Wittgenstein’s atomist picture, as embodied in his Tractatus, is initially
very appealing. However, it faces the famous colour-exclusion problem. In this paper,
I shall explain when the atomist picture can be defended (in principle) in the face of
that problem; and, in the light of this, why the atomist picture should be rejected. I
outline the atomist picture in Section 1. In Section 2, I present a very simple neces-
sary and sufficient condition for the tenability (in principle) of the atomist picture.
The condition is: logical space is a power of two. In Sections 3 and 4, I outline the
colour-exclusion problem, and then show how the cardinality-condition supplies a
response to exclusion problems. In Section 5, I explain how this amounts to a distilla-
tion of a proposal due to Moss (2012), which goes back to Carruthers (1990: 144–7).
And in Section 6, I show how all this vindicates Wittgenstein’s ultimate rejection
of the atomist picture. The brief reason is that we have no guarantee that there are
any solutions to a given exclusion problem but, if there are any, then there are far
too many.

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• Exclusion Problems and the Cardinality of Logical Space