Monday, January 15, 2024

Poirot Necessities

 In Agatha Christie's Death on the Nile, Hercule Poirot at one point comments that he knows how things must be, but does not know that they are so. Call this kind of 'must' a Poirot necessity. The essential feature of a Poirot necessity is that it is an alethic strong modality (i.e., necessity), but is not characterized by what in modal logic is often known as the M axiom or the T axiom. The M/T axiom says that "It is necessary that P" implies "P"; to put it sloganishly, necessity implies truth. Most of the time when we are talking about necessities, we are talking about strong modalities that can be characterized by this axiom. However, a Poirot necessity is a necessity (and therefore a strong modality) that is weaker than this. That something must be does not imply that it is.

The most commonly studied kind of modal logic which has a strong modality that works this way is one where the characteristic axiom is known as the D axiom. The D axiom says that necessity implies possibility. This is implied by standard systems with an M/T axiom, but you can have a system with a D axiom and no M/T axiom. In these systems, necessity does not imply truth. Systems of this sort are usually interpreted as systems for deontic logic; that is, the strong modality or necessity is interpreted as obligation. 'Must' is interpreted as 'ought'. Poirot necessities are not deontic; they are not any kind of obligation. They are alethic, that is, actual necessities. They are, in short, 'musts' that are not 'oughts', if by 'ought' you mean something about what we should do. 

Nonetheless, a D system of modal logic would fit Poirot necessities very well -- arguably better than it fits any intuitive notion of obligation. For instance, there is a a rule in a D system, interpreted as a deontic system, known as deontic necessitation. It tells you, roughly, that if something is a theorem (proven from the logical principles of the system itself), then it is obligatory. This often seems weird to people to think of logical theorems as obligatory. However, there's no weirdness at all if we are interpreting D systems as alethic systems rather than deontic systems. Obviously, if something is provable from logical principles, it is part of 'how things must be', and when something is proven, we can easily say it has Poirot necessity.

This ties into something I've argued off and on for quite a while, that what we usually call deontic logics are actually best seen as logics concerned with requirements for solutions to problems. A Poirot necessity fits this bill exactly. When Poirot says that he knows what must have happened, he is saying that, given the set-up of the problem, he knows what is required to solve the problem; then he correctly notes that, given this, he still does not know that this is what actually happened.

I think Poirot necessities are very common, and absolutely essential to inquiry. If you give a set-up to a physicist, he can tell you what must happen. In the right circumstances -- and 'the right circumstances' are very extensive for a field like physics -- you would expect that this is what did happen. But strictly speaking, from the physicist's 'what must happen' on its own, you don't know 'what happens'. Sometimes physicists give you the correct answer about what must happen given the problem set-up as understood, but ambiguities about the set-up, or disruption of it, or even just a completely new phenomenon, make it so that it does not. This is a common happening at the edge of inquiry. Almost the entire history of particle physics consists of physicists figuring out what must be the case before they actually discover it -- but, once they've determined that such-and-such particle must exist, that's not actually the end of the story. The 'must' is a Poirot necessity. They still have to get the actual evidence, just like Poirot does. And when they do get the actual evidence, they sometimes find that they were spot-on; more often, they find out that they were right to a certain level of precision that was the best they could originally do; sometimes, as in the history of the discovery of the neutrino, they find something that very loosely does something in the neighborhood of what they expected, but has features and behavior that are definitely what they had inferred it would have. And physics, of course, is not at all unique in this respect.