Classical logic says that something cannot be A & not A at the same time and in the same way. Every sentence in your above statement relies on that being true even as you argue against it, because if it's not true language becomes irrelevant. Without identity, communication can't really occur.
Quantum mechanics, however, opens up situations where something can be both a wave and a particle, can be in a state of existential limbo with two distinct outcomes, etc. This is different than A and not A being identical in all ways.
It is as far as science relates to logic at all. Because until recently, the wave-particle duality meant only that we could measure some system exhibiting wave-like properties, or particle-like properties, but not both. So whatever ontological interpretation of quantum mechanics one held still involved at best thought experiments in which something could behave one way if observed in a certain way, and another way if observed in some different manner. All experiments had one, and only one, outcome.
This is no longer true. The article I linked to concerns the only tool science has available to understand the workings of reality as more than subjective interpretations: measurements/observations which can be used to describe the ways in which a system does what it does, such that it can be characterized as having certain properties but not others. Neuroscience involves a lot of neuroimaging, and perhaps the most important method is fMRI, which allows scientists to observe activity in parts of the brain under controlled conditions to see what places are more active given some type of cognitive task. Only fMRI doesn't actually measure activity, it measures something which gives and indication of blood-oxygen levels, which in turn is a measure of metabolic activity, and thus (ideally) neural activity. So when a subject is required to judge the similarity between two objects, or whether a spoken (or written) word corresponds to something larger or smaller than another word does, the metabolic changes hopefully provide a usable measure of whether or not particular regions were significantly more activated for this task relative to another.
Same with medical diagnoses, genetic markers, and all of science. When you measure and then describe a system has having certain properties (like a brain having vision pathways in particular regions), it either does, or it does not. Someone either has X virus, or they don't. That's the foundation of the scientific framework: being able to understand the dynamics of a system in terms of what properties it has/exhibits, and those that it does not.
QM didn't change this entirely for some time because even though we had systems which could behave in seemingly contradicting ways, we had only one actual measurement to deal with. So the wave-particle duality was protected by uncertainty, and the logic which does not obey classical logic could be an approximate model.
More modern experimental techniques have, even very recently, made this less and less true. The study I linked to was not one in which some quantum system could end up behaving like a particle if measured in some way, but like a wave if measured in another (cloaking a reality of seeming contradictions behind an inability to actually observe both). New quantum coherence techniques provided the ability to get measurements of a contradictory nature out of the exact same system.
Of course one can still say "well, the quantum system has the ontological property such that it can behave like a wave or a particle", but this doesn't solve the problem for 2 reasons. The first is that we are still left with using the same tool (observation/measurement) used to understand what anything is in a ways that gets results which cannot both be true (something cannot be a wave and a particle). The second is that superpositional states allow both microscopic and macrospic systems to be in two different places at the same time, thus having two different yet identical instantiations. This is again a violation of classical logic, as if e.g., I see a person in one location, they can only be at that location. If I see two people in two different locations, they must be different people. This is no longer true of even macroscopic systems.
That the part highlighted in blue was even brought up in response to my post shows that the physical details are still being brought into a logical model.
This is true. The question is whether or not it is possible to ever create a logical model which does what ideal scientific models (at least until recently, and still to a large extent) do: explain how everything works in terms of physical parts and the forces acting on them. If it is not (as is argued for living systems in general), then it will never be brought into a logical model in a way other than those which alreadly exist.
Rather, I've brought up that free will seems internally inconsistent and fails early on at the definition stage prior to physical details even factoring into the conclusion.
Because this is a classical causal model, in which we have distinguishable states and causes, and these are reducible to physical parts and physical laws. It is also largely rejected as inadequate, insofar as there seem to be functional properties/processes of systems which cannnot be reduced to the laws of physics and the parts at any time t (i.e., at any "state" at all), and yet can determine how the system evolves in time. To say that free will is internally inconsistent in the way you seem to relies on the notion that, at least in principle, I can explain the state of any system at any time t in terms of the physical parts making up the system and the laws of physics. Again, this is widely regarded as wrong. In the weakest rejection, the states don't violate the laws of physics, but they still cannot be reduced to (and therefore not predicted by) the state of the system at some prior time.
Modern science and mathematics (and systems of formal logic) began as an effort to describe movement: planets, falling objects, etc. After a while, the increased success this approach of applying mathematics/logic with observation introduced the idea that everything can be described in terms of laws of physics and the dynamics of parts. It was assumed that, whatever we couldn't yet explain, would still be reducible."What determines what the conscious agent wills?
More specifically, what does it mean for the conscious agent to 'will' something?"
How can 'free will' be proposed to exist without first having a fairly tight logical model of what 'will' is?
We don't have "fairly tight logical model(s)" for metabolic-repair processes in cells. There is an ongoing debate, with competing logical models, over whether or not basic cells are computable because (it is argued) this process describes a "logical" function which takes as its domain most of the cell, and as its image all of the cells, yet can only be understood as both producing cellular activity while at the same time being a product of the same activity it is produced by. This can be easily modelled logically (it has been). But the logical models incorporate features which do not correspond only to the activity of parts and the laws of physics. Instead, they introduce a functional process which exists apart from (but somehow dependent upon) both the cell's dynamics and the laws of physics.
Emergence doesn't create a problem for logical models. But it does create a problem for "tight" logical models insofar as these can be mapped onto the physical properties of the system as well as physics. However, it is increasingly used across scientific fields to explain observed phenomena.
Basically, what you call inconsistent is only inconsistent (if I understand you) in that it doesn't fit into to the classical understanding reductionism and causation. We can't map the models onto specific physical processes, but rather rely on emergent ones. However, this is hardly something which applies only to the human brain, or to brains.
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