Earlier I discussed The Question, identifying this question as “what will I do?” I wish to present a systematic way of exploring this question, in order to motivate an exploration of the concept of value. I will begin, however, with value and a continuation of an attempt to place value in a mathematical context, an attempt which was begun by Jeremy Bentham with Utilitarianism.
A value quantifier is an amount of value which something may have. It says, in effect, “x has this much value.”
Value quantifiers have an ordering relation. Over value quantifiers there will be defined operators > and = which say that one value quantifier describes value greater than, or value equal to, the value described by another value quantifier. In addition there will be a value quantifier 0, which indicates no value.
We will stipulate that the ordering relation is a strict weak ordering: that is, for any two value quantifiers v and w, either v > w, w > v, or v = w. Having defined these operators it will be possible to place value quantifiers on a number line.
There are two types of value quantifier which a theory of value may include: positive value quantifiers (greater than zero), and negative value quantifiers (less than zero). Positive value quantifiers indicate something desirable, such as pleasure, happiness, or moral virtue. Negative value quantifiers indicate something undesirable, such as pain, sorrow, or moral vice.
Prima facie, a theory of value may include only negative value, or only positive value. For instance, in a pessimistic philosophy such as Buddhism we might have only negative value. However, if negative value exists then positive value also exists, because if negative value exists, then anything which gets rid of negative value will by virtue of this fact have positive value. Similarly, if only positive value exists, then anything which gets rid of positive value will by virtue of this fact have negative value. Therefore, to posit value at all will necessarily posit both positive and negative value.
A possibility space is a directed graph which represents some set of possible futures. Each node represents a possible state of affairs. When two nodes are joined by an arrow x -> y, this means, “y is a possible outcome of x.”
One node is called the “starting node,” and one or more nodes are called “ending nodes.” The starting node represents the initial or current state of affairs; the ending nodes represent final outcomes. The following rules apply, formalizing the notions of starting and ending:
1. There must exist exactly one node s such that there exist no arrows n -> s for all n. s is called the starting node.
2. There must exist at least one node e such that there exist no arrows e -> n for all n. Any such node e is called an ending node.
It follows that for every node n that is neither the starting node nor an ending node, there exists some node m such that m -> n, and some node o such that n -> o. Such nodes are called intermediate nodes.
A “solution” of a possibility space represents a single possible future. It is a sequence of nodes such that:
1. The first node is the starting node.
2. The last node is an ending node.
3. For all nodes n and m which are adjacent in the sequence with n preceding m, there is an arrow n -> m.
A choice space is an extension of a possibility space. It captures the notion that possibilities are determined by various actors.
An actor is either an individual or a factor. An individual is a person making decisions. A factor is something else which contributes unpredictability to the possibility space.
A choice space consists of a possibility space, a set of actors, and a “choice map.”
A choice map gives, for every node n:
1. A set of pairs (a, C) where a is an actor and C is the set of choices for that actor.
2. A map which takes a set of pairs (a, c) as input (where a is an actor and c is a choice) and produces as output a node of the possibility space. The input must give one choice for every actor which acts at this node. The output must be some node m such that n -> m: other words, the output node must be a possible outcome of the current node.
The map must map every permutation of actor-choices onto exactly one possible outcome. This means that the total number of actor-choices must be equal to the total number of possible outcomes of this node.
A choice space is equivalent to a game in game theory.
A solution to a choice space is the same as a solution to a possibility space.
A value schema ascribes value to the solutions of a possibility space. It is a function from possibility space solutions to value quantifiers.
The constructs just given — possibility spaces, choice spaces, and value schemas — are sufficient to formalize most human reasoning about decision making, and also most moral dilemmas. Let us take a few examples.
First consider Sartre’s dilemma. “Sartre  tells of a student whose brother had been killed in the German offensive of 1940. The student wanted to avenge his brother and to fight forces that he regarded as evil. But the student’s mother was living with him, and he was her one consolation in life. The student believed that he had conflicting obligations.”
Here the choice space can be drawn as follows:
There are two actors here: the student, and the external factors which determine whether or not he succeeds in avenging his brother and fighting evil in the war. The choice points are the student’s decision to care for his mother or to fight, and in the latter case the external factors determining whether he succeeds or fails. There are three solutions to this choice space. He cares for his mother, he fights and succeeds, and he fights and fails.
These solutions are both evaluated under two independent value schemas. One value schema is the student’s desire to care for his mother. The other value schema is his desire to avenge his brother and fight evil in the war. Each of these value schemas gives a value quantifier for each solution. The value quantifiers for the first value schema are given in blue, and the value quantifiers for the second value schema are given in red.
+A means the value the student’s mother derives from having him stay. -A means the loss of this value. +B means the value created by the student avenging his brother and fighting evil in the war. -B means the loss of this value.
If the student stays with his mother, then he gains the value of consoling his mother and loses the value he would have gained from vengeance and fighting evil. If the student fights and suceeds, then he loses the value of consoling his mother and gains the value of vengeance and fighting evil. If the student fights and fails, then he still loses the value of consoling his mother, and within the value schema of vengeance and fighting evil, nothing is gained or lost.
What we see is that here the dilemma is created by the conflict between two incommensurable value schemas.
Next let us consider Sophie’s choice. “In the novel Sophie’s Choice, by William Styron (Vintage Books, 1976 — the 1982 movie starred Meryl Streep & Kevin Kline), a Polish woman, Sophie Zawistowska, is arrested by the Nazis and sent to the Auschwitz death camp. On arrival, she is ‘honored’ for not being a Jew by being allowed a choice: One of her children will be spared the gas chamber if she chooses which one. In an agony of indecision, as both children are being taken away, she suddenly does choose. They can take her daughter, who is younger and smaller. Sophie hopes that her older and stronger son will be better able to survive, but she loses track of him and never does learn of his fate. Did she do the right thing? Years later, haunted by the guilt of having chosen between her children, Sophie commits suicide. Should she have felt guilty?”
The following choice space illustrates this situation:
Here there are three value schemas.
+S/-S: The value obtained by having the son live, or lost by having the son die.
+D/-D: The value obtained by having the daughter live, or lost by having the daughter die.
-Ra/-Rb: The negative value of the responsibility that Sophie has for having daughter and/or son die. -Ra is the value of the responsibility she has for the death of her children in failing to choose. -Rb is the value of the responsibility she has for the death of the child whom she chooses to let die. It is perhaps not clear which of these values is greater.
Now let us consider Pascal’s wager:
Here there are two simultaneous actors. The first actor represents my decision to believe in God, or not to believe in God. The second actor represents the external factor that it is either the case or not the case that God exists. The interaction of these two actors produces four solutions. These solutions are evaluated under two value schemas. One value schema represents the value of my salvation or damnation. One value schema represents the positive value of my holding the true belief regarding God, versus the negative value of my holding the false belief regarding God.
If God exists and I believe in God, then I am saved (+S) and hold a true belief (+T).
If God does not exist and I believe in God, then I am not saved (0) and hold a false belief (-T)
If God exists and I do not believe in God, then I am damned (-S) and hold a false belief (-T).
If God does not exist and I do not believe in God, then I am not saved (0) and hold a true belief (+T).
Hopefully these examples are sufficient to suggest that this type of scheme can be applied to many situations, and perhaps every situation, in which it is necessary to make choices. These concepts line up in essence with those of game theory and decision theory, and so there is nothing particularly novel about them.
I have found this type of scheme to very useful in organizing my thinking on practical decisions that I have to make in my life.
A recurring characteristic of moral dilemmas, which distinguishes them from more mundane choice problems such as the problem of selecting chess moves, is that all moral dilemmas seem to involve multiple incommensurable value schemas. This may in fact be a characteristic of difficult choice problems in general. If a choice problem involves only one value schema, then we can solve it by estimating the probabilities of various external factors and finding ideal choices given the choices of any other actors in the situation. The problem will be one belonging to decision theory or game theory. If we have multiple value schemas, then not merely the difficulty but the very nature of the problem changes.
H1. The most difficult choices are those which involve conflicting value schemas.
In order to solve a choice problem with multiple value schemas in the kind of mathematical way that we are able to solve choice problems with single value schemas, what we require is some way to unify the value schemas.
Suppose that, given a choice problem with multiple value schemas, we could construct a value schema T and a function F to map from a set of pairs (S,v) with S a value schema and v a value quantifier under that schema, such that the set represents all of the value quantifiers of a given solution to the choice problem, to a new single pair (T,u) with T the unifying value schema and u the unified value quantifier.
One way to construct F would be to simply take the sum of all of the value quantifiers. This, however, would require that the value quantifiers be assigned fixed points on a number line, which would be difficult or even absurd in some cases. For instance, in Sartre’s dilemma we would be required to place numerical values on the student taking care of his mother or on his avenging his brother. We may regard this as absurd. Similarly with Sophie’s choice, we would have to place numerical values on the value of her son living or her daughter living, and it seems outrageous to put a numerical value on the value of a human life. It is for these reasons that I have used undefined variables for value quantifiers.
These quantifiers can be discussed in a relative way, within a single value schema: e.g., two dollars are twice as valuable as one dollar. Or, if Sophie’s son living is worth N, then Sophie’s son dying is worth -N. But to relate the value quantifiers of one value schema to the value quantifiers of another unrelated schema is a task that appears in most cases to be impossible. For example, which is more valuable: two dollars or two seconds of inner peace? Or, which is more negatively valuable: Sophie’s guilt about choosing to let her daughter die, or Sartre’s student’s mother’s hopelessness when her son leaves to fight?
We say therefore that value quantifiers belonging to different value schemas are seemingly “incommensurable.” They seemingly cannot be compared to one another in any meaningful way. This forms a significant problem in making choices where conflicting value schemas are involved, and it is a problem that is deeper than merely logical.
If a choice problem does not involve multiple value schemas, then it can be solved by estimating probabilities and predicting the choices of other actors. If a choice problem does involve multiple value schemas, then it touches upon the almost profound problem of reconciling these value schemas.
The question becomes, “what is of value?” Furthermore, if we can answer this question, then answering the question “what will I do?” becomes “easy” in the sense of becoming merely a technical problem. Having now motivated it, I will explore the question “what is of value?” in the future.