Recall that the causal law for this world is that the number of instantiations of a particular value of charge, + or , within a levelone causal nexus must be either zero or odd. Thus, both I_{1.1} and I_{1.2} can only take on one of two states, either [+.] or [.+]; [+.+] and [.] are excluded from possibility by the causal law. Furthermore, note that the charge C_{0.2} is shared by both I_{1.1} and I_{1.2}. As a consequence, it figures into the constraint relations of both individuals. In particular, whatever value it has in the one must be consistent with the value it has in the other, so (for instance) if I_{1.1} took on the determinate state [+_{0.1}._{0.2}], the only possible state left for I_{1.2} would be [_{0.2}.+_{0.3}].
Taking all this into consideration, the only possible joint states for the charges once they have been bound up in the particular way described by I_{1.1} and I_{1.2} are {+,,+} and {,+,}. Six of their eight independently possible joint states have been filtered out by the causal law, in conjunction with the particular receptive structure they entered into. As it turns out, in this example that constraint on the joint states is still not sufficient to determine any particular value for any of the charges. According to the first joint state allowed by the causal law, ?_{0.1} can take on a + value, and according to the second it can take on a  value, and likewise for the other charges. Suppose for an instant that the causal law had been different, such that the only allowable joint states were {,+,} and {+,+,+}. In this case, the filtering of joint states would have forced C_{0.2} into the determinate value +, although C_{0.1} and C_{0.3} would have remained indeterminate.
Tier 2: receptive structure. Notice that the structure of the receptive connection plays an important role in causal processes beyond just determining which effective properties come to have direct causal significance for each other. The structure of receptivity can also make a difference to the manner in which effective properties are constrained. For instance, suppose the three charges were receptively bound within just one individual:
I_{1.3}: [C_{0.1}.C_{0.2}.C_{0.3}]
In this case, the possible joint states allowed by the causal law are different than those detailed above, as a direct consequence of the new receptive structure. The possible joint states of the charges for I_{1.3} are {+,+,+} and {,,}, rather than {+,,+} and {,+,} as above.
Tier 3: independently determinate effective properties. Let's now consider the original causal situation described above, but this time with asymmetric receptive connections. That is, suppose the receptive structure existing among the three charges is as follows:
I_{1.1'}: [+_{0.1} => _{0.2}]
I_{1.2'}: [_{0.2} => +_{0.3}]
Recall that if two individuals A and B have a symmetric receptive connection, then both receive causal constraint from each other; on the other hand, if A and B have an asymmetric receptive connection, e.g. [A => B], then B receives causal constraint from A but not vice versa. In Rosenberg's formulation, all it means for two individuals to be asymmetrically connected is that the constraining individual is already determinate when considered independently of the nexus of which it is a member. (So, for instance, in an individual X of the form [A => B], we could say that A is determinate when considered independently of X.) The reasoning is that if an individual is already determinate, it cannot receive further causal constraint by definition, because it is already maximally constrained. On the other hand, an indeterminate individual is always receptive to further constraint, and so is always on the receiving end of receptive connections.
Thus, the formulation for the two revised individuals above can be read as follows. The charge +_{0.1} is determinate when considered independently of I_{1.1'} (perhaps it has been forced to take on the + value because of constraints placed on it by other charges in other causal nexii). In essence, it is a 'given' in this multiple constraint satisfaction problem. The causal law and receptive structure active here, in conjunction with the fixed value of +_{0.1}, thus forces C_{0.2} to take on a  value. _{0.2}, then, is likewise determinate when considered independently of I_{1.2'}, and likewise forces C_{0.3} to take on a + value. The specific possibility space of values for effective properties given to a causal nexus, then, adds another tier of constraint to the space of possible values that can pass through the filter of causal laws in a given receptive structure. The more determinate the effective properties are when considered independently of a causal nexus, the fewer possible joint states there are, which in turn further constricts the possible values the effective properties can instantiate.
Higherlevel causation. The process of causal constraint described here for level 0 effective properties bound into level 1 individuals generalizes for higherlevel individuals. Suppose we have the following level 1 individuals:
I_{1.1}: [C_{0.1}.C_{0.2}.C_{0.3}]
I_{1.2}: [C_{0.4}.C_{0.5}]
I_{1.3}: [C_{0.4}.C_{0.6}]
(Note that C_{0.4} is a member of both I_{1.2} and I_{1.3}.) These can be receptively bound into a level 2 individual as follows:
I_{2.1}: [I_{1.1}.I_{1.2}.I_{1.3}]
The causal law on I_{2.1} can then act to exclude some of the independently possible joint states of the level 1 individuals, thus moving them and their constituents towards determinacy. Suppose that the causal law on level 2 individuals is a straightforward extension of that for level 1 individuals: i.e., the sum total of +s instantiated within the level 1 individuals bound within a level 2 individual must be either zero or odd, and likewise for the number of  values. The independently possible joint states for the level 1 individuals (also called the prior possibility space presented to I_{2.1}) are:
1) { [+.+.+]_{1.1}, [.+]_{1.2}, [.+]_{1.3} }
2) { [..]_{1.1}, [+.]_{1.2}, [+.]_{1.3} }
3) { [+.+.+]_{1.1}, [+.]_{1.2}, [+.]_{1.3} }
4) { [..]_{1.1}, [.+]_{1.2}, [.+]_{1.3} }
The causal law at work here eliminates 3) and 4) as potential joint states, thus constraining the space of the leve 1 individuals' possible effective states. An effective state for a level n individual I is defined as an ordered set of the level n1 effective properties which are bound within I. The notation we have been using to describe level 1 individuals, then, is basically just a manner of writing their effective states; thus [+.], [.+], [+.+.+], and [+...+....+] are some examples of what the effective state of a level 1 individual might look like. Notice that [+.] and [.+] are distinct effective states, because the ordering is different.
The ordering matters because each ordered item represents a unique 'slot' in an instance of receptivity, and the outcome of causal constraint in a nexus is sensitive to specifications of which individuals are bound to which instances of receptivity. For instance, suppose we have a receptive structure of two overlapping level one receptivities, e.g. [C_{0.1}.C_{0.2}] and [C_{0.2}.C_{0.3}.C_{0.4}]. C_{0.2} is a member of both individuals, and thus the value it takes on will affect the values of the other two charges. The two possible outcomes are [+._{0.2}] and [_{0.2}..], or [.+_{0.2}] and [+_{0.2}.+.+]. Without specifying an ordering here, we could not determine whether the second individual should be [+.+.+] or [..], even given that the other twoplace individual is already determinate.
Diagrams are presented in the text to illustrate instances of charge embedded in various kinds of causal circumstances. In these diagrams, charge is given a spatiotemporal context by being plotted against an axis of space and an axis of time. Also explicitly depicted in the diagrams is the underlying receptive structure connecting instances of charge in time and space, producing a directed graph where receptive connections are the edges and instances of charge are the nodes.
The following is a (crude) representation of what these diagrams look like in the text:
Code:
t3  +
 ^
t  
i  o
m t2   o> +
e 

t1  + oo 
L______________
space
The +s and s are instances of charge located in space and time. The extended lines between them denote receptive connections binding them into causal nexii. These receptive connections create three distinct levelone natural individuals in this diagram. (In the text, boxes are drawn around the individuals to explicitly denote their presence.)
The first individual, at time t1 (call this I_{1}), is an individual with spatial breadth. The two 'o's on either end of the receptive connection represent the 'slots' in the receptivity into which the + and  'fit.' The instances of charge in I_{1} are symmetrically connected, meaning that each receives causal constraint from the other. Notice that this is a causal relation existing across space rather than across time. Causal relationships are thought of as strictly temporal according to our common sense notions of causation, but this need not be the case in the more general theory of causal significance.
The next individual, I_{2}, is located at time slice t2, extending across space similarly to I_{1}. A notable difference is that the rightmost 'o' in the receptive connection has been replaced with an arrowhead. This signifies that there is an asymmetric causal link between the  and +. The + is receiving causal constraint from the , but the  is not receiving causal constraint from the +.
Finally, there is an individual I_{3} that stretches across time, from t2 to t3. I_{3} corresponds to the traditional notion of a temporally extended causal process. Notice that the  at t2 is a member of both I_{2} and I_{3}; individuals can be bound to more than one instance of receptivity.
Rather than reproduce diagrams like the one above for each illustrative example, instead I will use the more concise notation for natural individuals introduced in chapter 9. The general form of this notation for a symmetrically connected natural individual I_{m} is [I_{n1.i}.I_{n1.j}...I_{n1.k}]_{n.m}. The brackets and dots represent I_{m}'s receptivity, which symetrically binds I_{m}'s component individuals, denoted as the 'I_{n1.i}' terms. n is the level of nature at which I_{m} exists, and the i, j, k, and m subscripts are labels used to refer to each distinct individual at each level of nature. For instance, I_{1} from the diagram above is written as [+_{0.1}._{0.2}]_{1.1} using this notation. A generic individual name can be replaced with its value in the notation in order to make its value explicit; for instance, if a charge I_{0.1} is positive, it can be written +_{0.1} to signifiy this fact. If its value is indeterminate, I will sometimes write it as ?_{0.1} in the following to emphasize its indeterminacy.
In the case where component individuals are bound asymmetrically, we use an arrow (=>) in place of the dots, pointing from the constraining individual to the constrained individual. So, I_{2} from the diagram above would be written as [_{0.3} => +_{0.4}]_{1.2}, and similarly I_{3} is written as [_{0.3} => +_{0.5}]_{1.3}. Note that this notation does not differentiate between temporally and spatially extended individuals, and for this reason the examples that will be introduced below will be ambiguous about whether the causal processes involved are spatial or temporal, even though they are explicitly presented as one or the other in the text. However, this ambiguity will not matter for most of the examples' illustrative and theoretical purposes. For those examples where a spatiotemporal context can be helpful for one reason or another, I will explicitly provide one.
The Character of Causal Processes: Tiers of Constraint
First tier: causal laws. There is still some work to be done in constructing our toy world: We must specify a causal law that describes the compatibility relationships that obtain among instances of charge that are bound by a common receptivity. This will be a universal law that operates on each set of instances of charge that share a common levelone receptivity. The causal law determines which joint states for the effective properties in a nexus are permissible and which are impermissible. A simple causal law for this toy world is as follows: Each value of charge, + or , must have an odd number of occurences in any natural individual where that value of charge occurs at all. (Note that the diagram above conforms to this causal law.)
Causal laws, receptive structures, and effective properties combine to realize operations of causal significance. The causal law describes the general relations of compatibility for effective properties; receptivity creates an infrastructure that defines the scope of an effective property's causal significance for other sets of effective properties; and the independently possible values of effective properties occupying that receptive infrastructure constrain each other directly, as described by the causal law. As Rosenberg sums it up:
With these three tiers of constraint laid out, the causal significance view of causation is essentially in place. It represents the objective core of causation in terms of nature's solving the determination problem by turning it into a multiple constraint satisfaction problem. The constraints operate on a space of possibility, each effective property is cast as a variable whose potential values are possible solution values, and the receptive connections themselves emerge as operators on this space of possibilities that creates structure within which these variabls can be related to one another. (pg. 191192)
The process of causal constraint. Let us analyze how these principles apply to specific causal circumstances. Let's start with three instances of charge which are, as of yet, indeterminate. Call these C_{0.1}, C_{0.2}, and C_{0.3}. Given no receptive context, there are eight potential joint states for these charges: {+,+,+}, {+,+,}, {+,,+}, {+,,}, {,+,+}, {,,+}, {,+,}, and {,,}. These are the charges' independently possible joint states. Now suppose that these instances of charge are bound up into the following receptive structures:
I_{1.1}: [?_{0.1}.?_{0.2}]_{1.1}
I_{1.2}: [?_{0.2}.?_{0.3}]_{1.2}
Given a receptive context, causal laws can now operate. The manner in which causal laws operate is by placing constraints on effective properties' (EPs') independently possible joint states. The EPs themselves can only instantiate those values which are consistent with the joint states that are not filtered out by the causal law. In some cases, exclusion of certain joint states entails the exclusion of certain possible values for an EP, in which case the EP becomes more determinate.
Recall that the causal law for this world is that the number of instantiations of a particular value of charge, + or , within a levelone causal nexus must be either zero or odd. Thus, both I_{1.1} and I_{1.2} can only take on one of two states, either [+.] or [.+]; [+.+] and [.] are excluded from possibility by the causal law. Furthermore, note that the charge C_{0.2} is shared by both I_{1.1} and I_{1.2}. As a consequence, it figures into the constraint relations of both individuals. In particular, whatever value it has in the one must be consistent with the value it has in the other, so (for instance) if I_{1.1} took on the determinate state [+_{0.1}._{0.2}], the only possible state left for I_{1.2} would be [_{0.2}.+_{0.3}].
Taking all this into consideration, the only possible joint states for the charges once they have been bound up in the particular way described by I_{1.1} and I_{1.2} are {+,,+} and {,+,}. Six of their eight independently possible joint states have been filtered out by the causal law, in conjunction with the particular receptive structure they entered into. As it turns out, in this example that constraint on the joint states is still not sufficient to determine any particular value for any of the charges. According to the first joint state allowed by the causal law, ?_{0.1} can take on a + value, and according to the second it can take on a  value, and likewise for the other charges. Suppose for an instant that the causal law had been different, such that the only allowable joint states were {,+,} and {+,+,+}. In this case, the filtering of joint states would have forced C_{0.2} into the determinate value +, although C_{0.1} and C_{0.3} would have remained indeterminate.
Tier 2: receptive structure. Notice that the structure of the receptive connection plays an important role in causal processes beyond just determining which effective properties come to have direct causal significance for each other. The structure of receptivity can also make a difference to the manner in which effective properties are constrained. For instance, suppose the three charges were receptively bound within just one individual:
I_{1.3}: [C_{0.1}.C_{0.2}.C_{0.3}]
In this case, the possible joint states allowed by the causal law are different than those detailed above, as a direct consequence of the new receptive structure. The possible joint states of the charges for I_{1.3} are {+,+,+} and {,,}, rather than {+,,+} and {,+,} as above.
Tier 3: independently determinate effective properties. Let's now consider the original causal situation described above, but this time with asymmetric receptive connections. That is, suppose the receptive structure existing among the three charges is as follows:
I_{1.1'}: [+_{0.1} => _{0.2}]
I_{1.2'}: [_{0.2} => +_{0.3}]
Recall that if two individuals A and B have a symmetric receptive connection, then both receive causal constraint from each other; on the other hand, if A and B have an asymmetric receptive connection, e.g. [A => B], then B receives causal constraint from A but not vice versa. In Rosenberg's formulation, all it means for two individuals to be asymmetrically connected is that the constraining individual is already determinate when considered independently of the nexus of which it is a member. (So, for instance, in an individual X of the form [A => B], we could say that A is determinate when considered independently of X.) The reasoning is that if an individual is already determinate, it cannot receive further causal constraint by definition, because it is already maximally constrained. On the other hand, an indeterminate individual is always receptive to further constraint, and so is always on the receiving end of receptive connections.
Thus, the formulation for the two revised individuals above can be read as follows. The charge +_{0.1} is determinate when considered independently of I_{1.1'} (perhaps it has been forced to take on the + value because of constraints placed on it by other charges in other causal nexii). In essence, it is a 'given' in this multiple constraint satisfaction problem. The causal law and receptive structure active here, in conjunction with the fixed value of +_{0.1}, thus forces C_{0.2} to take on a  value. _{0.2}, then, is likewise determinate when considered independently of I_{1.2'}, and likewise forces C_{0.3} to take on a + value. The specific possibility space of values for effective properties given to a causal nexus, then, adds another tier of constraint to the space of possible values that can pass through the filter of causal laws in a given receptive structure. The more determinate the effective properties are when considered independently of a causal nexus, the fewer possible joint states there are, which in turn further constricts the possible values the effective properties can instantiate.
Higherlevel causation. The process of causal constraint described here for level 0 effective properties bound into level 1 individuals generalizes for higherlevel individuals. Suppose we have the following level 1 individuals:
I_{1.1}: [C_{0.1}.C_{0.2}.C_{0.3}]
I_{1.2}: [C_{0.4}.C_{0.5}]
I_{1.3}: [C_{0.4}.C_{0.6}]
(Note that C_{0.4} is a member of both I_{1.2} and I_{1.3}.) These can be receptively bound into a level 2 individual as follows:
I_{2.1}: [I_{1.1}.I_{1.2}.I_{1.3}]
The causal law on I_{2.1} can then act to exclude some of the independently possible joint states of the level 1 individuals, thus moving them and their constituents towards determinacy. Suppose that the causal law on level 2 individuals is a straightforward extension of that for level 1 individuals: i.e., the sum total of +s instantiated within the level 1 individuals bound within a level 2 individual must be either zero or odd, and likewise for the number of  values. The independently possible joint states for the level 1 individuals (also called the prior possibility space presented to I_{2.1}) are:
1) { [+.+.+]_{1.1}, [.+]_{1.2}, [.+]_{1.3} }
2) { [..]_{1.1}, [+.]_{1.2}, [+.]_{1.3} }
3) { [+.+.+]_{1.1}, [+.]_{1.2}, [+.]_{1.3} }
4) { [..]_{1.1}, [.+]_{1.2}, [.+]_{1.3} }
The causal law at work here eliminates 3) and 4) as potential joint states, thus constraining the space of the leve 1 individuals' possible effective states. An effective state for a level n individual I is defined as an ordered set of the level n1 effective properties which are bound within I. The notation we have been using to describe level 1 individuals, then, is basically just a manner of writing their effective states; thus [+.], [.+], [+.+.+], and [+...+....+] are some examples of what the effective state of a level 1 individual might look like. Notice that [+.] and [.+] are distinct effective states, because the ordering is different.
The ordering matters because each ordered item represents a unique 'slot' in an instance of receptivity, and the outcome of causal constraint in a nexus is sensitive to specifications of which individuals are bound to which instances of receptivity. For instance, suppose we have a receptive structure of two overlapping level one receptivities, e.g. [C_{0.1}.C_{0.2}] and [C_{0.2}.C_{0.3}.C_{0.4}]. C_{0.2} is a member of both individuals, and thus the value it takes on will affect the values of the other two charges. The two possible outcomes are [+._{0.2}] and [_{0.2}..], or [.+_{0.2}] and [+_{0.2}.+.+]. Without specifying an ordering here, we could not determine whether the second individual should be [+.+.+] or [..], even given that the other twoplace individual is already determinate.
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