Friday, January 04, 2019

Boole's The Laws of Thought as a Text on Natural Theology

George Boole's The Laws of Thought (the longer title being An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities) is one of the classic works in logic. Published in 1854 and showing that you could do a very large amount of logic using an algebra in which x2=x was treated as a fundamental principle, it made a significant splash, but some of the choices made by Boole -- in particular, his use of exclusive disjunction (which was deliberate, and which Boole would insist was necessary to the end of his life) and his allowance of uninterpretable terms struck most of his contemporaries as odd, and eventually led them to develop what we today call Boolean algebra, which works on Boole's ideas but throws out such oddities. This has led people, I think, not to fully understand what Boole was doing; what we call Boolean algebra is simply not appropriate to what Boole was trying to do to begin with.

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).

Thursday, January 03, 2019

I Ease You

by George Herbert

Jesu is in my heart, his sacred name
Is deeply carved there: but th’other week
A great affliction broke the little frame,
Ev’n all to pieces: which I went to seek:
And first I found the corner, where was J,
After, where ES, and next where U was graved,
When I had got these parcels, instantly
I sat me down to spell them, and perceived
That to my broken heart he was I ease you,
And to the whole is J E S U.

Wednesday, January 02, 2019

Capreolus on the Unity of the Virtues

One of the standard positions of traditional virtue ethics is what is often called the unity of virtues thesis, which is that virtues are interconnected so that to have one you have to have all of them. In practice, the position has usually been understood not to apply literally to everything that can be called a virtue but only to the cardinal virtues. To have prudence, the intellectual cardinal virtue, you must have the three moral cardinal virtues of justice, fortitude, and temperance; to have justice, you must have prudence, fortitude, and temperance, and so forth. It has also tended to be recognized that we often use virtue-terms to indicate acquired dispositions that are not strictly virtuous but virtue-ish, incomplete virtues, we might say, and that this incompleteness is due to imperfections in the integrity of one's character, so that there is a perfectly reasonable sense in which one can say that (for instance) someone is temperate but not just.

So the basic position really amounts to saying that to have any cardinal virtue (at least in a complete and proper form), you must have all the cardinal virtues (in a complete and proper form). This makes a certain amount of intuitive sense, given that you can think of ways in which one cardinal virtue might be necessary to the completeness of another -- for instance, someone who (we suppose) has justice but not fortitude is not going to be appropriately disposed for cases in which justice requires endurance in the face of death or grave danger, and someone who has (again, we suppose) temperance but no justice would presumably have difficulty finding the right point of moderation in matters in which justice was intimately involved. In addition, of course, there are good Aristotelian reasons for thinking that all virtues whatsoever depend on prudence in one way or another. However, even so, people tend to be quite shy about accepting the unity of the virtues thesis today, so it's worth looking a bit more closely at a more developed defense. Capreolus's defense of the Thomistic version of the unity of virtues thesis, from his Defensiones Theologiae Divi Thomae Aquinatis.

Capreolus analyzes the position into basic claims that could be disputed, with each given its own basic argument.

(1) No one can possess prudence without moral virtues. You can't have, by a sort of second nature, the disposition to judge rightly in a matter to which a virtue is relevant unless you can judge rightly with respect to the particular presuppositions of particular virtues, which vary according to the virtue. We see this in cases where someone desires badly; people who desire badly have a tendency to make false assumptions in their moral reasoning about those things they desire. People who have a vice are prejudiced in the direction of the vice whenever they reason, for instance. So it follows that prudence has to grow up with the other moral virtues, so that we will be reasoning rightly in the regions of human life with which those virtues are concerned.

(2) No one can possess moral virtues without prudence. This follows in the standard Aristotelian way:

No habit that chooses according to right reason can exist without a right reason that deliberates, judges, and prescribes. But this is the way in which moral virtue is related to prudence. Therefore it cannot exist without prudence. (p. 327)

(3) The four cardinal virtues are interconnected so that whoever possesses one possesses the others. This follows, of course, from the previous two claims.

These arguments suffice for the conclusion, but as Capreolus is a scholastic commentator, he does not consider a question to have been properly understood until one can see how objections may be handled, and despite the near-universal acceptance of the unity of virtues thesis up to Capreolus's day, he is able to find some fairly formidable objections in the work of John Duns Scotus. As with many disputes between Thomists and Scotists, the dispute is not so much about the central point but about load-bearing features in the rational account of that point. Scotus, like Aquinas, does accept the unity of virtues thesis in some form, even if qualified. However, if you think about the above two arguments, it's clear that the load-bearing element in these arguments is the virtue of prudence, so that the ground for the interconnection of the cardinal virtues is prudence itself. Scotus thinks that this is more complicated than it seems, and that because of this (1) is not strictly necessarily true, and that (3) is only going to be the case if we are talking about the character of someone who is moral, simply speaking. Obviously you need all the virtues to be moral, simply speaking, and obviously to have virtue you need prudence. But Scotus has the view that there is not a single species of prudence, and you need different kinds of prudence for different virtues, so even (2) is not accepted by him in an unqualified form.

Part of the reason for the additional complications is Scotus's desire to preserve freedom of will against all possible intrusion: on his account, you can judge rightly and not will at all in accordance with it. So it is in principle possible to have the habit of judging rightly (prudence) but have no moral virtues because you simply don't make the relevant choices that your right judgments indicate. Obviously its being in principle possible does not imply that this is at all the norm; I'm not sure how common Scotus thinks it actually is, in fact. But if it's in principle possible, then (1) certainly has to be weakened and (3) qualified.

A second reason is that Scotus thinks virtues could be complete in their own field even if they are not complete with regard to what they would have to be for a completely moral character. He uses the analogy of the senses. You could have perfect sight without perfect hearing; the two kinds of sensation are different. Obviously if you lack perfect hearing, you have imperfect sensory ability, simply from that very fact, but it doesn't directly affect the act of sight. In the same way, he thinks, if you lack (say) the virtue of temperance, you lack virtue in the sense of full moral character, but this would not affect acts of (say) fortitude, for which you still could have the right disposition of second nature.

Thomists have always regarded the Scotist account as effectively breaking the account of virtues, so that virtues are barely more than habits that happen to conform to some kind of right judgment; the relationship between virtues on Scotus's account, while real, is quite limited. The acquired dispositions for the virtues aren't interconnected on their own but only in the sense that they happen to be the acquired dispositions that are consistent with right reason. The unity is posterior to the habits, one might say, rather than integral to them in the way Thomists want to argue.

The Thomistic position on free choice, of course, differs from the Scotistic position in that Scotus treats the intellect as a purely natural faculty and puts all of the freedom in the will itself, whereas Aquinas divides human freedom in matters of choice between the intellect and the will, so that both are free faculties in a limited sense and freedom of choice in the full sense arises from their interaction. Because of this, Capreolus takes the position of the Scotists to over-assimilate prudence to universal practical knowledge; prudence is in fact concerned with deliberating, judging, and prescribing in these particular circumstances here and now, and it is impossible treat the actual choice as completely separate from this judgment -- it is of the nature of the thing that how the will chooses incorporates, so to speak, the intellectual contribution. This is not to say that prudence always issues in doing the right thing, which everyone agrees is not the case with any of the virtues, but when prudence fully prescribes something as the right course, it's impossible for the will to be in a state as if this had not happened or as if this were irrelevant, and since a virtue is a habit and not a single act, it's impossible for the intellect to have the habit of prudence, and the consistent right judgment that comes with that, without the will acquiring an appropriate disposition in response.

Capreolus, of course, simply denies the analogy with the senses; the two cases are not similar. It is true that there is no strict interconnection among the senses, but this is because the sense are straightforwardly independent: sight does not subserve hearing, nor vice versa, they just contribute different things. Sensory life does not, as such, depend on any particular ordering. This is not true of moral life: moral matters are structured such that (for instance) if you are intemperate, you are going to commit injustices, and we see this all the time in people failing to do justice to each other because of lust. Moral decision-making is not, for the Thomist, a matter of doing the right thing in relatively isolated domains; rather, doing the morally right thing in one domain will often overlap with doing the morally right thing in another domain. So prudence is in fact all one and the virtues have to grow up together, so to speak, into one integral moral character. Consistently good moral decision-making is holistic, and can't be broken into lesser parts. This doesn't change the fact, noted previously, that as the virtues are being developed, the interconnections might not be fully formed; but these failures will be precisely ways in which we have failed fully to develop the virtue in question.

Now, obviously I have somewhat simplified the points in question, and equally obviously Scotists will have responses to Capreolus. But as is generally the case with scholastic dispute, as one works through the dispute one gets a clearer picture of the conclusions that are being drawn.


John Capreolus, On the Virtues, White and Cessario, trs. Catholic University Press (Washington, DC: 2001).

Basil and Gregory

Today is the feast of Ss. Basil of Caesarea and Gregory of Nazianzen, Doctors of the Church.

From On the Holy Spirit (1.2) by St. Basil the Great:

To count the terms used in theology as of primary importance, and to endeavour to trace out the hidden meaning in every phrase and in every syllable, is a characteristic wanting in those who are idle in the pursuit of true religion, but distinguishing all who get knowledge of "the mark" "of our calling;" for what is set before us is, so far as is possible with human nature, to be made like God. Now without knowledge there can be no making like; and knowledge is not got without lessons. The beginning of teaching is speech, and syllables and words are parts of speech. It follows then that to investigate syllables is not to shoot wide of the mark, nor, because the questions raised are what might seem to some insignificant, are they on that account to be held unworthy of heed. Truth is always a quarry hard to hunt, and therefore we must look everywhere for its tracks. The acquisition of true religion is just like that of crafts; both grow bit by bit; apprentices must despise nothing. If a man despise the first elements as small and insignificant, he will never reach the perfection of wisdom.

From The First Theological Oration (section 3) of St. Gregory of Nazianzen:

Not to every one, my friends, does it belong to philosophize about God; not to every one; the Subject is not so cheap and low; and I will add, not before every audience, nor at all times, nor on all points; but on certain occasions, and before certain persons, and within certain limits.

Not to all men, because it is permitted only to those who have been examined, and are passed masters in meditation, and who have been previously purified in soul and body, or at the very least are being purified. For the impure to touch the pure is, we may safely say, not safe, just as it is unsafe to fix weak eyes upon the sun's rays. And what is the permitted occasion? It is when we are free from all external defilement or disturbance, and when that which rules within us is not confused with vexatious or erring images; like persons mixing up good writing with bad, or filth with the sweet odours of ointments. For it is necessary to be truly at leisure to know God; and when we can get a convenient season, to discern the straight road of the things divine. And who are the permitted persons? They to whom the subject is of real concern, and not they who make it a matter of pleasant gossip, like any other thing, after the races, or the theatre, or a concert, or a dinner, or still lower employments. To such men as these, idle jests and pretty contradictions about these subjects are a part of their amusement.

Of Easy Wind and Downy Flake

Stopping by Woods on a Snowy Evening
by Robert Frost

Whose woods these are I think I know.
His house is in the village though;
He will not see me stopping here
To watch his woods fill up with snow.

My little horse must think it queer
To stop without a farmhouse near
Between the woods and frozen lake
The darkest evening of the year.

He gives his harness bells a shake
To ask if there is some mistake.
The only other sound’s the sweep
Of easy wind and downy flake.

The woods are lovely, dark and deep,
But I have promises to keep,
And miles to go before I sleep,
And miles to go before I sleep.

Tuesday, January 01, 2019

Fortnightly Books Index 2018

Obviously there was a lot of Jules Verne this year, as I was trying to read through as many of his Voyages Extraordinaires as possible; but the non-Verne novels were quite a diverse group.

November 25: Jules Verne, Paris in the Twentieth Century; (with Michel Verne) The Lighthouse at the End of the World
Introduction, Review

November 11: Flann O'Brien, The Third Policeman
Introduction, Review

October 28: Jules Verne, 20,000 Leagues Under the Sea(s)
Introduction, Review

October 14: Mary Shelley, Frankenstein
Introduction, Review

September 30: Jules Verne, Family Without a Name
Introduction, Review

September 16: Harriet Beecher Stowe, Uncle Tom's Cabin
Introduction, Review

September 2: Umberto Eco, Baudolino
Introduction, Review

August 19: Jules Verne, The Mighty Orinoco; Invasion of the Sea
Introduction, Review

August 5: Frances Mossiker, Napoleon and Josephine: Biography of a Marriage
Introduction, Review

July 22: Jules Verne, The Begum's Millions [with Paschal Grousset]; Robur the Conqueror; Master of the World
Introduction, Review

July 8: Frederick Douglass, Narrative of the Life of Frederick Douglass; Booker T. Washington, Up from Slavery
Introduction, Review

June 24: Jules Verne, Journey to the Center of the Earth
Introduction, Review

June 3: George MacKay Brown, Magnus
Introduction, Review

May 20: Jules Verne, The Self-Propelled Island; A Castle in Transylvania
Introduction, Review

May 6: Isaac Asimov, The Complete Adventures of Lucky Starr
Introduction, Review

April 22: Sir Walter Scott, Waverley
Introduction, Review

April 8: The Poetic Edda; The Saga of the Volsungs and The Saga of Ragnar Lothbrok
Introduction, Review

March 25: Jules Verne, The Kip Brothers; Travel Scholarships
Introduction, Review

March 11: Sigrid Undset, Ida Elisabeth
Introduction, Review

February 25: Sir Walter Scott, Rob Roy
Introduction, Review, Timeline

February 11: Jules Verne, From the Earth to the Moon & Around the Moon
Introduction, Review

January 28: Elias Lönnrot, The Kanteletar
Introduction, Review

January 14: Jules Verne, The Adventures of Captain Hatteras
Introduction, Review


Fortnightly Books Index for 2017

Fortnightly Books Index for 2016

Fortnightly Books Index for 2015

Fortnightly Books Index for 2014

Fortnightly Books Index for 2012-2013

Monday, December 31, 2018

Music on My Mind

Connor Engstrom (ft. Joey Elrose), "Final Countdown".