8.7 Reasoning about Counterfactuals

The language ^→ can be used to reason about counterfactuals as well as defaults. Now the interpretation of a formula such as φ → ψ is "if φ were the case, then ψ would be true." In this section, φ → ψ gets this counterfactual reading.

Under what circumstances should such a counterfactual formula be true at a world w? Certainly if φ is already true at w (so that φ is not counter to fact) then it seems reasonable to take φ → ψ to be true at w if ψ is also true at w. But what if φ is not true at w? In that case, one approach is to consider the world(s) "most like w" where φ is true and to see if ψ is true there as well. But which worlds are "most like w"?

I am not going to try to characterize similarity here. Rather, I just show how the tools already developed can be used to at least describe when one world is similar to another; this, in turn, leads to a way of giving semantics to counterfactuals. In fact, as I now show, all the approaches discussed in Section 8.4 can be used to give semantics to counterfactuals.

Consider partial preorders. Associate with each world w a partial preorder ≽_w, where w₁ ≽_w w₂ means that w₁ is at least as close to, or at least as similar to, w as w₂. Clearly w should be more like itself than any other world; that is, w ≽_w w′ for all w, w′ ∈ W. Note that this means simple structures cannot be used to give semantics to counterfactuals: the preorder really depends on the world.

A counterfactual preferential structure is a preferential structure (for one agent) M = (W, , π) that satisfies the following condition:

Cfac^≽. If (w) = ≽_w, ≽_w), then w ∈ W_w and is the maximum element with respect to ≽_w (so that w is closer to itself than any other world in W_w); formally, w ∈ W_w and w ≻_w w′ for all w′ ∈ W_w such that w′ ≠ w.

Let ^pref_n consist of all (single-agent) counterfactual preferential structures. This can be generalized to n agents in the obvious way.

I have already given a definition for → in preferential structures, according to which, roughly speaking, φ → ψ holds if φ ∧ ψ is more likely than φ ∧ ψ. However, this does not seem to accord with the intuition that I gave earlier for counterfactuals. Fortunately, Theorem 8.4.5 shows that another equivalent definition could have been used, one given by the operator →′. Indeed, under the reinterpretation of ≽_w, the operator →′ has exactly the desired properties.

To make this precise, I generalize the definition of best_M so that it can depend on the world. Define

Earlier, best_M(U) was interpreted as "the most normal worlds in U"; now interpret it as "the worlds in U closest to w." The formal definitions use the preorder in the same way. The proof of Theorem 8.4.5 shows that in a general preferential structure M (whether or not it satisfies Cfac^≽)

That is, φ → ψ holds at w if all the worlds closest to or most like w that satisfy φ also satisfy ψ.

Note that in a counterfactual preferential structure, W_w is not the set of worlds the agent considers possible. W_w in general includes worlds that the agent knows perfectly well to be impossible. For example, suppose that in the actual world w the lawyer's client was drunk and it was raining. The lawyer wants to make the case that, even if his client hadn't been drunk and it had been sunny, the car would have hit the cow. (Actually, he may want to argue that there is a reasonable probability that the car would have hit the cow, but I defer a discussion of counterfactual probabilities to Section 8.8.) Thus, to evaluate the lawyer's claim, the worlds w′ ∈ W_w that are closest to w where it is sunny and the client is sober and driving his car must be considered. But these are worlds that are currently known to be impossible. This means that the interpretation of W_w in preferential structures depends on whether the structure is used for default reasoning or counterfactual reasoning.

Nevertheless, since counterfactual preferential structures are a subclass of preferential structures, all the axioms in AX^cond are valid (when specialized to one agent). There is one additional property that corresponds to the condition Cfac^≽:

C7. φ ⇔ (ψ ⇔ (φ → ψ)).

C7 is in fact the property that I discussed earlier, which says that if φ is already true at w, then the counterfactual φ → ψ is true at w if and only if ψ is true at w.

Theorem 8.7.1

AX^cond_n + {C7} is a sound and complete axiomatization for the language ^→ with respect to ^pref_n.

Proof I leave it to the reader to check that C7 is valid in counterfactual preferential structures (Exercise 8.50). The validity of all the other axioms in AX^cond follows from Theorem 8.6.3. Again, completeness is beyond the scope of the book.

Of course, rather than allowing arbitrary partial preorders in counterfactual structures, it is possible to restrict to total preorders. In this case, C5 is sound.

Not surprisingly, all the other approaches that were used to give semantics to defaults can also be used to give semantics to counterfactuals. Indeed, the likelihood interpretation also makes sense for counterfactuals. A statement such as "if φ were true, then ψ would be true" can still be interpreted as "the likelihood of ψ given φ is much higher than that of ψ given φ." However, "ψ given φ" cannot be interpreted in terms of conditional probability, since the probability of φ may well be 0 (in fact, the antecedent φ in a counterfactual is typically a formula that the agent knows to be false); however, there is no problem using possibility, ranking, or plausibility here. All that is needed is an analogue to the condition Cfac^≽. The analogues are not hard to come up with. For example, for ranking structures, the analogue is

Similarly, for plausibility structures, the analogue is

I leave it to the reader to check that counterfactual ranking structures and counterfactual plausibility structures satisfy C7, and to come up with the appropriate analogue to Cfac^≽ in the case of probability sequences and possibility measures (Exercises 8.51 and 8.52).