Based on my understanding of the debate in this thread, I believe (i) Incompatibilism is logically sound and Compatibilism is logically fallacious if the Incompatibilist's classic definitions of Determinism and Free Will are used, and (ii) Incompatibilism is logically fallacious and Compatibilism is logically sound if the Compatibilist's modernly modified definition of Determinism and description of Free Will are used
And yet again, the definitions used in (i) are not logically sound. They require the statement of "necessity" or "god", are based entirely on the ideas of such from antiquity either way. That people try to drop the omniscient god from the reason and keep the flawed
As long as you stick with "classical" ideas of things deserving treatment with formal, mathematical logic, you will find yourself running into nonsense.
Either way, as I have said before, we have proven in this thread
for a fact that pseudo-probabilistic events do clearly happen, observably, all the goddamn time. I have described the nature of a system which allows them to happen.
I will now go on a fairly long discussion about such events as are documented and which are metaphysically handled within math.
Such events as are observed in a "pseudo-probabilistic" manner each represent a surrender of agency to whatever produces enough "noise" to trigger "signalling on the noise".
These are used in engineering, principally, to
enforce uncertainty of action on
relatively distant autonomous systems over the behavior being performed, and have
nothing to do with the freedom of things other than in the sense that these mechanisms can be used to encrypt and force strategic elements in "unsolved" game theory to come into play.
As I have said to Mike a few times now, maybe this is the macrophysical probabilistic element that he is looking for!
An abstract example would be if your opponent can make a
suboptimal play in "perfect play terms" that is not well mapped by an opponent rather than taking an optimal play where the opponent has an easily known "win", however this requires chaos and entropy to occur, and good "active strategy"; in a setting where both sides have "full depth" of all play branches, there ends up being "unbeatable" play for games of finite depth, or at least "guarantee of win or draw" regardless of opponent play.
Because we know a large class of problems that arise in real life have very large search spaces such that the tree cannot be mapped, or where the "tree" can't be mapped at all without something quite literally "quantum" acting to generate calculations and where even THAT space is large, we know that there are situations where such entropy being added to a decision making process allows "gaming" someone except through "strategic" rather than "mapped" play. Some games have strategic solutions no matter how large the board becomes -- such as Nim.
Making the external reading of a system more complicated than the logical truth of the system can be useful to prevent 'gaming the player'.
This is textbook graph theory and game theory, but it is a feature of deterministic systems writ large, especially of systems exceeding a certain amount of complexity, or of particular sorts of component parts (such as transistors or neurons or what-have-you).
It's not actually a feature requiring "indeterminism" or rather "probabilistic systems", it's a feature of fully deterministic ones, too.
Generally, in signal processing math, a signal that has nothing to do with some problem space of interest, is called "noise".
There are many different kinds of "noise" in the world, from "random noise", which is noise that has nothing to do with anything observed in any other system at all, anywhere (which arguably does not exist), all the way to "simple noise" like a sine wave interacting with the problem space information.
If the argument is that you need NOISE for free will, it's all over everywhere, from simple deterministic noise to three body problems.
The problem here is that while noise can hide or stir up a decision making process for adversarial purposes, it can't actually make "good" decisions except in an adversarial strategic "confuse the other guy" sense, or in the sense of "we need to try a new direction, might as well try this one...".
We consider the parts of systems which access such noise as responsible for being vulnerable to the interference, as much as the source is responsible for producing it, especially when the thing held responsible for its vulnerability can directly alter the gain on that signal.
When the gain is a physical knob on a fairly "dumb" machine, we respond to the responsibility of the system's vulnerability by moving the knob ourselves, since it can't or won't.
When the "gain" is the function of whether an eyelid is open or not, for example, we can say "quit being a psychopath, open your eyes, and look at the goddamn road, get your hands on the wheel, and actually start driving before you get us all killed", where we directly can observe the responsibility as stated by the request for a change. But we should never have needed to say it and the fact we did may indicate a deeper problem and a more pressing responsibility.
While noise is useful for maintaining autonomy in very complex and potentially rare circumstances, it isn't really important to the discussion of high level autonomous action.
In fact, when noise is generated by some intentionally wacky source at sufficiently high levels to completely drown out problem-space information, this directly prevents processing in the system from taking place, preventing rational decision making. A direct and easy example of this would be when a human gets tazed or drugged.
Indeed, we can recognize this is not what we mean when we say "free will" because each of these things explicitly is an attack on free will.
Oftentimes we allow the influence on our agency of information (and often noisy information) relating to some problem space. In fact in the earlier example with the guy failing to drive; we kind of have to be influenced by information to make informed decisions. Still, we choose those influences, especially if we are driving and are not psychopaths.
Hopefully at this point we can put to rest whether and what role noise and "interference" and "randomness" play in human cognition: preferably as little as is necessary.
Instead, of you want to know what the will of a human is, you will have to reverse engineer the statements their brain is currently making, and how differences in how it is interacting modify those statements it makes to itself in regular ways.
The "freedoms" of the will, for which it is responsible for in its expression through structure, correspond to the logical "truth" states of the system, just as much as the expression through structure of transistors creates a logical truth, though the neuron is much more analog-ish, as this averages the quantification errors of its components over a large number of component parts.
In total, however, you can assess the momentary freedom of the will by looking at what goals and activities are scheduled at what priority level in the system.
If major goals are being constrained by some external momentary force, in that moment you can observe the responsibility there by reaching the exchange of forces, however subtle.
It is really no different from observing the responsibility of one process in a simulator putting together a message and sending it on, and tracing the message back to where it came from through a block view of system time.
But let's return back to this idea that compatibilist free will is not sufficient...
We can observe the admission that things are perfectly consistent when compatibilist definitions are used.
We can observe that none of that clearly sensible mathematical discussion of the role of noise, or of signaling, or of the "truth" of the system having substantive responsibility in the outcome in a directly describable way exist when you use Fatalist definitions. Instead, you get the notion that the infringement of noise is freedom rather than "literally having a seizure or being tazed or drugged" or some other loss of control.
Finally, reality itself contradicts the account in which infringing noise, or chaotic action within a system grants or upholds responsibility or responsible action except when trying to defeat some adversary too similar to yourself;
all of these theories bear fruit in applications all across the world. Experimental proof with the role of noise in computation directly invalidates the Fatalist and Libertarian conclusions as to how things really end up working. When we observe in reality, "unasked-for noise on the line" is "having your decisions rendered by a guy with a tazer such that you shit your pants", not "making decisions for yourself".
The complete disconnect from what would be suggested by atheist mathematical notions being assembled and engineered, vs the completely inverse view that Libs and Fatalists present offers a strong reason to suspect a major logical flaw in a foundational aspect of the construction of the Lib and Fatalist accounts.
In conclusion...
Noise is a thing. Noise
interferes with signals when not collected with the sole intent of being a confounding signal for an adversary. It doesn't render "best" decisions outside of that context, and it is easily and suitably generated even in deterministic systems. Instead, cognition is visibly the product of "signals", generated and exchanged through switch actions that
resist noise -- though signals may be hidden "under" noise or incorporate it so as to be inscrutable.