Compatibilists argue that determinism is compatible with human freedom, and that indeterminism is not compatible or at best incoherent. They feel (correctly) that there must be a deterministic or causal connection between our will and our actions. This allows us to take responsibility for our actions, including credit for the good and blame for the bad. As long as the agent is free from external coercion, they have freedom of action, which is the compatibilist freedom we have according to Thomas Hobbes and David Hume. Compatibilists (or "soft determinists" as they have been known since William James) identify free will with freedom of action - the lack of external constraints. We are free, and we have free will, if we are not in physical chains. But freedom of the will is different from freedom of action. Many compatibilists accept the view of a causal chain of events going back indefinitely in time, consistent with the laws of nature, with the plan of an omniscient God, or with other determinisms. As long as our own will is included in that causal chain, we are free, they say. And they think causality in nature is related to the very possibility of reason and logic. Without causality, they say, we could not be of the truths of our arguments. Compatibilists don't mind all their decisions being caused by a metaphysical chain of events, as long as they are not in physical chains. We think compatibilists should be classified according to the particular determinisms they think are compatible with human freedom. It is one thing to claim compatibility with physics, another to claim compatibility with God's foreknowledge, etc.
An increasing number of compatibilists, often reluctantly, accept the view that random quantum mechanical events occur in the world. Whether in the physical world, in the biological world (where they are a key driver of genetic mutations), or in the mind, randomness and uncaused events are real. Other compatibilists, Daniel Dennett, for example, simply insist that such genuine irreducible randomness is not needed for evolution or for human freedom. Others point out that even if strict determinism were true (which it isn't), compatibilist freedom of action, in David Hume's sense, would still exist. Quantum events introduce the possibility of accidents, novelty, and human creativity. Compatibilists who admit such indeterminism might very likely be convinced of a stronger argument for human freedom that still provides an adequately determined will. See Giving Determinists What They Say They Want.
Our Cogito model assumes causality and adequate determinism in the critical apparatus of the Macro Mind. From the Micro Mind however, as well as from the external world including other minds, come surprising and unpredictable events to feed the Agenda of possible thoughts and actions. The Cogito is compatibile with both determinism and uncertainty. It lives in Eddington's "halfway house."
Compatibilism explains the will. It cannot explain free. Our new mind model gets us both free (randomness) and will (adequately determined). The Cogito is genuine free will.
Note there is also incompatibilism. There are two kinds of incompatibilists, those who deny human freedom (usually called "hard" determinists) and those who assert it (often called voluntarists, free willists, or metaphysical libertarians - to distinguish them from political libertarians).
The first compatibilist was Carneades (214-129), the great Skeptic.