In Mark Richard's Structure, he talks about Stalnaker's unstructured view of propositions. One particular idea of Stalnaker's that he argues against early in his paper is that the acquisition of deductive knowledge is putting one's separate belief states together. He argues that this supposed advantage of unstructured propositions is not possible because it commits one to certain beliefs that one does not necessarily have. Richard sums it up as follows:
"When a collection of premises entails distinct propositions p and q, one may see one entailment, but not the other".
He gives one of Stalnaker's responses to this problem; that perhaps it is because the merging of belief states is done in a sequential manner that does not necessarily entail the not-believed conclusion. (I will illustrate with the same example from the text):
(1) Barbers shave only those who do not shave themselves
(2) The barber Jones shaved all those who attacked Lionel
(3) 1+2 --> No one who shaves himself attacked Lionel
(4) Anderson shaves himself
(5) 3+4 --> Anderson did not attack Lionel
Here, Stalnaker is showing that from 1, 2, and 4 "Smith" can deductively (through the sequential merging of belief states) conclude that Anderson did not attack Lionel (5) without realizing (or believing at some point) that Jones did not attack Lionel.
Richard thinks Stalnaker is mistaken. He thinks that according to the unstructured propositionalist, "Jones did not attack Lionel" cannot be avoided by sequential deduction. Richard's argument is:
(6) (1) and (2) entail the conjunction of (1) and (2), not (3)
(7) The conjunction of (1) and (2) entails that Jones did not attack Lionel (while at the same time entailing that Anderson did not attack Lionel when combined with (5))
(8) If the conjunction of (1) and (2) can lead Smith to deduce that Jones did not attack Lionel (even though they also entail that Anderson did not attack Lionel, then Smith must realize (and believe) that Jones did not attack Lionel
And of course, if (8) is the case then Stalnaker is stuck with the problem of deduction.
There are a few oversights in Richards objection that I would like to point out. Richard (and all supporters of this line of objection) seems to be assuming that the entire process of merging belief states takes place instantaneously (that once a part of a belief state starts to merge with another, the entire belief state is immediately merged). What if is the case that parts of belief states can merge quickly while other parts take more time to merge (more deductive reasoning, etc..). This would allow for certain deductions to be made while others do not occur to you right away. Or what if certain parts of belief states did not merge at all (like in the case of incompatible belief states).
Someone might try to point out that belief states are platonic in nature and are not restricted by time limitation in such ways. To this I would respond by pointing out two things:
A) For belief states to merge, there would have to be a time where they were not merged, which suggests some sort of being-in-a-state-at-time-x-versus-at-time-y distinction.
B) The part of our brain that pulls the magic trick of interaction with an abstract object is bound by spacetime limitations, and would take time to "catch-up" to an immediate belief state merger (and it is in this "catching-up" transition where certain deductions could happen non-uniformly, or maybe even not at all).
If belief states do not in fact merge instantaneously, then I think (7) and (8) of Richard's argument are not as sound as they seem. At the very least, it is reason to question the problem of deduction in the sequential deduction position.