Why Determinism is irrelevant to the argument from reason

Lewis's account of naturalism seems to imply that it does, although he mentions quantum-mechanical theories suggesting it does not. John Beversluis, in his treatment of Lewis's argument from reason, points out that Lewis does not consider or refute indeterministic forms of naturalism.

However, in dealing with more recent versions of the argument from reason the question of determinism is irrelevant, as I argued in a recent essay:

Exactly what does Lewis mean by naturalism? Very often the terms Naturalism and Materialism are used interchangeably, but at other times it is insisted that the two terms have different meanings. Lewis says,

“What the naturalist believes is that the ultimate Fact, the thing you can’t go behind, is a vast process of time and space which is going on of its own accord. Inside that total system every event (such as your sitting reading this book) happens because some other event has happened; in the long run, because the Total Event is happening. Each particular thing (such as this page) is what it is because other things are what they are; and so, eventually, because the whole system is what it is.”

As a presentation of naturalism, however, this might be regarded as inadequate by contemporary naturalists, because it saddles the naturalist with a deterministic position. The mainstream position in contemporary physics involves an indeterminism at the quantum-mechanical level. Lewis himself thought that this kind of indeterminism was really a break with naturalism, admitting the existence of a lawless Subnature as opposed to Nature, but most naturalists today are prepared to accept quantum-mechanical indeterminism as part of physics and do not see it as a threat to naturalism as they understand it. Some critics of Lewis have suggested that his somewhat deficient understanding of naturalism undermines his argument. Lewis, however, insisted on “making no argument” out of quantum mechanics and expressed a healthy skepticism about making too much of particular developments in science that might be helpful to the cause of apologetics.
However, contemporary defenders of the Argument from Reason such as William Hasker and myself have developed accounts of materialism and naturalism that are neutral as to whether or not physics is deterministic or not. Whatever Lewis might have said about quantum-mechanical indeterminacy, the problems he poses for naturalism arise whether determinism at the quantum-mechanical level is true or not.
Materialism or naturalism, as we understand it, is committed to three fundamental theses.
1) The basic elements of the material or physical universe function blindly, without purpose. Man is the product, says Bertrand Russell, of forces that had no prevision of the end they were achieving. Richard Dawkins’ exposition and defense of the naturalistic world view is called The Blind Watchmaker: Why the Evidence of Evolution Reveals a World Without Design not because no one ever designs anything in a naturalistic world, but because, explanations in terms of design must be reduced out in the final analysis. Explanation always proceeds bottom-up, not top-down.
2) The physical order is causally closed. There is nothing transcendent to the physical universe that exercises any causal influence on it.
3) Whatever does not occur on the physical level supervenes on the physical. Given the state of the physical, there is only one way the other levels can be.

These three claims can be true if "the physical" is deterministic or not. Even if there are no determining physical causes, if all that makes it undetermined and is nothing but brute chance, this hardly introduces libertarian free will or reason.