In chapter 4 of Articulating Reasons, Brandom argues that singular terms and predicates have a certain sort of inferential behavior. Predicate isn’t quite the correct label. Rather, he concerns himself with sentence frames, which include predicates (Fx) but also include predicates made of, e.g., logically complex sentences that are turned into predicates by removing an argument (from Fa->Pa, to Fx->Px) Singular terms are symmetrically substituted for and sentence frames are asymmetrically substituted in. There are three views to argue against: both are symmetric, terms are asymmetric while frames are symmetric, and both asymmetric. The first option is dismissed as wrong since both being symmetric would be structurally deficient. The other two are argued against together. The strategy is to show that terms can’t behave asymmetrically.

The argument begins by noting that if the inference from Pa to Pb, for terms a and b, is good, then the inference from Pb to Pa will not be good generally. If there is a way to construct a sentence P’x, for arbitrary sentence Px, such that P’x is inferentially complementary to Px, then this will show that there is no way to characterize the asymmetric terms. P’x is inferentially complementary to Px if when the inference from Pa to Pb is good but not conversely, the inference from P’b to P’a is good but not conversely. Such complements can be constructed given either conditional or negation. If the language has a conditional, then P’x is Px->S for some sentence S. If the language has negation, then P’x is ~Px. Now, given any terms with a inferentially stronger than b, and that the inference from Pa to Pb is good but not conversely, and that the language under consideration has negation or conditionals, a complementary P’x can be constructed so that the inference from P’b to P’a is good but not conversely. But, a was inferentially stronger than b, so the inference from P’a to P’b should be good. This means that the inferential strength ordering breaks down, so the terms can’t be asymmetric given certain expressive resources.

The argument depends on the notion of asymmetry Brandom uses. One way of understanding the asymmetry of substitution is by taking it to be that if you can infer from a to b in a frame, then generally you cannot infer from b to a in that frame and if you can infer from b to a, then you can’t from a to b. Brandom’s idea of asymmetry is different. He seems to use a partial ordering of inferential strength of terms such that if for all frames, if you can infer from a to b, then you can’t infer from b to a. I’ll label the former weakly asymmetric and the latter strongly asymmetric.

The argument rests on setting up inferentially complementary sentence frames, but the inferential behavior of these is weakly be asymmetric. The asymmetry is simply reversed from the normal direction; in terms of the example I’ve been using, instead of being asymmetric going from a to b, it is asymmetric going from b to a. Weak asymmetry is not eliminated. This does contradict strong asymmetry though. In terms of an ordering, the complementary frames reverse the ordering.

For the following point the ordering of the terms in the non-complementary frame setting will have positive polarity and the ordering in the complementary frame setting will have negative polarity. For the argument to work, it needs strong asymmetry as a premise, so that the positive polarity ordering is preserved in the negative polarity setting; in other words, the positive polarity ordering is the only ordering. Since the complementary frames are constructed using some bits of logical vocabulary, it is possible to discriminate them syntactically. It looks like Brandom tries to get around this by introducing new syntactically simple signs for the complementary sentences, defined using the conditional or negation. I think this is smoke and mirrors. The crux of the argument is the use of strong asymmetry rather than weak. I find strong asymmetry a little bit weird. Since it is stronger than weak asymmetry (not just in title), Brandom should supply some argument or explanation why strong asymmetry is a property of term substitution. We can always pick out the frames in which the polarity of the ordering has been reversed, but the strong asymmetry’s appeal seems to rely on not being able to do that. The weak point of Brandom’s argument seems to be the strong asymmetry premise. Given that this argument is the linchpin of a foundational chapter of Articulating Reasons, this looks bad for Brandom.