Thursday, March 06, 2014

Precedents and Modal Diamonds

I've been thinking about precedents recently and haven't been able to find anything on them that deal with the kinds or issues I think interesting, so these are some jottings as I work my way through some ideas. My basic idea is that the best way to characterize reasoning using precedent is in terms of modal logic.

A modal logic is one where there are operators that identify the way things are. A typical modal logic will have two kinds of operators: a strong modal operator, which we can call Box, and a weak modal operator, which we can call Diamond. Examples of Box would be: necessarily, always, everywhere, obligatorily. Examples of Diamond would be: possibly, sometimes, somewhere, permissibly. There can sometimes also be a neutral modality, which we could call Null, which is weak compared to Box but strong compared to Diamond. The most obvious example of Null is: actually or truly, which is between necessarily and possibly.

How Box, Null, and Diamond relate to each other varies considerably depending on the modality. It's fairly common, though, to allow one to infer Null from Box and Diamond from Box or Null. So, for instance, if something is necessarily true (Box), it is true (Null) and possible (Diamond); while if something is true (Null), it is also possible (Diamond). My basic idea for reasoning with precedents is that it uses Null to establish Diamond.

How do you establish that something is possible? For instance, if you are asked whether it is possible for there to exist a black swan, what would you do? There are a number of things that you could do, but the easiest thing would be to look around and see if you can find an actual black swan. What you are doing is finding a Null (actual black swan) from which you can conclude the corresponding Diamond (possible black swan). When we reason using precedents, we are doing exactly this, but the modality is not possibility but permissibility. Or, in other words, precedented is a Null modality implying permissible as a Diamond modality, and in reasoning based on precedents we are starting with Null and concluding Diamond.

Not just any kind of thing can serve as a precedent in this way. But it shows directly why precedents would be somewhat important in fields like law, which deal with what is permissible. And I think we can go beyond even this. Why is precedent so important in law, so that it is very difficult to find any area of law in which precedent doesn't have at least some weight? It's because law is a system in which the permissible cannot always be safely assumed as a default. In many cases with Diamond modalities, we take Diamond modalities to be quite cheap -- just assume that things are possible, for instance, until you have a good argument otherwise. But permissibility is harder, and there are always parts of a legal system where this is simply not acceptable to assume that things are permissible unless you can show they are. You can't assume, for instance, that it's permissible for the government to do anything until it is shown that it can't; that would be tyranny. So the government's actions have to be established as permissible. But, again, the easiest way to establish something as permissible is to establish it as permitted; or, in other words, to show that some proper authority operating in apparently the right way permitted it. Precedents can mislead, of course, but this is no different from the fact that truth claims can mislead; and they can be wrong, of course, but this is no different from the fact that truth claims can be wrong. They still function in much the same way.

One way to put this might be to say that precedent becomes important in matters where liberty cannot be assumed and moral safety (in the casuistic senses) is important. The casuists held that some things, whether permissible or not, were more or less safe than others. For instance, when we say that something is reckless, we are saying that, even if it was not wrong, it increased the danger of doing something wrong to a very high degree: it was relatively unsafe. One of the biggest disputes in moral philosophy in the modern period was the argument over whether it was ever OK to do something morally less safe if doing something more safe was an option. (The position that you can't is called rigorism or tutiorism; its contrary is laxism, which held that it was always OK to do the less safe thing as long as you had some kind of reason for it. There are a number of positions in between.)

There are lots of situations where safety is not that big an issue -- eating ice cream one way might give you an ice cream headache, but nobody's going to make a fuss if you do it. There are other situations in which any kind of recklessness could cause terrible damage. In those cases liberty can't be assumed, and safety is a major issue. Lots of areas in law have this sort of effect, so it's unsurprising that if something isn't precedented it might be reasonable to regard it with suspicion; and if a new situation comes up, one might have to use acceptable kinds of inferences to show that the precedent's Null modality also establishes a Diamond for the new kind of situation.

There are a number of complications I haven't worked out, but that's more or less where I am at this point.


  1. Ye Olde Statistician3:22 PM

    Don't know if this is relevant or not, but I will assume it is not recklessly permissible: Regarding "possible" diamonds, in probability theory (or at least in one sort of probability theory) we say that there is no such thing as a probability without a model. That is, the probability that an adult American is 6'0" tall or more is only answerable if you assume that adult male heights are distributed as for example a normal distribution. Assume a different distribution, and you get a different probability. I don't know that there is some parallel in modal logic, but the thought struck me as I was reading: possible with respect to what assumptions? Permissible with respect to what precedents?

  2. branemrys4:18 PM

    As far as I know there's no standard terminology for it in modal logic, but you're quite right -- modal logic is very assumption-sensitive, and the precise behavior of Diamond and Box can vary depending on what kinds of modalities and relations between modalities we're talking about. (How modal logic relates to probability theory is a controversial subject, but I've often thought that probability theory is, in a sense, a modal logic in which you have a method for enumerating distinct possibilities, at least in principle, and modal logic is in a sense a probability theory in which you can only handle them in a general way. I don't know how far that could be taken, but it has at least a superficial plausibility for a lot of things, so I would be inclined to take possible parallels between the two at least very seriously.) For instance, we can talk about possibility, but do we mean possible-according-to-natural-laws or logically possible or possible actions given constraints or some other kind? So what would count as the right kind of precedent would make a big difference to how one would handle things.

  3. Ye Olde Statistician5:19 PM

    probability theory is, in a sense, a modal logic in which you have a
    method for enumerating distinct possibilities, at least in principle,
    and modal logic is in a sense a probability theory in which you can only
    handle them in a general way.

    Sounds like
    Probability:Modal Logic::Geometry:General Topology

    General topology regards the question of when point a is "close to" set B without any actual measurement of distance.

  4. branemrys6:18 PM

    Yes, I would say that there's something to that analogy -- how much, I don't know for sure; but it seems to work reasonably well for lots of things.

  5. branemrys6:39 PM

    Actually, I can go further. One of the most famous discoveries in modal logic in the twentieth century was that standard systems of topology are related to one of the important modal logic systems, S4. (S4 is also important because Godel showed that it could be used as a logic describing mathematical provability, and so it could be used as a higher-order way of characterizing what can and can't be proven.) So that part of the analogy is very strong -- modal logics do work, broadly, like topology, because some just are the logic for various kinds of topology. The idea that (at least some) modal logics have this general relation to probability theory is the only thing that might be controversial. But it makes sense -- in probability theory you are in some way laying out the possibilities and measuring them with respect to each other, while Diamond in modal logic can represent possibility as a general feature.


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