Boole on Identity, Noncontradiction, and Excluded Middle

As the principles of identity and excluded middle...are true principles and are moreover either first principles or as consciousness informs us lie close to the very ground of the possibility of thought (it matters not here that they have been regarded by some as metaphysical and not logical and by others have been considered frivolous and trifling). A theory of Logic which does not both recognise them and give to them a very important place must be a very imperfect theory. It does not suffice that their existence should be casually noticed.

On the other hand it is very evident that the principles of identity, contradiction and excluded middle are rather principles which relate to conception and to judgment -- to the power by which [we] conceive of things as existing and as existing in relation expressible by propositions -- than to reasoning. To the latter they seem to belong only or chiefly in an implicit manner inasmuch as reasoning presupposes conception and judgment as the sources form which the materials upon which it operates are derived.

George Boole, Prolegomena, sects. 4 & 5, Selected Manuscripts on Logic and its Philosophy, Grattan-Guinness and Bornet, eds. Birkhäuser Verlag (Boston: 1997) 51-52.

A Poem Draft

Half Asleep in a Thunderstorm

I lie in bed at night,
a fan above my head;
my mind whirls round and round
and I dream that I am dead.

The darkness all around me
like a blanket on the brain,
my heartbeat in my ears
like the pounding of the rain,
I watch the world go by,
just leaves upon the gale,
seeing visions of lost time
and the lapse of every tale.

The darkness thunders softly
as I drift here in my bed,
half in the world and of it,
half out of it and dead.

Dark Satanic Mills

Mark Shea notes the oddity of the hymn "Jerusalem", and of its popularity. The hymn was fashioned from a poem in the preface to William Blake's Milton:

And did those feet in ancient time,
Walk upon England’s mountains green:
And was the holy Lamb of God,
On England’s pleasant pastures seen!

And did the Countenance Divine,
Shine forth upon our clouded hills?
And was Jerusalem builded here,
Among these dark Satanic Mills?

Bring me my Bow of burning gold;
Bring me my Arrows of desire:
Bring me my Spear: O clouds unfold:
Bring me my Chariot of fire!

I will not cease from Mental Fight,
Nor Shall my sword sleep in my hand:
Till we have built Jerusalem,
In England’s green & pleasant Land.

I think the popularity of the hymn has to do with its militancy. Militant hymns are striking and stirring in ways that many other hymns are not; they make you want to rise up and do something. And they invoke the sublime, and that is a rare quality in hymns, or, indeed, in any songs whatsoever. Most songs, of course, are better off not even attempting it; but a song that manages to do it without obviously descending into a sort of parody is a rare thing.

Of course, there are hymns that do these things much better than "Jerusalem" does. "The Battle Hymn of the Republic," for instance. But the Battle Hymn, for all its Unitarian provenance, can't be (as Shea notes "Jerusalem" is) a post-Christian hymn; it lacks, for instance, the neo-Pelagian view that progress is something you build by imposing your will on the world. In the Battle Hymn, it is not you who bring progress: it is God's truth that marches on, as inevitable and inexorable as the glory of the morning on the wave, it is He who looses the fateful lightning of His terrible swift sword, against which those who oppose the march of His truth cannot stand. And the only contribution the hymn exhorts people to make to this progress is this, that just as Christ died because that was what it took to make men holy, so we should be willing to die if that's what it takes to make men free, because God is marching on. So much easier to sing that you'll fight ceaselessly to destroy all of the effects of the Industrial Revolution on the basis of a story nobody believes....

Decisive Voting and Oxfam Alternatives

Harry Brighouse has some arguments that there is no strict obligation to vote. There are a few points in the post I agree with (e.g., I don't think anyone has a moral obligation to vote), but he also gives an argument that just gets on my nerves:

Finally, whereas campaigning really does matter in well structured institutions, voting is very strange – the chance that your vote will, in fact, be decisive is almost zero (usually). Suppose it takes you 30 minutes to vote. Even if you earn minimum wage, it would almost certainly be better to work that 30 minutes, and donate the proceeds to Oxfam. All I can say about this is that most people who don’t vote do not, in fact, spend the time they gain from not doing so working for a wage they donate to Oxfam.

I do agree that if you are the sort of person who is persuaded by this argument, you probably should not vote. Ever. It provides such an utterly ridiculous criterion for what would make a vote "matter" that you would think that people who give arguments like this would pause a moment and say, "Well, now that I say it, that doesn't sound quite right." Under what reasonable, rational conception of voting does voting only matter if your vote, in particular, stands a good chance of being decisive? So that's one part of the argument that gets on my nerves.

But the rest gets on my nerves, too; there is, I've found, a whole class of arguments (particularly popular among utilitarians, but not exclusively among them) which might be called "Oxfam alternative arguments", where an activity is compared to donating to Oxfam (or something similar), or working for a period of time and donating to Oxfam, and the comparison is always, of course, unfavorable to the first choice. But this is silly; there is no way to compare the two activities unless we know the field and standard of comparison. Suppose, for instance, that you value participation in the institutions and traditions of your nation. Then it would be utterly silly to say, in light of that value, that it would almost certainly be better to work to donate to Oxfam than to vote. Even if it were still better to work to donate to Oxfam, it would not 'almost certainly' be so; and it could only be so because some greater value took precedence over the value of participation. So such comparisons are meaningless. Which is better, throwing a birthday party for your daughter or giving all the money to Oxfam? Which is better, typing up a post about whether you should vote or spend that time working and donating the money to Oxfam? Which is better, getting eight hours of sleep, or only four while working the other four to donate to Oxfam? There is no meaning to such vague and indefinite comparisons.

Feast of St. Athanasius

Who, then, is this Christ and how great is He, Who by His Name and presence overshadows and confounds all things on every side, Who alone is strong against all and has filled the whole world with His teaching? Let the Greeks tell us, who mock at Him without stint or shame. If He is a man, how is it that one man has proved stronger than all those whom they themselves regard as gods, and by His own power has shown them to be nothing? If they call Him a magician, how is it that by a magician all magic is destroyed, instead of being rendered strong? Had He conquered certain magicians or proved Himself superior to one of them only, they might reasonably think that He excelled the rest only by His greater skill. But the fact is that His cross has vanquished all magic entirely and has conquered the very name of it. Obviously, therefore, the Savior is no magician, for the very demons whom the magicians invoke flee from Him as from their Master. Who is He, then? Let the Greeks tell us, whose only serious pursuit is mockery! Perhaps they will say that He, too, is a demon, and that is why He prevailed. But even so the laugh is still on our side. for we can confute them by the same proofs as before. How could He be a demon, Who drives demons out? If it were only certain ones that He drove out, then they might reasonably think that He prevailed against them through the power of their Chief, as the Jews, wishing to insult Him, actually said. But since the fact is, here again, that at the mere naming of His Name all madness of the demons is rooted out and put to flight, obviously the Greeks are wrong here, too, and our Lord and Savior Christ is not, as they maintain, some demonic power.

If, then, the Savior is neither a mere man nor a magician, nor one of the demons, but has by His Godhead confounded and overshadowed the opinions of the poets and the delusion of the demons and the wisdom of the Greeks, it must be manifest and will be owned by all that He is in truth Son of God, Existent Word and Wisdom and Power of the Father. This is the reason why His works are no mere human works, but, both intrinsically and by comparison with those of men, are recognized as being superhuman and truly the works of God.

From Athanasius, On the Incarnation (ch 8/sect. 48).

Indicative Conditionals and Material Conditionals

A nice little argument worth thinking about. The following is a valid inference.

(p → q) & (r → s), therefore (p → s) v (r → q)

But given this we can easily show that at least some cases of English indicative conditionals do not fit this scheme. Since they would if they were material conditionals, they are not material conditionals. Here's an example, taken from Graham Priest, who takes it from W. S. Cooper:

If John is in Paris he is in France & if John is in London he is in England.

From this we obviously cannot conclude

If John is in Paris he is in England or if John is in London he is France.

Another one. This is valid:

~(p → q), therefore p

But this is not:

It is not the case that if there is a good God the prayers of evil people will be answered. Therefore there is a good God.

Because, of course, it does not follow that there is a good God merely from the assumption that "if there is a good God the prayers of evil people will be answered" is false.

Malebranche on Mathematics

In a recent discussion of the philosophy of mathematics at "n-Category Cafe", Malebranche shows up in the comments, in the form of the following quotation:

L’étude des mathématiques est la plus pure application de l’esprit à Dieu.

I don't presently have the whole Malebranchean corpus at my fingertips, so although I suspected this is an abbreviated rather than exact quotation (such as one might pull from memory rather than directly from the text), I couldn't rule out an exact reference. But it would be a very Malebranchean thing to say (in a slightly more qualified form), and I did find something very similar to it in The Search after Truth, Book V, Chapter IV:

But very few know with certainty that to know the truth is to be joined to God as far as nature allows, that to contemplate the true idea of things is in a way to possess God Himself, and that the abstract perceptino of certain universal and immutable truths, which determine all particular truths, is achieved by the mind that attaches itself to God and rejects the body. Metaphysics, pure mathematics, and all the universal sciences that determine and contain the particular sciences as the universal being contains all particular beings seem chimerical to almost everyone--to the pious as well as to those who have no love for God. Consequently, I hardly dare claim that in applying itself to these sciences the mind applies itself to God in the purest and most perfect way of which it is capable, and that it is in perceiving the intelligible worlds that these sciences have as their object that God God knows and produces the sensible world that bodies depend on for their life as as minds depend on the intelligible. (Lennon-Olscamp translation, p. 367, my emphasis).

Malebranche says other interesting things about mathematics in the Search. For instance:

One can hardly do without at least a crude smattering and a genral knowledge of mathematics and nature. Everyone should have learned these sciences in his youth; they detach the mind from sensible things, and they prevent it from becoming soft and effeminate. The are rather useful in life, and they even direct our attention toward God; the knowledge of nature causes this in and of itself, and that of mathematics through the distaste that it inspires in us for the false impressions of our senses. (LO 292)

And of geometry:

Geometry, then, should be regarded as a kind of universal science that opens the mind, makes it attentive, and gives it the skill to control the imagination and to draw from it all the help it can give; for with the help of geometry the mind controls the imagination, and a controlled imagination sustains the mind's perception and attention. (LO 429)

And similarly, of the ancient practice of teaching mathematics to the young:

Apparently they knew that arithmetic and algebra extend the scope of the mind and give it a certain acuteness we cannot acquire from other studies, and that geometry rules the imagination so well that it does not easily grow confused. For this faculty of the soul, so necessary for the sciences, acquires through the practice of geometry a certain scope of accuracy that impels and preserves the mind's clear perception even in the most perplexing difficulties. (LO 483)

He then goes on to recommmend a course of study that would cover all the major mathematics known in his day, up to "the new differential and integral calculi, and to the methods drawn from them for the understanding of curved lines."

Examples could be multiplied; mathematics plays a very important role in Malebranche's theological rationalism, both on the rationalist side (since he uses it to mount a powerful challenge against any empiricist approach) and on the theological side (since on his view mathematics studies purely intelligible extension, which is found in God, being God's divine idea of body).