Sextus Empiricus on the Third Unproved

In his attack on Stoic Logic, Sextus Empiricus goes through each of the five Stoic unproveds and shows that they suffer from being redundant. An argument is redundant just in case it is valid if one of the premisses is removed: redundant arguments have one premiss too many. The Stoics argue that this flaw renders an argument invalid, and so Sextus concludes that, on these grounds at least, the five unproveds are not syllogisms. Sextus’ arguments can be made to work for the first, second, fourth, and fifth unproveds, but his argument does not work for the third. Sextus misstates the truth conditions of its leading premiss, and the argument does not go through under the correct truth conditions. Galen also had trouble accommodating the Stoic third unproved in his hypothetical syllogistic, rejecting the possibility of a hypothetical syllogism whose leading premiss is a Stoic negated conjunction. There is an error common to both authors, which is that they both require the leading premiss of the third unproved to be a proposition which announces an inference, with the remainder of the argument enacting the inference. Such an error betrays a misunderstanding of how the Stoic unproveds are supposed to work.