Little Flower

Today is the Feast Day of St. Thérèse of Lisieux, Doctor of the Church. She died in 1897 at the age of 24.

But how shall I show my love, since love proves itself by deeds? Well! The little child will strew flowers . . . she will embrace the Divine Throne with their fragrance, she will sing Love's Canticle in silvery tones. Yes, my Beloved, it is thus my short life shall be spent in Thy sight. The only way I have of proving my love is to strew flowers before Thee—that is to say, I will let no tiny sacrifice pass, no look, no word. I wish to profit by the smallest actions, and to do them for Love. I wish to suffer for Love's sake, and for Love's sake even to rejoice: thus shall I strew flowers. Not one shall I find without scattering its petals before Thee . . . and I will sing . . . I will sing always, even if my roses must be gathered from amidst thorns; and the longer and sharper the thorns, the sweeter shall be my song.

But of what avail to thee, my Jesus, are my flowers and my songs? I know it well: this fragrant shower, these delicate petals of little price, these songs of love from a poor little heart like mine, will nevertheless be pleasing unto Thee. Trifles they are, but Thou wilt smile on them. The Church Triumphant, stooping towards her child, will gather up these scattered rose leaves, and, placing them in Thy Divine Hands, there to acquire an infinite value, will shower them on the Church Suffering to extinguish its flames, and on the Church Militant to obtain its victory.

Therese of Lisieux, Story of a Soul

Jottings on the Modal Logic of Quantifiers

These are mostly me trying to work through some ideas, perhaps badly, but comments are welcome.

I was kept up a few nights ago thinking about modal logic and quantifiers. Quantifiers, put in broad, broad terms indicate whether we are allowing exceptions for a claim. For instance, if I say, "All dogs go to heaven" that (as usually understood) tells us that "Dogs go to heaven" has no exceptions. If I say, however, that "Some dogs go to heaven," this allows exceptions (as long as there is at least one thing that is not an exception), even though it doesn't guarantee them.

It is not quite this simple in practice, since this is a term-logic interpretation of quantifiers; modern logic introduces other complications with existential import, etc. But the key thing quantifiers do is track exceptionlessness or possibility-of-exception.

From this it actually follows that quantifiers can be represented by a modal logic. A modality is basically a qualification; if you are talking about the modality of a proposition or claim, you are talking about some qualification of its truth (or falsehood). For instance, a proposition might be necessarily true, or possibly false, or true in the past, or any number of other things. But there's nothing that requires that we apply modalities only to whole propositions, and, indeed, we regularly apply them in practice to predicates. "All dogs necessarily go to heaven" is subtly but importantly different from saying "It is necessarily true that all dogs go to heaven"; if you don't see this, just take my word for it for a moment.

In general modalities that qualify one thing (whether propositions, or predicates, or whatever) fall into two broad classes: Box and Diamond. Box is the strong one (necessity, obligation, always), Diamond is the weak one (possibility, permissibility, sometimes). Since quantification is a kind of qualification (in term-logical terms, it is a qualification of the subject term), it makes sense to think of it modally. And we do obviously have a strong modality and a weak modality here: universal quantification ("All dogs go to heaven") is Box-like and particular quantification ("Some dogs go to heaven") is Diamond-like. And they do tend to work like Box and Diamond.

This analogy has actually been known for quite some time; I think von Wright was the first one to note it. But it's usually read the other way, so that Box and Diamond can be treated as if they were quantifiers. This is valuable but (frankly) can get one into weird territory, because you always have to ask, "What are we quantifying over?" And so we get possible worlds and the like. But you can go the other way, too: treat universal quantification as Box and particular quantification as Diamond.

Now, the thing of it is, there are many, many, many systems using Box and Diamond, and they are all very different; Box and Diamond, when we aren't specifying a particular system, are really just broad categories of modality, not modalities themselves. So this raises the question of many different kinds of quantifiers, depending on the system in which we are interested. And indeed, we know this in practice: some quantifier-systems allow subalternation (you can get particulars from universals, e.g., you can conclude that some dogs go to heaven from the truth of "all dogs go to heaven"), while some do not.

Thus a subalternating quantifier-system is one in which a version of what is sometimes known as the characteristic D axiom is true: Box implies Diamond for whatever Box might apply to. Thus, 'A is obligatory' (Box) implies 'A is permissible' (Diamond), 'A is necessary' (Box) implies 'A is possible' (Diamond), and, apparently, "All A" (Box) implies "Some A" (Diamond) when we are allowing subalternation.

Further, you can have reverse-subalternating systems. An example would be Sommers's account of 'wild quantity' for singular terms, which is both subalternating and reverse-subalternating. In a reverse-subalternating system Diamond implies Box. (This is, if I recall correctly, the characteristic CD axiom.) So, Sommers argues, there is no significant difference between "Socrates is dead," interpreted as forbidding exception, and "Socrates is dead," interpreted as allowing exception, as long as we are talking about the same Socrates.

Likewise, you can have a quantifier-system allowing universal instantiation. To understand this, we have to note that there are many modal logical systems that allow you to have claims, terms, or whatever, without either Box or Diamond. These have what can be called a Null modality. In alethic systems, Box is necessity, Diamond is possibility, and Null is just plain truth. In universal instantiation we have what is sometimes known as the characteristic M axiom: Box implies Null. Thus, for instance, "It is necessary that all dogs go to heaven" implies "(It is true that) all dogs go to heaven". In a universal-instantiating quantifier-system you can conclude "Dogs go to heaven" from "All dogs go to heaven". The other major rules we typically find all have their modal axiom:

Universal Generalization: Null implies Box
Existential Generalization: Null implies Diamond
Existential Instantiation: Diamond implies Null

Obviously if these were all true without restriction all the distinctions among the modalities would simply collapse; but, in fact, there are different restrictions on when you can use them, thus preserving the difference between universal quantity and particular quantity.

But here's a bit of a puzzle. Modern logic is universal-instantiating but not subalternating. But in modal logic, standard systems with M are stronger than standard systems with D, so it would seem to follow that any universal-instantiating system is also a subalternating system, unless there's something very unusual about the system we are using. In fact, I think the denial of subalternation is often ambiguous when we are talking about modern logic (even setting aside the confusions created by talk about existential import): subalternating is taken as not merely assuming that Box implies Diamond but also that Diamond implies Null. There are restrictions on these, so if subalternation required both it would be subject to a number of different restrictions that would all have to be met. Likewise, when people talk about subalternation they tend to talk about the square of opposition, in which Universal Affirmative propositions imply Particular Affirmative Propositions; and in modern logic many universal affirmative statements are treated as hypotheticals, which is a further complication. But I don't think that these can be a complete account. It's still the case that the combination of universal instantiation and existential generalization should guarantee subalternation in any situations in which they are both allowed by the restriction -- in which case we should at most take modern logic to be a system with restricted subalternation, not as a system without subalternation. The only alternative is that the modal logic is unusual here. Or perhaps I'm missing something obvious here.

The traditional account, though, allows one to have a modal logic of quantifiers of any kind as long as it is D or stronger. Which one is chosen, of course, will affect what one can infer, of course.

A Poem Draft

Promethean

Two are the gifts the titan
gave us to break the night;
one with its heated glory
extends our mortal sight,
one through all unseeing
can carry us through the fight:
hope, which lives in darkness,
fire, which dwells in light.

Hierom the Thunderer

Today is the Feast of St. Eusebius Sophronius Hieronymus, better known as St. Jerome of Stridonium, Priest and Doctor of the Church. Jerome apparently never wanted to be a priest; he seems to have been ordained under heavy pressure from the bishop, and only have consented when promised he could continue doing what he wanted to do, which is living the ascetic life in the desert. After ordination, he studied under St. Gregory Nazianzen and then served as secretary to Pope St. Damasus; he got into extraordinary trouble in the latter gig, because he was unsparing in his criticism of everybody. This trait won many for the ascetic life, especially among the upper class women of Rome, but after the death of Damasus, he was accused of an improper relationship with a widow, and had to leave. He then studied in Alexandria under St. Didymus the Blind, finally arriving at his hermit's cell near Bethlehem, where for decades he wrote the works that have most made him famous. He also continued his practice of sharp criticism, which among other things led at one point to a band of Pelagians breaking into the monastic buildings in which he lived and setting fire to them.

Calling him 'The Thunderer' is not an old custom; it derives from a poem in the 1950s by Phyllis McGinley, which was turned into a very catchy blues song by Dion DiMucci, which made the designation famous.

Follow Me, Then, and Learn

Xenophon, the son of Gryllus, a citizen of Athens, was of the borough of Erchia; and he was a man of great modesty, and as handsome as can be imagined.

They say that Socrates met him in a narrow lane, and put his stick across it and prevented him from passing by, asking him where all kinds of necessary things were sold. And when he had answered him, he asked him again where men were made good and virtuous. And as he did not know, he said, "Follow me, then, and learn." And from this time forth, Xenophon became a follower of Socrates.

Diogenes Laertius, Life of Xenophon. Diogenes Laertius's epigram on Xenophon, which refers to his Anabasis, is also worth noting, since it is the single best summary of the Anabasis as a philosophical work (it is usually read as a mere history) that I have ever found. Somewhat paraphrastically:

Not only for Cyrus did Xenophon go up [aneben] to Persia,
But rather to find the way up [anodon] to Zeus's realm;
For, having shown that Hellenic deeds came from education [paideia],
He recalled how noble and good [kalon] was Socrates's wisdom.

The Game of Summit

This is what we'll be doing in my Ethics course tonight.

You have been selected as part of a national delegation to a major international summit whose purpose is to propose a universal declaration of human rights.

(1) Break up into groups (delegations) and introduce yourselves.

(2) Select someone to be secretary and someone to be the spokesperson to present your delegation’s proposal to the class.

(3) Each delegation must identify three (3) rights to propose as part of the declaration. It must be clear precisely what right you are proposing, and after you propose it to the class you will not be allowed to change it in any way. In other words: Formulate your proposed right very carefully and make sure the secretary writes it down correctly!

(4) Every right proposed must be supported by an argument or arguments agreed upon by your group. This will be your delegation’s official reason for why this right should be added to the universal declaration.

(5) The class will re-form as a general assembly. Each delegation will be allowed to present its proposed rights and give an argument for it. After each delegation has proposed all of its rights and given the delegation’s official arguments, the floor will be opened for a few minutes of debate on the rights, formulations, and arguments. The proposing delegation will be allowed to sum up its general rebuttal to any objections.

(6) There will be a general vote over each proposed right. Each delegation gets one (1) vote, to be determined by the majority of the members of that delegation. A tie within the delegation will be construed as a NO vote, i.e., a vote against the proposal. The delegation’s vote will be taken by the secretary and presented to the class by the spokesperson. Delegation members should base their vote on three things: whether they agree that the right is or should be a right, whether they agree that it is important enough to belong to an official declaration of rights, and whether they agree with the particular formulation proposed. Remember, bad formulations even of legitimate rights can come back to bite you later as people start using the letter of the law to protect themselves from the spirit of the law, or using the letter of the law to demand things that were not intended.

(7) Due to the delicate international situation, no right will be added to the declaration except on the unanimous agreement of all the delegations.

Unexpected but Predictable

The NonSequitur is a blog that investigates reasoning errors in the wild. This is admirable but tricky work -- apparent reasoning errors can turn on you suddenly if you don't watch out -- so I tend not to criticize even when I think their claims a little strained. But I cannot bear pointing out that this post is entirely wrong. The argument is that there is a contradiction, or at least a tension, to the following statement:

What Herman Cain did in Orlando this weekend was both completely unexpected and entirely predictable.

To this Scott Aikin replies that it can't be true. But of course it can: the situation in which something was predictable but unexpected is the situation in which it could have been predicted (and thus was predictable) but no one actually did predict it (and therefore it was unexpected). Since such things do happen, the original statement is entirely fine, logically speaking; any tension is the deliberate rhetorical tension that arises from foiling expectation. And it's clear from the rest of the paragraph in which this sentence is found that this was precisely the point: the idea is that nobody expected it, but given what it would take to win the straw poll, it was "in some sense inevitable".

I don't think this is a silly error, though, since you could very well read 'predictable' as saying that it was predicted (and thus expected). What it points to is that there is a whole class of ability-words and ability-phrases that are potentially ambiguous, being the sort that can be read either as indicating mere potential or as indicating activated potential. The best known and most widely used example is 'visible', which can mean either that something is see-able, or that something is seen (and thus see-able), but there are a number of others: sensible, intelligible, and so forth. Interesting, most of them are broadly cognitive in character, although you can occasionally find them in other contexts -- in the Aristotelian terminology of Aquinas's Third Way, for instance, the 'possible to be' and and 'possible not to be' is elsewhere shown to be precisely such a phrase, intended to be read in the activated-potential sense; it's a way of understanding the phrase we don't typically use anymore, which is why it takes some effort to read the Third Way in the sense it was intended to be read.