Small amendment arguments: how they work and what they do and do not show

The small improvement argument has been said to establish that the standard weak preference or value relation can be incomplete. We first show that the argument is one of three possible ‘small amendment arguments’, each of which would yield the same conclusion. Generalizing the analysis thus, we subsequently present a strong and a weak version of small amendment arguments and derive the exact rationality conditions under which they reveal incompleteness. The results show that the arguments (in any of their variants) need not reveal a problem for the possibility of rational choice. In fact, it can be argued that they only reveal such a problem if the underlying relation is complete rather than incomplete.