What does it mean to “heal” a fragmented mind?
Famously, Lewis used to believe Nassau Street ran roughly east-west; that the railroad nearby ran roughly north-south; and that the two were roughly parallel. Lewis’ three beliefs were mutually inconsistent. In classical logic, everything follows from an inconsistency. But not everything was true according to Lewis’ belief state, because he did not explicitly believe the conjunction of the three sentences. In one fragment of his mind, Lewis believed that Nassau Street ran roughly east-west and that Nassau Street and the railroad nearby were roughly parallel. In another fragment, Lewis believed that the railroad ran roughly north-south and that the railroad and Nassau Street were roughly parallel. But no fragment was such that the conjunction of each sentence in the inconsistent triple was true according to that fragment. Rather, the inconsistency occurs across fragments and not within fragments. When Lewis detected the inconsistency in his belief system, “the fragmentation was healed” and “straightaway [his] beliefs changed”. Consistency was restored, and he now thought that Nassau Street and the railroad both ran roughly northeast-southwest.
Assume that we have an agent who believes p in fragment F1, but ~p in fragment F2, where p is a contingent proposition.# For Lewis, fragments are represented by sets of possible worlds. So for each world in F1, p is true, and for each world in F2, ~p is true. As a first pass, healing a fragmented mind consists in conjoining F1 and F2 into one fragment F. But of course, since F is just another set of worlds, we need worlds at which both p and ~p can be true. Worlds at which both p and ~p are true are impossible worlds, where an impossible world, following Restall, can be thought of as the superimposition of two possible worlds. We can superimpose two worlds, w* and w**, to get w as follows (following Reschner and Brandom): for all worlds w* and w**, and for any proposition A, w is the world such that A is true at w if and only if A is true at either w* or w**. As easily seen, if p is true at w* and ~p is true at w**, then there is a w such that, by superimposition, p and ~p is true at w. Simplifying, let us assume that F1 consists of just one world w*, at which p is true, and that F2 consists of just one world w**, at which ~p is true. Then F consists of the impossible world w, at which p and ~p are true, and which results from superimposing w* and w**. So, by conjoining F1 and F2, the agent ends up believing an explicit contradiction p and ~p in F. Assuming our agent is minimally rational, but not a paraconsistent logician, contradictions are recognized as bad. So, to restore consistency in F, the agent has to give up one of the beliefs in F1 or F2.
Lewis, however, was a firm denier of impossible worlds. On his view, the above agent cannot heal his fragmented mind by: (i) conjoining the two fragments F1 and F2; (ii) noticing the contradiction in F; and (iii) giving up one of the beliefs in F1 or F2. So, rather than prescribing:
(A) conjoin F1 and F2 into F –> aware of contradiction in F –> restore consistency in F,
it seems that Lewis has to prescribe something along the lines of:
(B) aware of contradiction across F1 and F2 –> restore consistency between F1 and F2 –> conjoin F1 and F2 into F.
Unfortunately, Lewis does not explain what it means to heal a fragmented mind. But if it is right that he has to prescribe something like (B), then how exactly would such an explanation of successful healing look like? In particular, if an agent has a fragmented mind, how can he be aware of a contradiction across fragments without actually believing it? Do we have to postulate a super agent, who does not believe either p or ~p, but just monitors the belief system of the agent? Or can an agent, as it were, gaze at his fragments from above, without any of them being the dominating one, at least for the moment?
# Since each fragment represents a set of possible worlds, and since each possible world verifies all logical and mathematical truths, fragmentalization will not help explain why we can fail to believe such truths.
JCB and WHT
October 13th, 2007 at 4:05 am
A technical aside: given that “A is true at w if and only if A is true at either w* or w**”, how do you get that (p and ~p) is true at w, given that neither disjoined world contains the contradictory proposition? (Or do you instead mean that both p is true at w and ~p is true at w.)
October 13th, 2007 at 7:38 am
If belief in a proposition just means that this proposition is true at every world in the epistemic set, then “giving up one of the beliefs in F1 or F2″ means throwing in worlds that aren’t in the intersection of the two sets. It’s really very hard to see what the right way to throw these worlds in is. The same problem arises on a syntactic account of belief, if it’s assumed that beliefs are closed under logical operations, because then you can’t just remove one proposition, but have to remove some together with lots of their logical consequences.
October 13th, 2007 at 4:02 pm
Thanks Richard, your comment in the bracket is the right one: both p is true at w and ~p is true at w. If we want to have that (p & ~p) can be true at w as well, we have to say something about compositionality. For instance, if p is true at w and ~p is true at w, then (p & ~p) is true at w. But then we need to say something about the consequence relation holding for impossible worlds. In particular, to avoid explosion, we might want a non-classical consequence relation. But yes, the wording might have been bad: if an explicit contradiction means that (p & ~p) can be true at w, then we only have a semi-explicit contradiction: p is true at w and ~p is true at w. In this sense, agents can believe p and believe ~p, or believe p, ~p, within a fragment, but not (p & ~p); check out Restall and Reschner and Brandom for further information.
Speaking of it, there seems to be three ways to go in dealing with beliefs in contradictions:
(1) Believe (p, ~p) in F.
(2) Believe (p & ~p) in F.
Without compositionality for impossible worlds, believing a contradiction is given by (1), and with compositionality, believing a contradiction is given by (2). Lewis cannot have either. Intuitively, we might think of (1) as saying that you believe that you are in a world in which p is true and in which ~p is true, whereas (2) says that you believe you are in a world where p & ~p is true. You might think the difference between (1) and (2) is a difference in logic and not in content. Another way one might go is given by (3):
(3) in F, aware that believe p in F1 AND believe ~p in F2.
This approach will introduce something like an awareness operator, which ascribes awareness to an agent of a contradiction between two fragments; maybe by consciously “highlighting” both F1 AND F2 in F. A belief in a contradiction would amount to be aware of F1 AND F2. As such, compositionality does not arise between propositions, but between fragments using the English AND, OR, NOT, NECESSARILY etc.
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