Robustness

When we obtain new information, this new information may require us to revise our beliefs. And it may be that the impact of this new information on a certain belief is very different from the impact on another belief. For example, it may happen that the new information, F, reduces the subjective probability of a belief, S1, substantially without reducing the subjective probability of another belief, S2. Frank Jackson ("On Assertion and Indicative Conditionals", Philosophical Review (1979), pp. 565-89; Conditionals (1987), Oxford: Blackwell) describes such a situation as one where S2 but not S1 is robust with respect to F. Put another way, your belief in S is robust with respect to F when, or to the extent that, you would not abandon your belief in S were you to learn F.

Let us take an example due to Jackson. Suppose I read in the paper that Hyperion won the election. You ask me who won the election. I say "Either Hyperion or Hydrogen won." In saying so, I am not saying a false thing; but everyone would agree that I have done the wrong thing. I have misled you deliberately. In this case, the disjunction is not robust with respect to the negation of the first disjunct. If I come to know that in fact Hyperion was not the winner I would have to abandon the disjunction. Now, consider the following modification to this case. What I read is "H------- won". The name is too blurred for me to do more than pick out the initial letter. However, I happen to know that Hyperion and Hydrogen are the only to horses in the election whose names begin with 'H'. You ask me who won the election and I say "Either Hyperion or Hydrogen won". In this modified case, the disjunction is robust with respect to the negation of both its disjuncts taken separately. I would still believe the disjunction even if I come to know that Hyperion was not the winner, or even if I come to know that Hydrogen was not winner.

Jackson defines robustness more precisely in terms of the conditional probability. If we accept the plausible thesis that the impact of new information is given by the relevant conditional probability, then "A is robust with respect to B" will be true just when Pr(A) and Pr(A|B) are close and both high. If we are concerned with cases where Pr(A) is high enough to warrant assertion, "A is robust with respect to B" means that Pr(A|B) is high.

Robustness and Conditionals

The reason robustness is important to conditionals is that it plays a part in Jackson's (1979, 1987) defense of the material conditional account of indicative conditionals. Jackson holds that the indicative conditional (A→C) has the same truth conditions as the corresponding material conditional (A⊃C). But Jackson insists that the semantic truth about the 'if' of indicatives is not exhausted by the equivalence thesis that A→C = A⊃C. According to him, there is more to its meaning than the thesis and the two conditionals are not synonymous. When someone says A→C, Jackson holds, she assert only Asup;C, so that if the latter is true she has spoken truly; but she also conveys to her hearers something further that he does not assert but merely implicates.

For example, compare the two sentences "A and B" and "A but B". The truth conditions for "A but B" are the same as those for "A and B". But the use of "A but B" carries a conventional (not conversational) implicature that "A and B" does not. "A but B" suggests that A and B stands in some kind of contrast. (For the difference between Jackson's conventional implicature and Grice's conversational implicature see chapter 5 of Jackson's Conditionals (1987) and pp. 35-8 of Bennett, A Philosophical Guide to Conditionals (2003).).

Another such an example is the word 'even'. The two sentences "Joe can solve this problem" and "Even Joe can solve this problem" have the same truth conditions. But it is evident that there is some difference between them; while the first is nice about Joe, the latter is not so nice about Joe. These examples show that in our language there are some devices or conventions to signal or suggest things without their being outright asserted.

Jackson maintains that 'if' of indicatives is such a device. So the indicative conditional conveys, in addition to their truth conditions, a conventional implicature to the effect that the assertion as a whole is robust with respect to the antecedent. That is, when the speaker says A→C, she says that A⊃C, but she also signals her confidence that, should the antecedent prove to be true, her utterance as a whole still stands. In this way, according to Jackson, robustness is an important ingredient in assertibility of indicative conditionals. As a part of its meaning, he holds, the indicative conditional also has an assertibility condition that the material conditional lacks, i.e. robustness with respect to its antecedent. (Similarly, if "A or B" is assertible, then it should remain assertible upon learning that A is false or upon learning that B is false. That is, "A or B" is assertible when it is robust with respect to both not-A and not-B.) Thus, Jackson's theory is a "supplemented" equivalence theory of conditionals. Although A→C and A⊃C have the same truth conditions, there is something more to say about A→C, namely, that in using it you explicitly signal the robustness of A→C with respect to A.

By linking assertibility of indicatives with a notion of robustness, Jackson is able to support or explain a thesis due to Ernest Adams (1975, The Logic of Conditionals), the thesis that the assertibility of an indicative conditional is the conditional probability of its consequent given its antecedent. (Assertibility of A→C is Pr(C|A)). According to Jackson's account, it is proper to assert A→C only when A→C is robust with respect to A, that is, when Pr (A→C|A) is high. Given that A⊃C is equal to ¬A∨C, Pr(A→C|A) is equal to Pr(¬A∨C|A), which amounts to Pr(C|A). Hence, according to Jackson's account, A→C is assertible only if the conditional probability Pr(C|A) is high.

Another attractive feature of Jackson's theory is that it has the resources to deal with the paradoxes raised against the equivalence thesis. It is objected, for example, that the equivalence thesis implies that "¬A; therefore, A→C" and "C; therefore, A→C" are valid inference patterns, which is counterintuitive. Jackson's reply is that the paradoxes arises through confusing truth with assertibility, i.e., through maintaining that certain conditionals are not true because they are not assertible. Jackson maintains that "¬A; therefore, if A, then C" is valid, i.e., truth preserving, but not assertibility preserving. We can infer the truth of A→C merely from ¬A, but we cannot assert A→C merely on the ground that ¬A is true. This is because our assertion of the conditional would not be robust with respect to A. The upshot is that (1) there is a discrepancy between truth- and assertibility-preserving inference involving indicative conditionals; and (2) our intuitions about valid reasoning with conditionals are apt to concern the latter, and so to be poor evidence about the former.

Robustness₁ and Robustness₂

So far, robustness has been introduced by what was in effect a double definition: one in terms of probability, the other in terms of what would happen if something were learned. Lewis (1986, Postscript to "Probabilities of Conditionals and Conditional Probability" in Philosophical Papers II, pp. 152-6) distinguishes robustness1 and robustness2 respectively:

A is robust₁ with respect to B iff P(A) and P(A/B) are close, and both are high.

A is robust₂ with respect to B iff P(A) is high, and would remain high even if one were to learn that B.

Robustness₁ is robustness as Jackson officially defines it; and it is the implicature of robustness1 that explains why assertibility of conditionals goes by conditional probability. But Lewis says that our reason for wanting to say what is robust, and for needing signals of robustness, seem to apply to robustenss₂. Most of the time, the two are typically the same or nearly the same. But when the two senses of robustness come apart in special cases, which one does the indicative conditional signal? Lewis says what really matters is robustenss₂, so it would be more useful to signal that. Relatedly, Stalnaker uses robustness₂ to define conditional belief, and seeks to align it with robustness1 through a general stipulation of rationality. (1984, Inquiry, pp. 104-5. Also see Bennett (2003), pp. 119-22.)