## Sunday, January 09, 2022

### The Two Barbara Problem

A longstanding puzzle about Aristotle's modal logic is what is known as the Problem of the Two Barbaras. It's a pretty obvious aspect of Aristotelian syllogistic that a Barbara syllogism is valid:

All M is P
All S is M
Therefore, all S is P.

What happens when we modalize some of these propositions, however? This is the context in which the Two Barbara Problem arises. If we take both of the premises and make them necessary, we can get either a necessary or merely true conclusion, as we wish, and the argument will also be valid. Things get more complicated if we only necessitate one of the premises. Aristotle is very clear that if we necessitate only the major (All M is P), we can still get a necessary conclusion by a valid argument; but if we necessitate only the minor (All S is M), we cannot.

These are the Two Barbaras. There are several issues involved with this view of Aristotle's. Most Aristotelians, in fact, accept the Theophrastian Rule: When you modalize the premises of a valid syllogism, you can still get a valid syllogism as long as the the modality of the conclusion is not stronger than the modality of any of the premises. This is a nice, clean, very powerful rule. (It would make both of the Two Barbaras invalid, if we are assuming a necessary conclusion.)

However, there are some things that do make sense about Aristotle's division. And, indeed, I think probably most of the confusion in the Two Barbara Problem is due to confusing different kinds of modality. The Theophrastian Rule is certainly right if we are talking about propositional modality -- which most people today do, and which people are often doing if they are talking about fiddling the modality of premises. But it's pretty clear that this is not what Aristotle is doing; he is assuming that we are talking about predicate modality. The major premise gives the predicate to the conclusion, so if we necessitate the major premise, its necessity-modality carries over to the conclusion; if we necessitate the minor premise, however, the predicate of the premise is the middle term, and thus the predicate modality will not carry over to the conclusion. This does not necessarily address all problems that have been raised with respect to the two Barbaras, but it's a fairly reasonable explanation of why Aristotle says there is a divergence.