## Modality Three Ways

I’m currently at Advances in Modal Logic, in Budapest, one of my favorite conferences; I’ve been a regular attendee since 2004 (only missed the one in Australia!) and have given historically-oriented papers every time, usually to great interest. In fact, the only drawback of the conference is how few historical papers there are. So here’s an early plug for AiML 2018: We need more medieval modal logic there!

Hanging out with a bunch of people for whom “history of [particular subfield of logic]” generally starts around 1930, or maybe, if you’re lucky, in the 1890s, is always interesting when you’re a medievalist. I particularly enjoy when people react to this “history” and discuss how what they are doing is new/interesting/different. Some of them even recognize that what they are doing is old/interesting/different, i.e., people who explicitly reject Frege and return to Aristotle.

One speaker did so this morning, pointing out what many people do in fact know but rarely incorporate into their formal systems, that English (and in fact many natural languages) has two types of negation — sentential and term. One can say both “It is not that case that AaB” and “Aa non-B”, and these are not equivalent. It’s not hard to move from this observation to a similar one about modality, in that one can have both internal and external modal operators, and the difference between them is usually explained via the de re/de dicto distinction.

But one thing I rarely see people discuss is the fact that given a categorical sentence, there are in fact three places in which one can put the modality (or the negation): At the beginning of the sentence, before the copula, and after the copula. It’s the difference between:

• $\square AaB$
• $A\square aB$
• $Aa\square B$

When I mention this to people, most often the response is that these three aren’t in fact all distinct, or to cast doubt on modal properties like being “possibly B”. This first may be true, the latter may be reasonable. But all three types are syntactically distinguished in medieval discussions of modality, so I am reluctant to dismiss them out of hand. For example, Paul of Venice in his chapter on syllogisms (Tract II, cap. 13) in the Logica Magna distinguishes such constructions:

• Omnis prima causa de necessitate est deus
• Contingenter omne creans est deus
• Omnem primam causam necesse est esse deum

I am not entirely sure which of the formal versions above these map to, but it may be (using $\bigcirc$ for contingency):

 $\square AaB$ Omnem primam causam necesse est esse deum $A\bigcirc aB$ Contingenter omne creans est deus $Aa\square B$ Omnis prima causa de necessitate est deus

On the other hand, I could be persuaded it is:

 $\square AaB$ Omnis prima causa de necessitate est deus $A\bigcirc aB$ Contingenter omne creans est deus $Aa\square B$ Omnem primam causam necesse est esse deum

I’m not sure anyone has investigated this with any systematicity. This would make a great topic for a paper for AiML!