# Mathematics for Ethics

## Inspired by the work of Jeremy Bentham

### Introduction

Ethical reasoning requires the consideration of values. If we represent the values numerically, we can use standard mathematical operations to manipulate them, and hopefully produce an answer which we can use to guide our actions.

In this paper, knowledge of positive, negative, and fractional numbers, addition, subtraction, multiplication and division is assumed. Multiplication is represented by "*", division by "/", and priority by round brackets "()".

### The scale

The initial problem is one of representation - how can we represent things like pain and pleasure with numbers? Presumably, the same way we can represent distances, temperatures, or a multitude of other things as numbers - by using a scale. There are standard scales for distance (eg. metres, kilometres, and miles etc) and temperatures (eg. the Celsius (aka centigrade) and Fahrenheit scales). There are no standard scales for pleasure and pain (that I know of) so we have to invent our own. The important thing is that we use the scale consistently, and that we make sure it is linear and absolute. I may not know exactly how far a mile is, but, because the mile scale is absolute, I know that 0 miles is no distance at all; and because it is also linear, I can be certain that one mile is exactly half the distance of two miles. Because we invent the scale, we can set it as we wish - to suit our particular problem - and it works fine so long as we use it consistently, just as we might use metres to measure the length of a person's stride and miles to measure the separation of two cities.

For example, if I have two chocolate bars, and I am considering the ethics of the situation, I might assign the value of the pleasure I get from eating a chocolate bar to the number "1". (This does not imply that "2" is the pleasure I get from eating both chocolate bars because, after the first, I've had my fill of chocolate for a while. [That is, the principle of Declining Marginal Utility applies.]) If we have an absolute and linear scale, "0" must be no pleasure at all, and "2" must be twice the pleasure I get from eating a chocolate bar, which is the same as the pleasure of two people each eating a chocolate bar, assuming that they like chocolate as much as me.

If we have a scale such that pleasures have a positive numerical value, then we can choose to represent pain and suffering on the same scale using negative numbers. For example, if I had a blister on my foot, I might think the pleasure from eating a chocolate bar isn't worth the suffering it would take to walk to the shop and buy one - in which case, I can say the value of the suffering caused by me walking on a blistered foot is less than (ie. more negative than) -1.

### Felicific Calculus

Once we have decided on the units (ie. the scale), we can attempt to think about the problem mathematically. The factors for us to consider when evaluating interests are:
• intensity
• duration
• extent
• certainty 

#### Intensity and duration

The value of an interest depends on the intensity of it, and the duration of it. For example, an intense pleasure is prima facie more valuable than a less intense one; and a pain of a given intensity is worse the longer it lasts. When choosing the scale for the value of interests, it makes sense for 0 to represent "no interest", positive numbers to represent positive interests (pleasures or avoided pains), and negative numbers to represent negative interests (pains or missed pleasures).

On some occasions, as I did above, we might immediately estimate the value of a certain interest - combining the intensity and duration factors automatically. At other times, we might consider them separately. For example, in the ongoing chocolate bar example, whether or not it is worth limping to the shops depends on how long it will take me to get there (ie. for how long I'll have to suffer to get the chocolate bar). We can model this mathematically using multiplication:
total suffering = average suffering per unit time * amount of time.
Notice by using the average suffering per unit time, I have not committed myself to the assumption that the suffering is constant - I may well find that suffering per unit time of walking on a blistered foot depends on how much walking I've already done on it. I can calculate the average amount of suffering per unit time over a given period of time using:
average suffering per unit time = (suffering per unit time at the beginning of the period + suffering per unit time at the end of the period) / 2
as long as we assume a constant rate of change in the period.
If I know the amount by which the suffering will increase in each unit time, then:
suffering per unit time at the end of the period = suffering per unit time at the start of the period + (rate of change of suffering * the length of the period)

#### Extent

The extent of the interest is the number of individuals to which the interest applies. Because we consider all equal interests equally ("Each counts for one, and none for more than one") this is also modelled using multiplication:
total suffering = average suffering per individual * number of individuals.
If the interest is different for each person, we might prefer to think of this as
total suffering = suffering of first individual + suffering of second individual + suffering of third individual etc etc, for all concerned.

#### Certainty

Because we are often attempting to calculate the value of a given action before we have performed it, there is usually an element of uncertainty as to what the actual consequences will be. There is a standard mathematical technique for dealing with uncertainty and probability, and I will attempt an explanation here.

We usually consider that the actual outcome will be one of a possible set of (mutually exclusive) outcomes. If we represent the probability of the outcome occurring as a fraction (with 1 representing certainty and 0 representing impossibility, 0.5 representing a 50% ie. 1 in 2 chance etc) then the sum of the probabilities of the set must be 1 - if it was less than one, then either there is an alternative outcome we have not considered, or we have underestimated the probabilities of some of the outcomes - because we assume that one of the outcomes must actually occur (ie the probability of the result being one of the possible results is certainty, ie. 1). Notice that in this case the probability of A or B occurring is the probability of A occurring + the probability of B occurring.

When we deal with more complicated situations, ie. when there are secondary outcomes which become more or less likely depending on the initial outcome, then it may be helpful to draw the possibilities as a tree, where each branch represents an outcome, and the hierarchy indicates which possible outcomes follow from previous outcomes.

##### An example
I am considering committing an illegal act. I know that I can perform this act: the problem is how to handle the uncertainty as to whether I'll get caught by the police and punished. I consider that there are three possible outcomes (assuming that I decide to do this act).
• Outcome 1: I perform the act and do not get caught by the police.
• Outcome 2: I perform the act, get caught by the police, and get found guilty and punished by the courts.
• Outcome 3: I perform the act, get caught by the police, but get found "not guilty" by the courts.
The other alternative is that I choose inaction (outcome "0"). The situation is modelled by this tree: The tree consists of nodes and links. A node represents a situation, and a link indicates which situations can follow from which previous situations. The triangular nodes (called "payoff nodes") represent outcomes, and are numbered as above. The square node at the top (the root node "r") represents my position. It is square because it represents a choice I can make (it is a "decision node"): I can choose to do the act and go down the path to "a", or I can choose inaction and go down the other. The circular nodes represent situations which are resolved by chance, and are called "chance nodes". Chance node "a" represents the situation in which I choose to do the action; it leads to node "1" where I get away with it, and to node "c" where I get caught. The number by a link from a chance node indicates the (estimated) probability of that link being followed, given that we are already in the position above. (For example, it only makes sense to talk about the probability that I get caught in the case where I do choose to do the action - I don't get to position "a" until I've decided to act; and nodes "2" and "3" can only be reached from "c" - they can't try me if they don't catch me.)

The next step is to estimate the value for each situation represented by a payoff node. When estimating, we must consider intensity, duration, and extent of the interests involved (as compared to a baseline: usually inaction) - basically everything except certainty. I consider that, if I choose not to do the act, then everything continues as before: inaction has the value 0. I consider the value of my action (ignoring any cost to me) to be 1 , the value of being tried as -0.2 (due to the inconvenience etc), and the value of being tried, found guilty and punished as -2.  This gives the payoff values

• Node "0": 0
• Node "1": 1
• Node "2": 1 + (-2) = -1
• Node "3": 1 + (-0.2) = 0.8
We can now calculate the tree value, and from this learn which "branch" I should pick: I could, in a larger analysis, have many other branches (links) from the root node, indicating things I could choose to do instead of the particular act that I'm thinking of. I would then choose the node with the highest value.

The values are calculated from the bottom up. A decision node has the largest of the values of the nodes that follow from it - it is assumed we always choose to do what we think is most valuable. A chance node has an "expected" value  which is sum of the values of its sub-nodes, multiplied by their respective probabilities.

In this example, the expected value of "c" is:
(the probability of reaching node "2" * the value of node "2") + (the probability of reaching node "3" * the value of node "3")
= (0.75 * -1) + (0.25 * 0.8)
= -0.75 + 0.2
= -0.55

The expected value of "a" can now be calculated as:
(the probability of reaching node "1" * the value of node "1") + (the probability of reaching node "c" * the value of node "c")
= (0.8 * 1) + (0.2 * -0.55)
= 0.8 + (-0.11)
= 0.69

The next node up being a decision node, its value is simply the largest of node "a" and node "0":
max(0.69, 0) = 0.69.

Since the expected value for node "a" is higher than that for node "0", and "a" represents what happens when I do this action, the tree therefore indicates that I should do the action (in preference to not doing it). If I could create a tree for a different course of action which (when using the same scale) returns a value greater than 0.69, then obviously I ought do that instead .

The procedure which has been described here, followed carefully, should avoid a number of the most common mistakes in moral mathematics.

### Notes

1. I describe fewer factors than did Bentham. "Propinquity" appears to be a factor used to model the way an interest appears more or less valuable, depending on how far away it is in space or time (see Chapter XIV rule XVI), and since were are here concerned with actual rather than apparent value, I ignore it. The remaining factors ("fecundity", and "purity") can, I believe, be properly accounted for using the existing factors, providing we take care to allow for all effects, including "side effects", "indirect effects" etc.

I make no account of Mill's contribution: "quality". To the extent that the concept is intelligible, which isn't much, it seems entirely accountable with "intensity" and "fecundity". (I assume that there is a consensus on this - otherwise, we would hear people say such things as "I suffered a great deal, but it was only low quality suffering so it wasn't too bad" or "I had a lot of fun, but it gave only low quality happiness so it wasn't very good".) As an analogy to what I think Mill meant: we might assume that some pleasures were more valuable than others, in the same way that a given mass of gold might be more valuable than another given mass of gold of a different "quality" - perhaps the second mass is 9 carat, and the first is 18 or 24 carat. However, this doesn't really work because 1 unit of 9 carat gold isn't really 1 unit of gold - it is a nine "twenty-fourths" of a unit of gold, the rest of the mass being other metals. So under this interpretation quality and quantity (e.g. of real gold) cannot be distinguished in this way, which is why quality of pleasures can be accounted for by intensity.

Another possible interpretation comes from Mill's explanation of a higher quality pleasure being chosen in preference to a lower quality pleasure. This may mean simply that we like the idea of a certain means to happiness, more than we like the idea of some other means to happiness - we may like to think ourselves as someone who does the former but not the latter activity. But again on this interpretation Mill fails, since we are not here evaluating how much we like the idea of the experience of happiness gained in this way, but how much we like the experience of happiness itself. Any positive feelings associated with the idea of the thing are valuable (and evaluable) quite separately from the value of the thing itself.

2. ie. I'm setting the (arbitrary) units of the scale to be the same as the benefit of my action - all other values will be measured in comparison to this benefit.

3. Notice that the punishment is of uncertain type, duration etc, and thus disvalue, but that I have given it a definite value anyway: we can often continue analysing probabilities indefinitely, so at some point we simply estimate the value at a given level. If deeper analysis is required, this node should become a chance node with links to several other nodes, each representing different punishments.

4. The name "expected value" might be considered to be a mis-nomer: we do not actually expect this value to emerge - in fact is it generally impossible for it to do so (only the values of the payoff nodes can actually be achieved) - but the expected value is a kind of average of the possible values after taking into consideration their likelihood.

5. This is does not contradict the principle of utility, as Derek Parfit explains in his thoroughly recommended book "Reasons and Persons": consequentialists generally use and distinguish two different meanings of right and wrong, and ought and ought not. There is the sense, as in the principle of utility, that what we ought do (what it is right to do) is to maximize utility. However, we generally do not know what the consequences of our actions will be, so what we aim to do is act in such a way as to maximize expected utility. (As explained in the previous note, the expected value of an act rarely emerges - only the values of the payoff nodes can do so, assuming that the model is accurate). So there are two different questions here: 1. what is the best thing to do? 2. what seems to be, or is likely to be, the best thing to do? Unfortunately, we often cannot find the answer to the first question, which is why the second question needs answering.

The name given to these two senses - of something being really right, and of something being probably or seemingly right (i.e. what we have most reason to believe will be right) - is "objective" and "subjective" respectively; though this is perhaps unfortunate in that it suggests a relation to the question of whether ethics are objectively or only subjectively valid, and this is a separate issue entirely. Often, we hope, what is subjectively right is also objectively right - the two coincide. However it is obvious that this is not always the case: we can sometimes do the (objectively) wrong thing (i.e. most harmful) even though we had tried to do the right thing (we had weighed up the consequences, applying equal consideration etc). And vice-versa: someone might do something that appears to everyone (himself included) to be wrong (i.e. it is subjectively wrong), but actually has the best consequences.

Examples displaying the difference between the objective and subjective meanings include these:

1. A man finds some children who have come into difficulty whilst swimming in a lake. He jumps in to try to save them, but gets into difficulty himself and they all drown. Clearly the man has acted heroically in risking his life for others - it was subjectively the right thing to do - even though, as it happened, it might have been better if he had done something else (it might be objectively wrong).
2. A man acting out of malice and perversion decides kills a child: clearly subjectively wrong. However, the person who the child would've grown into was a mass-murderer, or a second Hitler say, and in killing him the psychopath has actually saved many lives and relatives from the loss of their loved ones. So his act is, quite accidently, objectively right.
Clearly, if there is any use in it at all, it is generally subjectively wrong acts which are blameworthy and should be punished, and subjectively right acts which should be rewarded. One who takes needless risks with the wellbeing of another might well require punishment even if, in the event, the other is unharmed.
1998, 1999, 2000 