Hybrid systems

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

The reader has probably noticed similarities between the desire-based and happiness-based systems derived in the previous two posts. A comparison of some of the basic features of these two systems is given in Table 13.

Desire-based Happiness-based
Observable desires happiness
Measurement scale intensity emotional value
Communication between observables logical* action

Table 13. The main features of our desire-based and happiness-based systems.

Desires and happiness are separate observables, but they are dependent on each other through actions. When it is stated that desire A fulfills desire B, this means that any action taken to fulfill desire A is also an action that fulfills desire B. Conversely, when it is stated that desire A thwarts desire B, this means that any action taken to fulfill desire A is one that prevents any action that would fulfill desire B. This relationship is depicted in Figure 3, which represents the desires and emotional value in a single brain. In this figure, an action is performed that satisfies a particular desire, causing an increase in emotional value.

Figure 3. A brain (b1) contains desires and an emotional value. A particular desire (desire B in this case) is fulfilled by performing an action (action b) through muscle movement. Action b is detected by the senses and causes an increase in the emotional value.

Of course, the systems we have considered generally involve multiple brains. We add a brain to our graphical representation in Figure 4. Here, the action that the first brain causes to be performed is sensed by the second brain and affects its emotional value.

Figure 4. The situation depicted in Figure 3 is repeated here, with the addition of another brain (b2) which observes action b and undergoes its own change in emotional value. Click on the figure to enlarge.

The moral problems posed in our desire-based discussion were solved without explicitly referring to specific actions. But we can see from Figure 4 that actions are the mechanisms by which desire-desire relationships are realized. What is special about the desire-based system, then, is that it makes generalizations about actions. The claim that desire A fulfills desire B means that any action that fulfills desire A simultaneously fulfills desire B. The claim that desire A thwarts desire B means that any action that fulfills desire A simultaneously thwarts desire B. Problems would thus arise for the desire-based system if the fulfillment of desire A through one action fulfilled desire B, while the fulfillment of desire A through a different action thwarted desire B.

The happiness-based system, on the other hand, makes explicit reference to which action is performed. It does not, however, acknowledge explicitly which desires are fulfilled or thwarted by a given action, it simply considers how that action influences emotional value.

Can we unify the two systems? If so, we must make a proper link between desires and emotional value. Perhaps the most obvious way forward is to link the intensity of desires with emotional value. We could simply let the emotional value of a brain be equivalent to the sum of the desire intensities in that brain. However, emotional value is a measure of the degree of happiness in a brain, and it must therefore depend on which desires are presently fulfilled or thwarted, as well as those that await future fulfillment or future thwarting. Thus far, when discussing desires, we have implicitly assumed that these are desires that await their fate, namely those that may be fulfilled or thwarted at some later time once another desire is fulfilled. These desires are thus better related to the possible changes in happiness we discussed in Problem 3. In Problem 4 we included the initial (present) emotional value which, if it is be related to desires, must be related to both the history of desire fulfillment in the brain and on desires awaiting fulfillment. This, I feel, is a complex matter that is not easily addressed by adapting the preceding analyses, and I consider it no further.

However, we can unify the desire-based and happiness-based systems in the context of Problem 3, which makes no reference to the present emotional state. To begin, let us make a formal definition:

Definition 3.1 Equivalence of Emotional Value and Desire Intensity

The change in emotional value brought about by a particular action is equivalent to the total intensity of the desires fulfilled or thwarted by that action.

This definition essentially puts desire intensity and emotional value on the same scale: it states that our two systems of units introduced previously, namely desirons and emotons, are the same. In other words, a neuroscientist measuring changes in emotional value is observing the same thing as the neuroscientist who measures the intensity of desires. This definition also unifies the goals of Problems 2 and 3: the goal of Problem 2 was to maximize the net intensity of desires and the goal of Problem 3 was to maximize the change in happiness. These two goals are now identical. We must, however, choose which of the two variants of this goal (desires or emotional value) we wish to use from this point forward. So as not to discard any information from the problem, let us retain the more detailed variant, namely the desire-based one. Emotional value, according Definition 3.1, is a sum of desire intensities and it therefore carries no information about the individual desires. If we wish to retain this information, we must continue to use the language of desires, rather than emotional value.

We thus have two key components in our new system: desires and actions. A particular desire, if fulfilled by a particular action, will cause other desires to be fulfilled or thwarted. As with all the systems considered until now, we must establish relevance: in this case, we must establish which desires and which actions are relevant. I take the same approach to actions as I did for Problem 3: it is more practical to consider some relatively small, prescribed set of possible actions than the almost inexhaustible set of actions that could have some influence on desire fulfillment. Given this set of actions, then, we must determine which desires are relevant. In Problem 2, we defined relevance by starting with a focal desire and determining which desires could be fulfilled or thwarted by that desire. However, under our unified system, it is actions that fulfill or thwart desires. We must therefore seek that set of desires that can be either fulfilled or thwarted by any of the chosen actions.

Definition 3.2 Relevant Desires

Relevant desires are those which can be fulfilled or thwarted by the performance of at least one action in the set of relevant actions.

Finally, we must also determine which brains are relevant. As before, this is important because the intensity of a desire can vary from brain to brain. Consequently, the performance of a particular action might have different effects in different brains. Since the neurological observables we are working with are desires, it makes sense to use Definition 1.4 to specify the relevance of brains, namely that relevant brains are those containing at least one of the relevant desires.

We are now in a position to pose a similar problem to Problems 2 and 3:

Problem 5.

We are given:

  1. A set of actions,
  2. The set of brains relevant to this set of actions,
  3. The set of desires relevant to the set of actions, and
  4. The set of intensities of these desires in the relevant brains.

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

Notice how the language of the question posed here is a hybrid of the questions from Problems 2 and 3.

Before offering a solution to Problem 5, we return to the examples used in previous sections. When working with desire-based systems, we used an example involving four desires (A, B, C, and D), and three brains (b1, b2, and b3). We use these desires and brains again here, including the same desire intensities, which were introduced in Table 4, which we reproduce for convenience below.

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”).

As we did before, we can 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 introduced in Table 5, which we reproduce below.

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.

Another component of our desire-based system was the relationship between desires. Now, however, we need to relate desires and actions. This involves a switch to the happiness-based system. In this system, we associated each action with a change in emotional value. Now, we associate each action with a set of desires that would either be fulfilled or thwarted by that action. To represent such relationships mathematically we simply use the same scheme introduced by Definition 1.1, with some small changes in wording:

Definition 3.3 Desire-Action Relationships

a. If action a tends to fulfill desire A, then it has a value of 1.

b. If action a tends to thwart desire A, then it has a value of -1.

c. If action a has no influence on desire A, then it has a value of 0.

Note that I adopt lower case letters to describe actions, and upper case letters to describe desires, in keeping with Figures 3 and 4. We see that a particular action will have different values with respect to different desires, and we can therefore draw up a table (Table 13) of desire-action values similar to the table of desire-desire values (Table 1).  In this case, we assume that there are five relevant actions (a, b, c, d, and e).

A B C D
a 0 1 1 -1
b -1 0 0 0
c 1 -1 -1 -1
d 0 1 1 0
e -1 1 1 1

Table 13. A complete description of action-desire relationships for our example problem.

To take a specific example from Table 13, we see that the performance of action a will fulfill desires B and C. The performance of action a will also thwart desire D and will have no influence on desire A.

From Table 13, we can see that there is a special relationship between desires B and C, namely that any action that fulfills desire B also happens to fulfill desire C. This would allow us to make the general statement (of the kind we made in the desire-based system) that desire B fulfills desire C, regardless of the specific action that communicates this relationship. (This is not, in fact, quite as big a generalization as it sounds: I should really say that the relationship holds for any relevant action. There may be irrelevant actions that contradict the relationship.)

We can now offer a solution to Problem 5, modeled after the solution to Problem 2.

Solution 5.

  1. Multiply each entry in the action-desire 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.

The first step of this solution is to multiply the total intensities in Table 5 by the values in Table 13. The result is shown in Table 14.

A B C D
a 0 4 37 -13
b -49 0 0 0
c 49 -4 -37 -13
d 0 4 37 0
e -49 4 37 13

Table 14. Desire-action values from Table 13 multiplied by total intensities from Table 5.

The second step of our solution is to compute rankings by summing the values in each row, as shown in Table 15. We also compute normalized rankings for ease of interpretation.

A B C D Ranking Normalized Ranking
a 0 4 37 -13 28 0.86
b -49 0 0 0 -49 0
c 49 -4 -37 -13 -5 0.49
d 0 4 37 0 41 1
e -49 4 37 13 5 0.60

Table 15. Value-intensity table with rankings and normalized rankings.

Action d has the highest ranking. Performing this action results in two fulfilled desires (B and C) and no thwarted desires. This is not an overwhelmingly positive result, but it is certainly better than the other actions: all other actions thwart at least one desire. Indeed, actions b and e actually thwart the most intense desire (desire A). The only action (action c) that fulfills the most intense desire also thwarts the remaining three desires, so its ranking is relatively low.

There is only one action (action e) that fulfills a greater number of desires than the highest ranking action. This would certainly recommend it, but unfortunately action e also thwarts the most intense desire, as already noted. Action e thus obtains only a moderately high normalized ranking (0.60).

Action a has the second highest ranking. Like the highest ranking action, action a fulfills two desires. However, it also thwarts a desire, which explains its failure to take the number one spot.

Action b has the worst ranking. It thwarts the most intense desire (desire A), and has no influence on the remaining desires.

I close this post by noting that the hybrid system just developed contains more information about the relevant variables than either the pure desire-based system or the pure happiness-bases system. The pure systems both adopt generalizations that are not necessarily warranted. The desire-based system, by stating that desire A fulfills desire B, assumes implicitly that any relevant action that fulfills desire A also fulfills desire B. On the other hand, the happiness-based system, by using emotional value, omits information about the intensities of individual desires.

For this reason, I generally favor the hybrid system in the discussion that follows. We must remind ourselves, however, of the original assumption that made this system possible, namely that changes in emotional value are dictated by the intensity of desires that stand to be fulfilled or thwarted.  If this assumption ceases to hold, the hybrid system collapses.

Next post on moral calculus.

Previous post on moral calculus.


*One desire fulfills or thwarts another according to logical rules.

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