Let's suppose you're having a discussion with someone about free will. They believe that you have it and you believe that you don't. Most of the time when these discussions occur, people are just talking past each other, so you ask them to define what they mean by "free will". They say that they can't define it, so you suggest that they spend some more time thinking about it and that you resume the discussion when they produce a definition. They claim that they don't think there's any way they could ever define it, so you leave the discussion there and conclude that they are being unreasonable.

But are they? Let's suppose someone asks you to define "a pile". This is clearly a useful concept, but it's hard to define. 100 grains of rice is a pile, 1 grain isn't, where do we draw the line? We could pick a number arbitrarily, but perhaps the number of items required to counts as a pile varies depending on whether we are talking about rice or sand or waterbottles with smaller items needing a greater number and larger needing a lesser number. And maybe there's other contextual factors too that you've left out? The word doesn't have a nice crisp definition, you simply learn how to use it from experience. It's kind of the same for words like red and orange where the boundaries are also extremely fuzzy.

This isn't perfectly analogous to the free will situation, but it does highlight the difficulty of definitions. Let's suppose both you and the person you have talked to work together and you manage to identify five models of reality. He argues that the first two would have free will and that the last three wouldn't. That seems all well and good, but what if there's another three models of reality that neither of you had considered and if they had thought of them, they would have considered the first one to clearly be free will and the second one to be a case where you could reasonably call it either way. That would certainly complicate things, wouldn't it? The point is that expecting a complete definition may require someone to have thought of all the possible models of reality, which is a pretty massive, arguably impossible, burden to put on someone.

Nonetheless, I think the discussion can proceed. Even if the first person can't perfectly define what they mean by free will, they should be able to produce some kind of imperfect attempt. They could define a paradigmatic case of what would count as free will for them and what wouldn't count as free will. They could provide a definition that attempts to capture most of it, even if it doesn't completely match the way that they would draw the boundary. Or they could endorse a definition purely for the purpose of this discussion. For example, I might say that I think that grains of rice would be a reasonable cutoff for a pile, even if my actual model is more complicated with 20 grains definitely being a pile and 10 mostly being a pile and 3 grains only kind of being one.


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A collection of grains of rice is a pile if and only if (1) a majority of the rice grains are supported by other grains rather than whatever surface the pile is on and (2) you can get from any grain to any other grain by stepping from grain to grain, each step happening between grains whose surfaces are no further apart than 10% of the diameter of a median rice grain in the pile.

Of course I don't claim that this is The One True Definition of "pile", but it kinda annoys me that heaps/piles have been the standard example of fuzziness about number for centuries, and everyone assumes that if you did lay down a somewhat arbitrary definition then it would be of the form "at least N objects", when in fact I think what fuzzily distinguishes piles from non-piles is something else -- you can definitely have a pile with 6 objects in it, and you can definitely have 100 objects of the same type that don't form a pile -- and one can give somewhat-plausible clear-cut definitions with no arbitrary numbers-of-objects in them.

This is mostly a mostly-irrelevant tangent, but it's maybe worth pointing out explicitly the phenomenon that when you have some notion that you leave undefined it's possible to be mistaken about what sort of notion it is. If you're having an argument about piles or heaps then you will likely go astray if you start asking "well, how many grains of rice do we need to have before we definitely have a heap?". Compare: "how complex does a nervous system have to be before we assign moral significance to the organism whose nervous system it is?" which is fairly clearly wrong in the same sort of way as treating "pile" as purely numerical.

One thing I found helpful was realizing I can't expect mathematically precise definitions about the territory; you can only get those for definitions of things that exist within the map. Otherwise there is always a gap where uncertainty and fuzziness seep in, but that's okay. If you expect that what you're doing is less like defining mathematical terms or programming and more like painting a picture so that it evokes the expected thoughts (e.g. "oh, I guess that must be a flower"), you'll suffer a lot less strife and be more effective at using and explaining words in ways that are useful.

A productive purpose of giving definitions to your interlocutor is to incite cognitive activity that's relevant to your own thoughts, as levers that bring about thinking on the right topic. Apt definitions are great, when available, but demanding them when they are not is losing sight of the purpose of the whole activity. Formulating definitions (or researching existing ones) might be a good subproblem to focus on though.

For vague ideas it makes little sense to use definitions that sharply delineate their instances. A definition should instead give a degree of aptness (centrality) to its potential instances, and definitions should match based on the whole distribution of aptness they assign.

With free will, and similarly with possible worlds, I think a crucial point missing from discussion on LW is that these are semantic notions, and it's possible to consider radically different semantics for the same syntactic thing. A great illustration is semantics of programming languages, where it's possible to understand what a given program is doing in terms of very different semantic constructions. To give some examples, there's the straightforward sets of possible values, related worlds in Kripke semantics, formal theories of observations (that are not even maximal) in Scott domains, plays and strategies in game semantics. Closer to normal mathematics, there's internal languages of categories that let us interpret a term as something incarnated in very different situations, notably sheaf toposes where you can interpret a construction (as in discussion, argument, proof, term, type) as varying continuously on some space and saying different things for different places, all at the same time.

So when considering a thought experiment, assuming that what's going on is that there is some discrete set of possible worlds is very limiting. There are other ways to think about things, and thus prematurely locking in a discussion under a technical definition can be a problem, manifesting as failure to notice or urge to ridicule the notions arising from other definitions and inapt for this one. (Not formulating technical definitions is another problem.)

How definitional issues affect the debate on free will specifically:-

Laypeople sometimes assert that free will is straightforwardly refuted by the absence of a ghost in (and able to drive) the machine. Call this the homuncular definition. The argument depends on an implicit definition of free will, where the vital criterion is whether there is an essential inner self pulling the levers of the brain and body. The philosophy community continues to debate free will, not because they are unaware of science, but because they are using different definitions of free will, having different criteria.

The majority of philosophers are compatiblists. The compatibilist notion is designed to be almost trivially true. It asserts that person is free whenever they are not under someone else's compulsion. The compatibilist concept of free will is practical. It tells you under which circumstances someone can be held legally or ethically responsible. It does not require global additions about how the laws of the universe work. Only when compatibilist free will is asserted as being the only kind does it become a metaphysical claim,. The existence of compatibilist free will isn't worth arguing about, since it's designed to be compatible with a wide variety of background assumptions.

So compatibilism is obviously true -- about compatibilist free will, at least -- and homuncular or "contra-causal" free will is scientifically false. Between these two extreme lie other definitions that are worth debating because they are neither obviously true nor obviously false.