What Boole was doing was in the title itself: The Laws of Thought is not a discussion of a pure abstract system, but a mathematical investigation of an empirical phenomenon, thought, on analogy with mathematical physics. Boole recognizes that there are differences between the mental and the physical, and that these can sometimes be relevant to the course of the investigation, but the analogy on its own makes obvious that we should not worry about uninterpretable terms. In the mathematical derivations used in physics, nothing requires that the mathematical steps be describing physical steps. As long as your starting point describes something physical and your ending point describes something physical, you are fine; the rest may be physically interpretable or it may just be mathematical bookkeeping. And a number of features of Boole's argument can only be properly understood in light of this approach. For instance, in one passage that is often skipped over by people who are only interested in the logical methods, Boole makes appeals to how poetry works. If you're only interested in the mathematical method, you might treat this as a quaint example. But a close examination of the brief discussion makes clear that it is actually a major pillar of Boole's argument. Poetic language is expression of thought that is not encumbered by ordinary practical constraints; it is intelligible but free. Showing that poets regularly make use of commutativity (e.g., treating adjectives as switchable) gives us reason to think that it's reasonable to treat commutativity as part of the underlying structure of thought; if we find commutativity-violating cases, we can look for the additional factors (practical limitations, for instance) that make those cases different from the most unrestricted case. The Laws of Thought is about the laws of thought.
If you read the whole work, however, it becomes obvious that there is a lot of natural theology in it. Theology was one of Boole's major interests; he read extensively in it. Many of the examples given in The Laws of Thought are theological in character, and there is no doubt that one of the purposes of this is to show the value of Boole's methods in analyzing arguments even in quite difficult fields of thought. But here again, focusing on the structure of the method alone overlooks certain important structural features of the argument itself. One of the chapters devoted to showing the power of the method -- Chapter XIII -- uses algebra to discuss arguments in Clarke's Demonstration of the Being and Attributes of God and Spinoza's Ethics. You could very well treat this as a matter of happenstance, due to Boole's interests, or as just a handy set of examples, but you would be wrong. Boole does not confine himself to simply showing that you can characterize the arguments using his algebraic methods, but instead takes the trouble to use that as a platform for deeper considerations. Boole, remember, is not barely interested in the use of algebra to do something relevant to logic; he is trying to make progress in understanding the constitution of the mind, the laws of thought, analogous to the way the physicist seeks to make progress in understanding the constitution of the world and the laws of nature. And this is precisely what he tries to do. Boole has an obviously high opinion of Clarke and a low opinion of Spinoza, but his overall thrust is to argue that the a priori approach they both favor is not really tenable. In Spinoza's case, Boole thinks the algebraic method shows that there is not much happening beyond some rather trivial manipulation of terms. Clarke, he thinks, is generally doing things that are more subtle. But in both cases, the overall conclusion that can be derived is that the argumentative part of the arguments is not doing the actual work. While Boole has a high opinion of Clarke's logical acumen, the discovery of truth does not primarily consist in the logical transformations that are traced by the algebra.
When Joseph Butler was a student, he wrote a number of letters to Samuel Clarke with objections to certain points in Clarke's Demonstration; the correspondence has been preserved, and serves an important role in Boole's analysis. The correspondence established that some of Clarke's argumentation depended on the principle that whatever is absolutely necessary is necessary at all places and times. Butler objected that necessary existence seems to require only that the necessarily existent exist somewhere and at some time, assuming, of course, that all existing is existing somewhere and at some time. Commenting on the correspondence, Boole says,
The objections of Butler are precisely those which would occur to an acute mind impressed with the conviction, that upon the sifting of first principles, rather than upon any mechanical dexterity of reasoning, the successful investigation of truth mainly depends. (p. 200)
This conviction attributed to Butler is also Boole's. (It's notable, incidentally, that every one of the several mentions of and allusions to Butler in the work are favorable and express Boole's agreement with Butler's approach.) Boole takes a priori approaches in theological matters to put the cart before the horse. Discussions of these difficult metaphysical topics must begin with analogy and probable induction, because an inquiry into the laws of thought makes clear how limited our pure thinking-capacity is:
To infer the existence of an intelligent cause from the teeming evidences of surrounding design, to rise to the conception of a moral Governor of the world, from the study of the constitution and the moral provisions of our own nature;--these, though but the feeble steps of an understanding limited in its faculties and its materials of knowledge, are of more avail than the ambitious attempt to arrive at a certainty unattainable on the ground of natural religion. And as these were the most ancient, so are they still the most solid foundations, Revelation being set apart, of the belief that the course of this world is not abandoned to chance and inexorable fate. (pp. 217-218).
That this is all not just a digression becomes clear given the remaining course of the book. Boole immediately goes on to discuss probability, with the ultimate conclusion that Algebra, Logic, and Theory of Probability all have extensive analogies with each other, and the subordinate conclusion that when we think probabilistically about causal order, the algebra inevitably involves terms to which we cannot definitely assign a numerical value. Probable reasoning about causes is only possible by assuming hypotheses; but this does not mean that the causal order is itself merely hypothetical. Rather, there are steps that are not measurable but real. And this is precisely the point that Boole draws in the final chapter of the work, when he reiterates that he is interested in understanding the constitution of the mind. Boole takes it, as it is a question of the Order of things, that we can only draw probable inferences involving hypotheses, but it is an error to treat this as making the inferences insignificant. (This is as true when we are talking about laws of thought as when we are talking about laws of nature.) In every kind of causal reasoning, the make-up of our minds is playing a key role. How do we move from particular data to general conceptions? "It is the ability inherent in our nature to appreciate Order, and the concurrent presumption, however founded, that the phaenomena of Nature are connected by a principle of Order" (p. 403). This means that necessary truths should probably be seen more as limiting suppositions than as things we know distinctly in themselves; our approach should be generally a posteriori.
Further, Boole returns to the question of a priori method in metaphysical and theological questions, and the argument again shows that he is interested in the constitution of the mind and not just the algebraic method. Building on his algebraic methods, he tries to explain why the a priori approach always recurs in the history of thought. In Boole's algebra, the number 1 is found all over the place. It represents what he calls the 'universe of discourse', the ultimate subject of discussion. Thus, if I say, "There is nothing in the refrigerator", this can actually mean many different things, depending on whether we assume that the overall discussion is about milk, food in general, or matter. If the universe of discourse represents material objects, there is a material void in the refrigerator; if it represents food, there may be things like air but not things like pickles; if it represents milk, there may be pickles, but there won't be milk. The universe of discourse shows up as 1 in Boole's algebra, and Boole takes this not to be an accident: logical universe is closely analogous to numerical unity, and human thought is so constituted as to trace everything it considers to some kind of primal unity (p. 411). Thus we get various mysticisms of The One, and so forth. And the fact that we do tend to think this way suggests the probable hypothesis of there being some sort of Primal Unity such as natural theology indicates. But it also shows that what is occurring in a priori approaches to natural theology is the assumption that the natural world has the unity-based structure we are inclined by the constitution of our thought to assume for it, combined with the assumption that we have direct insight into how the primary unity is to be interpreted. But as we ascend from particulars to general conceptions, this interpretation is really something we could only get by analogy and probable induction. And similar considerations can be traced with other foundational elements in the laws of thought as Boole has traced them out algebraically.
Boole concedes that some people might regard all of these theological and metaphysical discussions as a digression, but they are not; this in fact how he ends the book:
To some they will appear foreign to the professed design of this work. But the consideration of them has arisen naturally, either out of the speculations which that design involved, or in the course of reading and reflection which seemed necessary to its accomplishment. (p. 424)
And this is confirmed by Mary Everett Boole, George Boole's wife, who in a published letter usually titled "Indian Thought and Western Science in the Nineteenth Century" explicitly lays out precisely this interpretation of The Laws of Thought. First, the originary idea:
My husband told me that when he was a lad of seventeen a thought struck himsuddenly, which became the foundation of all his future discoveries. It was a flash of psychological insight into the conditions under which a mind most readily accumulatesknowledge. Many young people have similar flashes of revelation as to the nature of their own mental powers; those to whom they occur often become distinguished in some branch of learning; but to no one individual does the revelation come with sufficient clearness to enablehim to explain to others the true secret of his success....But by the help of a learned Jew in Lincoln he found out the true nature of the discovery which had dawned onhim. This was that man's mind works by means of some mechanism which "functions normally towards Monism."
This, she says, was the point in the writing of The Laws of Thought:
If he had stated it in words, he would have been entangled in an unseemly theological skirmish. He presented the truth to the learned, clothed in a veil so transparent that it is difficult to conceive how any human being could have been blinded by it; he proved that by the mere device of always writing the symbol 1 for whatever is the "Universe of Thought" for the time being, the whole cumbersome mechanism then known as 'Logic' could be dispensed with.... He said in the book that this law was a law, not of facts or of essential reason, but of the human mind (Laws of Thought, p. 4).
Mary Boole, I think, has a tendency to oversimplify a bit when talking about her husband's work in matters involving her own interests, but her interpretation here is borne out by the facts of the case. The Laws of Thought discuss the application of algebraic methods to logic and probability theory; it does this explicitly to investigate the laws of thought that constitute the mind. Obviously a lot of different issues are discussed in doing this -- thus we get the 'veil' -- but the structure of the work is quite clearly a twofold argument: that the constitution of the mind "functions normally towards Monism", i.e., has a tendency to trace everything back to primal unity, and that this is a law "not of facts or of essential reason, but of the human mind", and thus the significance of it really requires an a posteriori rather than an a priori approach.
Arguably, Boole's attempt to avoid being "entangled in an unseemly theological skirmish" resulted in a veil less transparent for the learned than Mary Boole suggests. Certainly most people fail to grasp the structure of the work at all, and most of the theological side of the work is treated as either merely an example for the logical apparatus or a digression. Nor can they be entirely blamed, because when Boole brings these things up, he always ends up talking about suggestion and analogy, and saying he can't fully pursue the question here, which makes it easy to overlook that these things may in fact have been among the things he primarily had in view. It also, I think, leads to a number of inadequately filled gaps in Boole's argument. Much of his diagnosis of a priori approaches, for instance, really depends on hypotheses for which he has provided only very brief argument, or even only gestures at arguments. As Boole notes with respect to natural theology itself, that an inference depends on hypotheses is not in itself a flaw, but the avoidance of "skirmish" has the result that some important parts of the argument are not thoroughly examined. Nonetheless, there can be no doubt, I think, that a major goal of The Laws of Thought is the critical investigation of methods of reasoning in matters such as the existence of God; it comes up not just as a set of examples but as essential to the structure of the work.
George Boole, The Laws of Thought, Prometheus Books (Amerst, NY: 2003).