Desire-based systems

This is the first of five posts making up my essay on moral calculus.

I will focus here on a specific observable phenomenon, namely desire. Desires are brain states, and therefore observable – for this reason, the word “desire” is the only concept related to morality that will appear in this section. (It turns out that my justification for choosing desires stems from Alonzo Fyfe’s moral theory of desire utilitarianism, which states that our only reasons for action are desires. However, this information is not relevant to the logical steps that I shall take below – indeed, these steps would be the same no matter what reason I had given for choosing desires as my basic observable.)

1.1 Relationships between desires

Since desires are the basic observables for what follows, it behooves me to describe some of their properties:

  1. Desires are located in brains, which may or may not be human brains.
  2. A single brain may contain many desires.
  3. A single desire may be found in many brains.
  4. Desires have intensity: they can be weak or strong.
  5. The intensity of a desire can vary with time.
  6. Fulfillment of a desire may influence the likelihood of another desire being fulfilled.

The above properties fall into two classes. The first class contains properties that can be defined independent of other desires, such as intensity and location (by “location” I mean which brains the desire is found in). The second class contains properties that are only defined in relation to other desires (such as “desire A tends to fulfill desire B”). I begin by considering this second class. I will return to the first class later.

Desires can be either fulfilled or thwarted. Which of these two outcomes occurs might depend on whether other desires are fulfilled or thwarted. Consider the following numerical system which quantifies these inter-desire relationships:

Definition 1.1 Inter-desire Relationships

  1. If the fulfillment of desire A tends to fulfill desire B, then it has a value of 1.
  2. If the fulfillment of desire A tends to thwart desire B, then it has a value of -1.
  3. If the fulfillment of desire A has no influence on desire B, then it has a value of 0.

Note that the values thus assigned are not moral values, they are numeric values. As stipulated above, the current discussion is not concerned with moral language. Furthermore, these values are not absolute, but are defined relative to other desires. Thus, for instance, if desire A fulfills desire B but thwarts desire C, then desire A has a value of either 1 or -1 depending on which of desire B or C is being considered.

A little house-keeping is in order here. Below, I will use the following sort of language: “Desire A has a value of 1 with respect to desire B.” instead of using the more bulky language of Definition 1.1. Also, I adopt the convention that the value of a desire relative to itself is 0.

Definition 1 gives us conditions under which a desire has a direct influence on another desire. But what of indirect influence? For instance, if desire A has a value of 0 with respect to desire C, then it has no direct influence on that desire, according to the third part of Definition 1. But what if the value of desire A with respect to desire B is -1, and the value of desire B with respect to desire C is 1? In this case, desire C can be indirectly fulfilled by desire A.  We encapsulate this information in the following definition:

Definition 1.2 Direct and Indirect Influence.

  1. Desire A has a direct influence on desire B if it has a value of 1 or -1 with respect to desire B.
  2. Desire A has an indirect influence of desire B if it can be linked to desire B through an unbroken chain of desires, each with a direct influence on the next.

Let us use an example to explore what we have developed thus far. Suppose there are four relevant desires: A, B, C, and D. If A fulfills B, thwarts C, and has no influence on D, then according to Definition 1.1, desire A has values of 1, -1, and 0 with respect to these desires. We can write this information in tabular form as follows:

A B C D
A 0 1 -1 0

The number in the first column is 0 because it is the value of desire A with respect to itself. The next three numbers are the values of A with respect to B, C, and D, respectively. These relationships are depicted graphically in Figure 1.

Figure 1. The values of desire A with respect to three other desires (B, C, and D) are indicated by numbered arrows

A similar set of four numbers can be written down for each desire in the example. For instance, if desire B fulfills A and C, and thwarts D, then we can add a row to our table for desire B to give:

A B C D
A 0 1 -1 0
B 1 0 1 -1

Here, the 0 in the second column reflects the fact that the value of B with respect to itself is 0. This information is added to our graphical representation in Figure 2.

Figure 2. The values of desire B with respect to the other desires are now included. Note that in this example, desires A and B are mutually fulfilling, since each has a value of 1 relative to the other.

Similarly we could, given enough information, add rows to our table for desires C and D. Our description of the inter-desire relationships in this example would then be complete.

Before continuing, I must define what I mean by “relevant” desires. The word “relevant” implies that these desires have some relationship to a particular object. Since desires are our only observables, it makes sense for that particular object to be a desire. Therefore, desires are relevant if they can be influenced by a particular starting desire, which I call the focal desire*. Here then, is a concise recipe for constructing a set of relevant desires:

Definition 1.3 Relevant Desires

  1. Establish a focal desire. This is some desire that we are particularly interested in, for whatever reason.
  2. Find that set of desires that can be influenced directly or indirectly by the focal desire, as determined by Definition 2.
  3. Add the focal desire itself to the set.

Consider as an example the set of desires in Figure 2. If desire A is the focal desire, then desires B, C, and D are all relevant. Although desire A has no direct influence on desire D, it does have an influence on desire B, which in turn has an influence on desire D, satisfying the rule in Definition 1.2b concerning chains of desires. I could, if I wished, state that only those desires directly influenced by the focal desire are relevant. This would render desire D irrelevant in the example. However, I have no good reason to exclude indirect influence, and I must therefore include it. Practically, this approach could lead to an extremely large, complex web of indirectly connected desires. However, we cannot reject such an outcome purely out of distaste for messiness. For now, let us accept it and move on.

The above definition of relevance is in line with the approach taken in scientific theories. A scientist limits himself only to those observables that have some influence (direct or indirect) on the variable in question (the “focal” variable). For instance, if the scientist wishes to study the growth rate of oak trees, she is not going to be interested in monitoring the flight patterns of migratory birds. Instead, she will limit her study to those variables that influence plant growth, such as soil nutrient content, precipitation rate, and even ocean currents, which, through their relationship to climate conditions, may have an indirect influence on tree growth.

It is a matter of purely academic interest (at least for now) to determine how different desires compare to each other given their list of interrelationships. In particular, we might wish to know which of a group of desires tends to fulfill the greatest number of the remaining desires. Or, perhaps, we might wish to know which of a group of desires tends to thwart the greatest number of the remaining desires. More generally, we might wish to rank all of the relevant desires in order of their ability to fulfill one other. Here, we set ourselves the following particular problem:

Problem 1.

Given a set of desires containing a focal desire and all the desires relevant to it, how do these desires compare with one another in terms of their ability to fulfill other desires in the set?

Here is a possible solution to this problem:

Solution 1.

Define the rank of each desire as the sum of its values relative to the other desires in the set. The desire with the highest rank is the one which is most successful at fulfilling other desires in the set.

To see how this works, let us return to the example used above. We previously provided the values of desires A and B relative to all four desires in the set, and now we complete the problem by adding rows to the table for desires C and D:

A B C D
A 0 1 -1 0
B 1 0 1 -1
C 0 1 0 1
D -1 -1 1 0

Table 1. A complete description of inter-desire relationships for our example problem.

Note that, as we expect, the third value in the row for desire C is equal to zero and the fourth value in the row for desire D is zero, in keeping with the definition that the value of a desire with respect to itself is zero.

Solution 1 suggests that we rank each desire according to the sum of its values.  We thus have:

A B C D Ranking
A 0 1 -1 0 0
B 1 0 1 -1 1
C 0 1 0 1 2
D -1 -1 1 0 -1

Table 2. Inter-desire relationships, together with rankings computed as the sum of all values in a given row.

It is often difficult to see the proper relationship between rankings such as these, since some are negative, some are positive, and they cover a somewhat arbitrary range. We can simplify matters by normalizing the rankings, namely by scaling them to fall within the range of 0 to 1.  We do this by applying each ranking R to the following formula:

Rnorm = (RRmin) /(RmaxRmin)

where Rmin and Rmax are the minimum and maximum rankings, respectively.  In our example, we obtain the normalized values shown in Table 3:

A B C D Ranking Normalized Ranking
A 0 1 -1 0 0 0.33
B 1 0 1 -1 1 0.67
C 0 1 0 1 2 1
D -1 -1 1 0 -1 0

Table 3. Inter-desire relationships with rankings and normalized rankings.

We can now see clearly that if, as before, the focal desire is desire A, then our focal desire takes third place among the four rankings. Desire A is thus the third best desire in terms of its ability to fulfill the other desires in the set. The first place ranking goes to desire C, while the last place ranking goes to desire D. The normalized ranking of the worst and best desires will always be 0 and 1 respectively.

Has Solution 1 given us what we are looking for? Looking at Table 3, we can see that desire C fulfills two other desires (B and D) and thwarts none. Desire B is the only other desire that fulfills two desires (A and C), but it also thwarts one desire. It therefore makes sense that C should achieve the highest ranking. Similarly, we see that desire D, while fulfilling one desire (C), thwarts two desires (A and B), so it makes sense that it should achieve the lowest ranking.

It is important to note that the rankings obtained by Solution 1 are not dependent on which desire is designated as the focal desire. The focal desire is only important for choosing which desires should be included in the set (via Definition 1.3). Once the set has been established, the rankings are independent of which desire is the focus.

1.2 The Intensity of Desires

In the previous section, we introduced the following properties of desires:

  1. Desires are located in brains, which may or may not be human brains.
  2. A single brain may contain many desires.
  3. A single desire may be found in many brains.
  4. Desires have intensity: they can be weak or strong.
  5. The intensity of a desire can vary with time.
  6. Fulfillment of a desire may influence the likelihood of another desire being fulfilled.

We looked at possible computations that could be made if one assumed that inter-desire relationships (property 6) were of sole importance. In this section, we assume that one more property is also important: the intensity of desires.

The intensity of a particular desire may vary from one brain to the next. In the previous section, we were not concerned with variations from brain to brain because relationships between desires hold no matter which (or how many) brains contain them. For instance, the desire to diet tends to fulfill the desire to lose weight, regardless of how many people hold these desires or how strong they are (much like an arrow is able to maintain its direction no matter how long it is). This made our calculations in the previous section relatively easy.

The intensity of a desire, however, cannot be evaluated without making reference to the brains that contain that desire. In other words, we cannot say with any confidence that desire A is always more intense than desire B, no matter what brains contain these desires. We also have no justification in looking at the intensity of desires in just one individual’s brain. We must therefore consider the intensity of desires in all relevant brains. Our first step, then, must be to determine which brains are relevant:

Definition 1.4 Relevant Brains

Relevant brains are those containing at least one of the relevant desires.

(Relevant desires, of course, are defined in Definition 1.3.) Once we know which, and how many, brains we are dealing with, we can make definitions such as the following:

Definition 1.5 Total Intensity

The total intensity of a desire is the sum of its individual intensities in all relevant brains that contain it.

Consider, for example, a set of three relevant brains, which we label b1, b2, and b3. We observe that desire A is present in brains b2 and b3. In brain b2, desire A has an intensity of IA2 (the first subscript refers to the desire and the second subscript refers to the brain). In brain b3, desire A has an intensity of IA3. According to the above definition, the total intensity of desire A is IA = IA2 + IA3.

There are, of course, other ways in which we could represent the intensity of a particular desire. Instead of adding the intensities across the relevant brains, we could have averaged them instead, used the median value, etc. I make no claim that the summing of intensities is necessarily the best approach, since we currently have no metric against which the different approaches can be measured and compared.  My choice can therefore be considered arbitrary, at least for the moment.

One other important note is that all brains are implicitly assumed to be of equal importance. In other words, if the intensity of a particular desire is the same in all the relevant brains, then each brain contributes equally to the computation of total desire – no brain is conferred a special advantage. Once again, we have no information that would justify the special treatment of one brain over another, and we must therefore treat them equally.

As before, we find it of purely academic interest (for now) to inquire about the relationships between relevant desires. This time we are interested in how the intensity of desires affects their relationships. In particular, we wish to pose the following problem:

Problem 2.

We are given:

  1. A set of desires containing a focal desire and all the desires relevant to it,
  2. The set of brains relevant to this set of desires, and
  3. The set of desire intensities for each desire-brain combination.

How do the desires rank in terms of their ability to fulfill other desires with the greatest net intensity?

A very simple example serves to demonstrate what is being asked here. Consider only two desires (A and B) that are present in only one brain. Let us invent an arbitrary unit of intensity, which we henceforth call the “desiron”, that can later be converted into whatever units a neuroscientist would use when measuring the intensity of a desire in the brain. We observe that the intensity of desire A is 8 desirons, while the intensity of desire B is 27 desirons.  Furthermore, desire A has a value of 1 with respect to desire B, and vice versa. In other words, the two desires are self-fulfilling: A tends to fulfill B and B tends to fulfill A. Problem 2 asks us to rank these two desires in terms of their ability to fulfill other desires with the greatest intensity. In this case, it is desire A that should be ranked first, since it tends to fulfill a desire with a relatively high intensity (the intensity of desire B is 27 desirons), whereas desire B only tends to fulfill a relatively weak desire (the intensity of desire A is only 8 desirons).

This calculation is easy enough for a simple two-desire, one-brain problem, but what about more complex problems?  Let us set up such a problem, starting with the same set of four desires that we considered in the previous section. To recap, here are the basic relationships between those desires as reported in Table 1:

A B C D
A 0 1 -1 0
B 1 0 1 -1
C 0 1 0 1
D -1 -1 1 0

Table 1. A complete description of inter-desire relationships for our example problem.

Now we introduce the brains. Suppose there are three brains, b1, b2, and b3. Table 4 describes the intensity of each of the four desires as measured in each of the three brains.

b1 b2 b3
A 0 14 35
B 0 0 4
C 0 27 10
D 8 0 5

Table 4. The intensity of the four desires in the three brains, as measured in our arbitrary units (“desirons”).

A value of zero in Table 4 simply indicates the absence of a relevant desire in that particular brain. Thus, for instance, brain b1 contains only one of the four relevant desires (desire D), while brain b2 contains only two of the relevant desires (A and C).

We can now compute the total intensity of each desire. According to Definition 1.5, this is achieved by adding up the numbers in each row of Table 4. This yields the values shown in Table 5.

b1 b2 b3 Total Intensity
A 0 14 35 49
B 0 0 4 4
C 0 27 10 37
D 8 0 5 13

Table 5. Individual desire intensities, together with total intensities computed as the sum of all values in a given row.

We now offer a solution to Problem 2:

Solution 2.

  1. Multiply each entry in the desire-relationship table by the total intensity of the desire corresponding to the column that entry is in.
  2. Compute rankings in the manner prescribed by Solution 1, namely by summing the elements in each row of the table.

Let us start, for demonstration purposes, by considering the first entry in the bottom row of Table 1, which has a value of -1. This tells us that the value of desire D with respect to desire A is -1, i.e. desire D tends to thwart desire A. Step 1 of Solution 2 requires us to multiply this value by the total intensity of the desire corresponding to its column in Table 1, namely the total intensity of desire A.  We thus have -1 ´ 49 = -49. This is the new value that goes into the table. We can repeat the same procedure for all entries in the table, yielding Table 6.

A B C D
A 0 4 -37 0
B 49 0 37 -13
C 0 4 0 13
D -49 -4 37 0

Table 6. Desire values from Table 1 multiplied by total intensities from Table 5.

Step 2 of Solution 2 asks us to sum the values of each row to obtain our final rankings. This yields Table 7.

A B C D Ranking
A 0 4 -37 0 -33
B 49 0 37 -13 73
C 0 4 0 13 17
D -49 -4 37 0 -16

Table 7. Value-intensity table, together with rankings computed as the sum of all values in a given row.

Once again, our rankings take on a range of large negative and positive values, so it helps to normalize them, yielding the values in Table 8.

A B C D Ranking Normalized Ranking
A 0 4 -37 0 -33 0
B 49 0 37 -13 73 1
C 0 4 0 13 17 0.47
D -49 -4 37 0 -16 0.16

Table 8. Value-intensity table with rankings and normalized rankings.

We see that desire B now has the highest ranking whereas in the previous section, desire C had the highest ranking. Does this make sense? Previously, we were only concerned with how many desires were fulfilled or thwarted by a particular desire. Desire C therefore obtained the highest ranking because it fulfilled two desires (B and D) and thwarted none. Now, however, we see that desires B and D have relatively low intensities (desire B is only found in one brain, with an intensity of only 4 desirons, while desire D is found in only two brains, with respective intensities of only 8 and 5 desirons). So, although desire C still fulfills two desires, these desires are weak.

Does it make sense that desire B should obtain the highest ranking? In the previous problem, it got the second highest ranking, because it fulfilled two desires (A and C) but thwarted one (desire D). Now, however, we see that desires A and C happen to be the two most intense desires, while desire D is relatively weak. This distribution of intensities helps to boost the ranking of desire B.

Overall, the intensity of desire A is quite a lot higher than that of the other desires. This inevitably means that any desire that fulfills desire A is going to achieve a high ranking (e.g., desire B) while any desire that thwarts desire A is going to achieve a low ranking (e.g., desire D).

Before moving on to the next chapter, I note that the two problems offered above are by no means exhaustive. For instance, we could enquire how to rank desires in terms of their ability to fulfill desires with the greatest average desire intensity, rather than the greatest total intensity.

Next post on moral calculus.


*I use the term “focal” because it has fewer value-laden connotations than words such as “primary” or “principal” that might otherwise have come to mind. The focal desire is not one that is more important in some way than the other desires, it is simply the desire we happen to be interested in considering at the moment.

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