The argument from reason challenges the proposition that everything that exists, and in particular thought and reason, can be explained solely in terms of natural or physical processes. It is, therefore, an argument against materialist philosophy of mind. According to the argument, if such theories were true, our thoughts, and so also our reasoning, would be determined on the molecular level by neurochemistry, leaving no role for the free exercise of reason.

The basis of the argument is, then, that if materialism were correct, our thoughts would be the product of physical processes which are in themselves devoid of any purpose or intentionality (in line with the axioms of materialism, which holds that everything in the Universe is the product of physical laws and product of non-intentional and non-purposive processes). However our ability to engage in rational inference presupposes the existence of intentionality and purpose. Reasoning involves forming beliefs based on evidence, making logical inferences, and seeking to arrive at a true understanding. If our thoughts were merely the result of physical causation, they would lack the ability to either genuinely apprehend truth, or to be rationally justified in making truth claims.

The Target of Criticism
In order to clearly frame the argument from reason, it is necessary to understand what it is opposed to. This is usually said to be ‘naturalism’, but I will instead propose that its target is better named physicalism or materialism.

Thomas Nagel uses the term "the materialist neo-Darwinian concept of mind" ¹ to refer to a view that combines materialist philosophy of mind with the principles of neo-Darwinism, which is the modern synthesis of Darwinian evolution and genetics. Advocates include Daniel Dennett, Alex Rosenberg, D.M. Armstrong, J C C Smart, and others. These are strict reductionists, claiming that states of mind are brain-states, and therefore ultimately reducible to physical laws. For example, Daniel Dennett, speaking of the organic molecule, says that ‘An impersonal, unreflective, robotic, mindless little scrap of molecular machinery is the ultimate basis of all the agency, and hence meaning, and hence consciousness, in the universe’²

C S Lewis
The current form of the argument from reason was popularised by C S Lewis in 1947, subsequently revised and reformulated after criticism from G.E.M. Anscombe. The argument is as follows:

1. No belief is rationally inferred if it can be fully explained in terms of nonrational causes.

This is because rational inference requires insight into logical relations. A process of reasoning (P therefore Q) is rational only if the reasoner sees that Q follows from P, and accepts Q on that basis. Thus, reasoning is veridical only if it involves a specific kind of causality, namely, rational insight. If a belief can be fully explained by nonrational causes, such as a physical influence then it cannot be said to be the product of reason. Lewis gives the example:

We can say, 'Grandfather is ill today because he ate lobster yesterday.' We can also say, 'Grandfather must be ill today because he hasn't got up yet (and we know he is an invariably early riser when he is well).' In the first sentence 'because' indicates the relation of Cause and Effect: The eating made him ill. In the second, it indicates the relation of what logicians call Ground and Consequent. The old man's late rising is not the cause of his disorder but the reason why we believe him to be disordered. There is a similar difference between 'He cried out because it hurt him' (Cause and Effect) and 'It must have hurt him because he cried out' (Ground and Consequent). The first indicates a dynamic connection between events or 'states of affairs' i.e. 'eating lobster caused him to be ill'; the first, a logical relation between beliefs.

Now a train of reasoning has no value as a means of finding truth unless each step in it is connected with what went before in the Ground-Consequent relation. If our B does not follow logically from our A, we think in vain. If what we think at the end of our reasoning is to be true, the correct answer to the question, 'Why do you think this?' must begin with the Ground-Consequent 'because' — C S Lewis, Miracles, Chap 3

2. If naturalism is true, then all beliefs can be fully explained in terms of non-rational causes.

Support: Naturalism holds that nature is all that exists, and that all events in nature can in principle be explained without invoking mental causation. As a matter of definition physicalists claim that all events must have physical causes, and that therefore human thoughts can ultimately be explained in terms of material causes or physical events (such as neurochemical events in the brain) that are nonrational. In Lewis' terms, this would entail that our beliefs are a result of a physical chain of causes, not held as a result of insight into a ground-consequence relationship.

3. Therefore, if naturalism is true, then no belief is rationally inferred (from 1 and 2).

4. We have good reason to accept naturalism only if it can be rationally inferred from good evidence.

5. Therefore, there is not, and cannot be, good reason to accept naturalism.

(As noted, I feel the argument is better expressed by the term 'physicalism' rather than 'naturalism', as today's naturalism is not so clearly physicalist as it was at the time of Lewis' argument, although that in itself is an interesting discussion point.)

My view of the argument

I myself am not a particular admirer of C S Lewis (although I enjoyed a collective biography of him and his confrères, The Inklings, P & C Zalenski - the other Inklings being Owen Barfield, C.S Williams, and Tolkein). Nor do I present the argument as any kind of 'proof of God' (God forbid). However, I'm convinced that the basic drift of the argument is sound, because, as it says, reasoned inference comprises wholly and solely the relationship of ideas - not of neurochemicals across synapses.

'Hang on', you might say, 'if we but had sensitive enough instrumentation, would it not be possible to detect the neural configurations of such ideas in the operations of the brain?' To which I would point out that there's an inevitable circularity in such an attempt. This is because to establish any kind of logical relationship between the data, and the exercise of reason, would itself be an exercise of reason. I would argue that reason is internal to the act of thought - that it can't be detected in external data, such as neurological data, without drawing on the very faculty which you're seeking to explain. So I would argue that there's a problem of recursion - you can't see reason 'from the outside', as it were (another of Thomas Nagel's arguments.)

That's all for now - there are many more points to discuss, but it's already a long OP so will leave it there pending any comments.

--------
1. Thomas Nagel, Mind and Cosmos: Why the Materialist Neo-Darwinian Conception of Nature Is Almost Certainly False. In this book, Nagel makes the case that this is, as it were, the default philosophy of the secular intelligentsia. A detailed abstract is available here (NY Times).
2. Daniel Dennett, Darwin’s Dangerous Idea: Evolution and the Meanings of Life (New York: Simon and Schuster, 1995), 202-3.