I suppose this is a more mathematical than philosophical post, but it’s related to issues that have come up in my thoughts on probability and decision theory.
When talking about orderings, we often use three connected relations, symbolized by the three symbols (<, ≤, ≈). The standard axioms are:
- (Ir)reflexivity: it is not the case that x<x, and it is the case that x≤x and x≈x.
- (Anti)symmetry: If x<y then it is not the case that y<x; if x≤y and y≤x then x≈y; if x≈y then y≈x.
- Transitivity: If x<y and y<z then x<z; if x≤y and y≤z then x≤z; if x≈y and y≈z then x≈z.
We usually say in these conditions that < is a partial ordering of the strict type, ≤ is a partial ordering of the weak type, and ≈ is an equivalence relation.
When the relevant ordering involved is total (also called a linear ordering), we add the additional axiom:
- Trichotomy: For any x and y, either x<y or y<x or x≈y; either x≤y or y≤x.
Of course, if we’re using all three symbols together, we normally want one more condition:
- Coherence: If x<y is true, then x≤y is true and x≈y is false. If x≤y is true, then either x<y or x≈y is. If x≈y is true, then x<y is false and x≤y is true.
Given this coherence condition, we can reduce the number of axioms drastically by taking coherence to be a definition of one or two of the symbols in terms of the others, rather than an axiomatization of them. In the presence of trichotomy, it works just to lay down the axioms for ≤, and then define > as the negation of this relation, and ≈ as the symmetric part of it. However, without trichotomy, it becomes slightly tougher.
As a result, Jim Joyce uses both ≤ and < in defining the preference ranking of an agent in his book The Foundations of Causal Decision Theory, because he doesn’t want to presume the ordering is total, and thus he can’t define the strict relation as just the negation of the converse of the weak one. (Though he does define the equivalence relation as the intersection of the weak relation with its converse.)
However, it seems to me that a more natural way to define < and ≈ in terms of ≤ is to use antisymmetry of ≤ to define ≈, and then say that x<y holds just in case x≤y does and x≈y doesn’t. This latter definition is the same as saying that x<y just in case x≤y holds and y≤x doesn’t – in the case of a linear ordering we just needed to ask that y≤x doesn’t hold in order to get x<y, but here we also need to require x≤y. I believe there are a few results Joyce could have phrased more economically with this definition, instead of defining a ranking by using two independent relations.
However, there might be reasons to keep both definitions separate. For instance, given a preference ranking on the potential outcomes of actions, there’s a natural way to get at least some rankings on the actions themselves. (I will let s denote a possible state of the world, f and g denote possible actions, and f(s) and g(s) denote the corresponding outcomes of these actions.)
- Dominance: If for every state s we have f(s)<g(s) then we should have f<g; if for every state s we have f(s)≤g(s) then we should have f≤g; If for every state s we have f(s)≈g(s) then we should have f≈g.
However, as it stands, this dominance principle leaves some preference relations among actions underspecified. That is, if f and g are actions such that f strictly dominates g in some states, but they have the same (or equipreferable) outcomes in the others, then we know that f≥g, but we don’t know whether f>g or f≈g. So the axioms for a partial ordering on the outcomes, together with the dominance principle, don’t suffice to uniquely specify an induced partial ordering on the actions.
The natural solution to this situation would be to take one or another of the three dominance principles to be basic and to use the resulting defined relation to fully specify the other two by coherence. The most natural way to do this is to use ≤-dominance and ≈-dominance to define the two corresponding relations, and then say that f<g just in case ≤ holds and ≈ doesn’t. This corresponds to the common supposition that one action is strictly better than another if in some situations it is strictly better, and in no situations is it worse. One should be indifferent between them iff one is indifferent between their outcomes in every state.
However, we normally feel that if two actions differ in outcome only on states with a total probability of zero, then the difference isn’t significant. This contradicts the idea that if the only difference is on a set of probability zero, and one action is strictly better than the other on that set, then the former action should be strictly preferred. But there are reasons to be indifferent in this case (not least of which is the fact that we can often permute the set of states in some seemingly unimportant way and transform one action into an action the recommendation of the previous paragraph would strictly prefer). As a result, it seems that we have to live with the incompleteness given by these dominance principles, at least until we come up with some other means of filling out the preference relation on actions (which is of course the goal of decision theory). Knowing about probabilities will certainly help here, but we may have to make these distinctions in circumstances where probabilities are difficult or impossible to come by.