Compatibilism is the belief that free will and determinism are compatible ideas, and that it is possible to believe both without being logically inconsistent.
It is held by mainstream philosophers like Daniel Dennett (argumentum ad populum)
Laplace’s demon was the first published articulation of causal or scientific determinism by Pierre-Simon Laplace in 1814. According to Laplace, if someone knows the precise location and momentum of every atom in the universe, their past and future values for any given time are entailed; they can be calculated from the laws of physics.
According to ecologist Robert Ulanowicz, in his 1986 book Growth and Development, Laplace’s demon met its end with early 19th century developments of the concepts of irreversibility, entropy, and the second law of thermodynamics. In other words, Laplace’s demon was based on the premise of reversibility and classical mechanics; however, under current theory, thermodynamics (i.e. real processes) are thought to be irreversible in practical terms.
The final nail on the coffin of Laplace’s demon was put by David Wolpert in 2008. He used Cantor diagonalization to disprove Laplace’s demon. He did this by assuming that the demon is a computational device and showing that no two such devices can completely predict each other.
Laplace’s Demon can’t always accurately predict the future, even in a completely deterministic universe, even given complete knowledge of all the physical laws and states. Or rather, he can’t predict it without doing the equivalent to observing the system’s evolution.
The reason for that is computational: a sufficiently complex system may be regarded as a universal computer; being able to uniquely predict its evolution then would be equivalent to being able to solve the halting problem, which is unsolvable. Take a ‘ballistic’ computer, essentially an n-body problem; set it up appropriately, such that its evolution performs some computation. The question of what state the system will be in after a certain time can then always be re-interpreted as the question of whether or not the computation will halt. Even Laplace’s Demon, provided he does not have access to hypercomputational means, will have no other choice, in order to ‘predict’ the system’s evolution, to observe the system’s evolution, though possibly in some simulated form — which, however, does not change anything about the system’s essentials: whatever states the system goes through, the simulation must also undergo.
Now, humans are systems of the required complexity; thus, in order to find out whatever a human will do, the only way is to either observe their actions, or simulate them and observe the simulation — which, however, would be entirely equivalent: the simulated human would make the decision in just the same way the ‘real’ one does; he would know no difference. In this sense, then, the will may be its own ultimate cause, while still there is no possibility of ‘could have done otherwise’ — indeed, the notion does not make sense in this analysis: there is no fact of the matter of what one could have done, as the only factually certain thing is what one actually has done; and as Dennett observes, the ‘same exact situation’ is something unique, so it makes no sense to talk as if one could revisit it and ‘change one’s mind’.
This still does not solve the problem of moral culpability — here, I think, Dennett’s concept of ‘evitability’ bears fruit: one can’t escape the outcome of any given situation, but one can modify the likelihood of some undesirable outcome occurring in a host of similar situations. Like the golf player can train himself to make the put in more cases than the amateur, although in any given case, his success is precisely determined by the physical variables, so can a moral agent ‘train’ himself to raise his ‘evitability’, i.e. his ability to evade consequences considered morally undesirable in any given situation. The judgement of being ‘morally good’ is thus put on the same, unambiguous footing as the judgement of being good at golf; criminal prosecution and punishment constitute ‘training’ towards greater moral capacities, towards higher evitability.