On Defining your Terms

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.