It's commonly accepted that the universe began with a "big bang" which defined time t=0. A big unknown with respect to the big-bang creation of the universe is how and why non-uniformity burst forth. But given that it did (insert magic here), strong determinism is the assertion that this initial non-uniformity, acted upon by, and solely by, the laws of physics that also burst forth (more magic here), completely defines the universe at all states t>0. That of course leaves no room for free will. But how can that jive with our strong sense of having free will?
My little personal thought experiment on the matter is this. Imagine a standard computer program (the quintessential example of deterministic activity). Computer programs make decisions all the time. A simple "if condition then action1 else action2" statement is making a decision to do one thing or another depending on some sensed condition. Tracing through the execution of such a program, when the code takes one branch versus another due to a conditional statement, it could be asked "could the computer have taken a different branch"? And of course the answer is no. Given the same initial state and set of inputs, The computer would make the exact same decision every time. So it didn't really make a "decision" per se. The "decision" it made was inevitable given the initial state and set of inputs. It just required some calculation (deterministically carried out by the universe) to make.
Obviously I'm suggesting (as does strong determinism itself) that human minds are nothing more than highly sophisticated biological analog computers, and therefore the same assertion applies. Each and every decision we make is inevitable given the state of the universe at the time. But that doesn't help explain why we feel so strongly that we do have some sort of free will, that we could have made a different decision, state of the universe be damned. Here's where my thought experiment gets interesting.
Imagine again a computer program, but this time one that has a set of goals programmed into it. And imagine that its decision-making process is a complex one, involving attempts at making sense of inputs by concocting models of the outside world that produces them, and then constructing simulations of these models, which it can then run various decision alternatives through in order to determine which ones result in getting closest to achieving the desired goals. Now imagine that the computer decides on the optimal course of action, executes it, and experiences a set of results that weren't predicted by its simulations. It knows that the models it had concocted weren't a sufficient representation of the reality it's attempting to make sense of. But the actual set of results is new information that it didn't have before. Imagine that this computer also has sophisticated-enough programming that it's able to now use this new information to adjust its models, and then run the same simulations again (this time of past events) to see if this time they arrive at decisions that lead to better results. In this way the program is essentially adjusting and fine-tuning itself to be better prepared for future decisions.
Now imagine this computer being the human mind. The human mind has an uncanny ability to whip up models of the outside world and pass them through simulations. Every conscious thought involves doing this. We run quick simulations through our head, allowing us to make decisions one way or another. And then comes the learning. When our decisions don't lead to what we expected or hoped they'd lead to, we try adjusting our models of the outside world in various ways and then go back and run simulations on these adjusted models to see if any of them lead to more effective decision paths. It may not even be just the models that we adjust in this way; it could also be the heuristic algorithms for choosing the decision alternatives themselves. Keep in mind this this is all still deterministic computation. But it's my belief that consciousness is an emergent property of our simulation engine. It's the trick that the brain has evolved to allow it to perform all of these simulations, both for the purpose of decision making and for the purpose of model adjustment.
So here we have a conscious mind that's constantly going through simulations of what could have occurred differently had it made different decisions. Even though this is all happening in an entirely deterministic world, within the simulation-engine subsystem of the human mind there exists the notion of these various possible decisions and their consequences. And since that simulation-engine subsystem is our consciousness, we can't help but perceive these "choices" as being a fundamental part of our interaction with the universe.