Analogies and Iconoclasm

Dale Tuggy has an interesting post in which he notes the claims of certain theologians that analogies should not be used except to show what the Trinity is not. And he concludes by suggesting that this would be opposed by a Catholic:

This is interesting; a hard-core catholic traditionalist could accuse both of departing from the tradition, which has long used various analogies, with the standard caveat that one should take care not to be mislead by any one of them, and taking care to multiply and diversify them.

I think this is right, but I think the point can be put much more strongly than Dale does; you don't have to be Catholic, and you don't have to be a particularly hard-core traditionalist to find yourself in strong opposition to this approach. And the reason is this: it's a form of iconoclasm. If the Trinity cannot be taught with verbal similitudes or figures it certainly cannot be taught with icons. Icons are pictures; they present a sensible representation. But a claim that does not deal with something directly sensible can only be pictorially represented if it can be thought of in terms of a sensible analogue -- and if it can be thought of in terms of a sensible analogue, it can be taught by verbal analogy based on that analogue. The analogy isn't what it represents; proper understanding of how it represents depends on recognizing this fact. But if you cannot represent the doctrine of the Trinity with verbal similitudes you cannot represent it with pictures, and if you cannot represent the doctrine of the Trinity with pictures, you cannot pictorially represent the Incarnation, and that is iconoclasm, and if you're Catholic or Orthodox you don't even have to be very traditionalist to have a problem with it.

And in both Catholic and Orthodox cases this will link up eventually with the iconicity of Scripture. Analogies do have a negative function: they can easily show that a particular claim that X and Y are inconsistent is false at least up to a particular level of description. But the Scriptural passages themselves from which we have the doctrine of the Trinity are rich with analogies: light, word, wind, flame, son. The doctrinal definitions that followed upong them are based on being able to recognize that they can teach us about the Trinity: unity in distinction, origination without subordination, that perichoresis by which (in Augustine's famous formula) "each is in each and each is in all and all are in each and all are in all and all are one." We don't have any sort of doctrine of the Trinity without them. If Scripture shows the way to teach the doctrine, as any iconodule would hold, at least some analogies must be good ways to teach at least some parts of the doctrine.

Friday Random Ten

1. Zager and Evans, In the Year 2525
2. Styz, Renegade
3. Noel Coward, Mad Dogs and Englishmen
4. Dion & the Belmonts, The Wanderer
5. Nightwish, Amaranth
6. Nek, Laura No Está
7. The Hooters, All You Zombies
8. Sufjan Stevens, Casimir Pulaski Day
9. Dion Dimucci, The Truth Will Set You Free
10. Deas Vail, White Lights

A rather atypical selection this time around.

No Clear Demarcation

My ethics class discussed usury for a bit last night, which brought this to mind.

Canto XLV
by Ezra Pound

With usura hath no man a house of good stone
each block cut smooth and well fitting
that design might cover their face,
with usura
hath no man a painted paradise on his church wall
harpes et luthes
or where virgin receiveth message
and halo projects from incision,
with usura
seeth no man Gonzaga his heirs and his concubines
no picture is made to endure nor to live with
but it is made to sell and to sell quickly
with usura, sin against nature,
is thy bread ever more of stale rags
is thy bread dry as paper,
with no mountain wheat, no strong flour
with usura the line grows thick
with usura is no clear demarcation
and no man can find site for his dwelling.
Stone cutter is kept from his stone
weaver is kept from his loom
WITH USURA
wool comes not to market
sheep bringeth no gain with usura
Usura is a murrain, usura
blunteth the needle in the maid's hand
and stoppeth the spinner's cunning. Pietro Lombardo
came not by usura
Duccio came not by usura
nor Pier della Francesca; Zuan Bellin' not by usura
nor was 'La Calunnia' painted.
Came not by usura Angelico; came not Ambrogio Praedis,
Came no church of cut stone signed: Adamo me fecit.
Not by usura St Trophime
Not by usura Saint Hilaire,
Usura rusteth the chisel
It rusteth the craft and the craftsman
It gnaweth the thread in the loom
None learneth to weave gold in her pattern;
Azure hath a canker by usura; cramoisi is unbroidered
Emerald findeth no Memling
Usura slayeth the child in the womb
It stayeth the young man's courting
It hath brought palsey to bed, lyeth
between the young bride and her bridegroom
CONTRA NATURAM
They have brought whores for Eleusis
Corpses are set to banquet
at behest of usura.

Sommers Notation, Part VI

(Part V)


Propositional vs. Predicate Modality

When we have a modality, like necessity, should we take it as modifying the predicate or as modifying the whole proposition? In language we sometimes seem to do one or the other. For instance, we might say,

It is possible that the new animal is a mammal.

And that seems to be propositional modality. However, we might also say,

The new animal is possibly a mammal.

This makes it look like a predicate modification. One could, perhaps, hold that they are equivalent; but this seems very clearly not to be true. A more abstract example might help. Compare:

Necessarily, no B is nonB.
No B is necessarily nonB.

'Necessarily' is clearly doing different work here. In the one, "No B is nonB" is taken as a necessary truth. The second, however, does not imply this. It doesn't rule out the possibility that some B is nonB; the only possibility ruled out is that some B is necessarily nonB. It could be true, for all that we can tell from this sentence alone, that some B is non-necessarily nonB. Very different sorts of modality.

The distinction noted here is often called the de re/de dicto distinction for modality; I'll instead continue to use the terms I've already introduced and call it the distinction between predicate and propositional modality.

Predicate and Propositional Modality in Sommers notation

How would we go about incorporating predicate modality into Sommers notation? It turns out that this is very easy. A proposition with predicate modality is simply a proposition with a modalized predicate. Thus, we can simply add modal functors to our predicates (for typographical reasons I will use M for possibility and L for necessity; we could also use diamond for possibility and box for necessity, as in the image above). Then we can simply use our ordinary Sommers notation rules, just adding a few basic modal laws (e.g., if +S+P, we can conclude +S+MP). Let's put up two axioms:

Axiom 1. If xL, then x.
Axiom 2. If x, then xM.

For instance, +S+P implies +s+MP; +S+LP implies +S+P. With these we can handle all the important logical relations that hold for predicate modality.

Propositional modality is a more complicated affair, and to see how we could handle it, we should perhaps take a moment to look at domains again. Every proposition is put forward relative to a domain of discourse; to contradict each other "There are dragons" and "There are no dragons" have to presuppose the same domain (if the domain of one is fictional beast mentioned in Piers Anthony novels and the domain of the other is real animals, for instance, they simply don't conflict). Any logical relation between two propositions, in fact, requires some connection of domains. We could state this by saying that every statement denotes its domain; and what a statement signifies is what it says of that domain. "Dragons are blue" says of the domain of the sentence that it includes blue dragons. When we say a statement is true, we mean that the domain it denotes includes the feature it signifies.

Putting it this way, however, simplifies matters considerably, because some statements, while having a domain, don't have a determinate domain -- or, if you prefer, their domain is a meta-domain, a domain of domains -- and are therefore harder to pin down. As it happens, all propositionally modalized propositions are of this sort. "Possibly at least one S is P" can't be interpreted as saying that at least one P-characterized S is in a particular domain; rather, it has to be interpreted as saying that there is at least one domain that includes at least one P-characterized S. Likewise, "Necessarily at least one S is P" doesn't mean that there is at least one P-characterized S in a particular domain; it means that in every domain there is at least one P-characterized S.

When we recognize this, it turns out to be fairly easy to see that the contradictory of L(Every S is P) is not, as one might think, L(Some S is not P), but instead, M(Some S is not P); and the contradictory of L(No S is P) is M(Some S is P); which is actually rather similar to what one finds in predicate modality (the universal + necessity is contradicted by the particular + possibility). If we take Sommers notation and add the following axioms, we can handle all propositionally modal propositions using necessity and possibility:

Axiom 3. If Lx, then x
Axiom 4. If x, then Mx

And, it turns out, we can combine our two types of modality if we recognize that the following two statements are true:

If Lx, then xL
If xM, then Mx.

This gives us a truly lovely square of opposition or, perhaps more accurately, square of implication, for modal statements, which I borrow from Englebretsen. You can find it above.

Each arrow indicates an implication or possible inference, and gives us an excellent sense of the strongest and weakest modal claims, and, for stronger claims, what weaker claims they entail. Claims that are both necessary and universal are at the top; claims that are both possible and particular are at the bottom; everything else is somewhere in-between.

Inferences with Modality in Sommers notation

With our square of opposition we have already said something about how one would handle simple inferences in a modal version of Sommers notation. But we need a way to handle mediate inference. Since predicate modalities are just modalized predicates, DDO, or the rule of mediate inference, is sufficient to handle them. But propositional modalities are more slippery here as elsewhere. DDO is not sufficient. We need to add another rule:

If Lx and Ly, then L(x and y)

All this only gives a quick glimpse of a vast and important extension of basic Sommers notation. In any case, any system that cannot distinguish predicate modality from propositional modality is hampered in its handling of real-life modal inferences; and one of the nice things about modalized Sommers notation is that it can handle this distinction very well.

(Part VII)

Plane Crash in Austin

We're having an exciting morning in Austin today. A Cessna crashed into the Echelon building on Research Boulevard. Research Boulevard is the frontage road for U.S. Highway 183; Echelon is near the interchange for 183 and Mopac (Loop 1), so it has apparently gummed up traffic. There is actually more than one Echelon building in that complex; I'm not sure which number it is, although it is one visible from the road. [Added Later: Apparently it's Echelon 1, although some sources keep saying it's Echelon 3.]

I'm up in Cedar Park this morning, north of Austin, but Echelon is just a few minutes south of where I live, and it must have happened as I was driving up. There are a lot of government offices in the area, including FBI, IRS, and (they say) CIA, but (as far as I know) none in that building itself, which consists only of commercial office suites -- Echelon is mostly full of things like yoga classes, software company branch offices, and travel agents. The buildings themselves are nondescript; I suspect most people who see them don't notice them (they are just south of the Arboretum, right near the much more visible back of Grace Covenant Presbyterian Church, for those who ever drive 183; Dave & Buster's is across 183 and a little south).

[UPDATE I: There have been rumors all morning that this was deliberate, and that the target were the IRS offices in the area. So far there hasn't been much in the way of evidence.]

[UPDATE II: St. Edward's University apparently has an adult education center in the building; all the students and faculty were evacuated, with none harmed.]

[UPDATE III: Things are beginning to get more specific. There were IRS offices in the building that was hit. The pilot may have set his house on fire and stolen a plane.]

[UPDATE IV: So far no casualties, only a few hospitalizations. They've apparently found Joseph Andrew Stack's suicide note.]

A Confraternity of Dust and Ashes

I know, I know the nations past
I know, I know they rust at last
They tremble with the nervous thought
Of having been, at last, forgot

Sufjan Stevens,
They Are Night Zombies!! They Are Neighbors!! They Have Come Back from the Dead!! Ahhhhh!

B-Side Dominance

It's easy to find things that were intended to be clever and great that just end up being totally stupid. But while they are rare, here and there you can also find something that was intended to be very stupid that, contrary to expectation, turned out splendid. I was put in mind of this because something or other as I was walking back from the grocery store put me into mind of one of pop music's most famous cases -- a song quite literally written never to be listened to.

In the days of vinyl, singles were generally sold in pairs -- there was an A-side single and B-side single. Because the record actually had to be physically flipped over to hear the song, one side would always be listened to more. Thus one tried to pair songs in such a way that took this into account. What is more, if you were really trying to put focus on a song, you wanted its B-side counterpart to be a throwaway song that didn't require much and that you wouldn't mind if people never listened to. So when Leka, and DeCarlo were told that the record company, Mercury Records, wanted to use all their suggested singles as A-sides, they had to scramble to find B-sides. One of the ones they chose was an old one they had done in a previous band, The Chateaus, with another person, Freshuer; but it was too short. So when Leka, DeCarlo, and Freshuer re-recorded it (they basically just threw it together, using a drum track instead of drummer), they put in the easiest and stupidest chorus they could think of, "na, na, na, na, na, na, na, na," which they used when they were composing lyrics and couldn't think of anything at the moment.

None of the three wanted to be associated with such a stupid song, so they made up a band, "Steam," on whom it could be blamed. Mercury Records was so horrified by the song that they decided not to release it under their Mercury label -- they passed it down to Fontana, a subsidiary, and it got put on a record as B-side trash, a song made never to be listened to.

One day a radio DJ in Georgia turned the promo record over, however. People who heard it began to call in to request it; it became one of that radio station's most commonly requested songs. Other radio stations began to play it, and soon it was heard all over the South. Wind got back to Mercury Records; they were startled, but being record company executives in 1969 they were not stupid. They did what record companies do with songs they think are going to make them money: they bought up enough copies to guarantee them a place on the Billboard magazine lists. Radio stations that watched Billboard listings began to play it. And it became one of the best-selling songs of 1969. It started being used as change-over music in baseball; in the late 70s, the fans started singing along, and they have been singing along ever since. From baseball it spread to other sports.

It was far from high art. So far, in fact, that nobody bothers to judge it by the standards of high art -- they just judge it by how catchy and fun it is.

There are bound to be examples of similar things elsewhere -- things designed to be just plain stupid, not even for comic effect, that turned out to be just plain fun instead. But they're rare enough I can't think of any; can you?

Cannot Live on Tomorrow's Bread

Democracy
by Langston Hughes

Democracy will not come
Today, this year
Nor ever
Through compromise and fear.

I have as much right
As the other fellow has
To stand
On my two feet
And own the land.

I tire so of hearing people say,
Let things take their course.
Tomorrow is another day.
I do not need my freedom when I'm dead.
I cannot live on tomorrow's bread.

Freedom
Is a strong seed
Planted
In a great need.

I live here, too.
I want freedom
Just as you.

The Death of Descartes

A truly bizarre line of thought about Descartes's death. Descartes died very suddenly, so rumors about poisoning began almost at once. But the reason historians say he died of pneumonia is that all of the definite evidence is that he did, in fact, die of pneumonia, probably with the contributing factor of being severely weakened by aggressive bloodletting as part of his treatment: he had a fever and congestion of lungs, and the circumstances under which he became ill are consistent with pneumonia. It was also diagnosed as pneumonia a the time.

Of course, if one wants to say that Descartes was poisoned, Baillet (who is not always reliable) does tell us that the doctor who was put in charge wanted to see Descartes dead -- a far more plausible candidate to accuse of being a poisoner than anyone else in the whole scenario, and indeed, the rumors of poisoning at the time were rumors that he had been poisoned by Weulles (Johan van Wullen). By the time Viogué was called in, Descartes was already far gone -- Viogué was called in to do last rites when it was already clear that Descartes was unlikely to survive, and there are far more plausible enemies who had far more plausible opportunities to do damage to Descartes. The proposed motive for Viogué doesn't even make sense -- Descartes himself believed in transubstantiation, he was widely believed to be trying to convert Queen Christina to Catholicism, and even if Viogué thought Cartesian view too like Calvinism, it wouldn't have mattered because Christina was Lutheran, not Calvinist.

But it's possible, too, that there's more to the argument than just the one point, briefly mentioned, about the urine; newspapers are not good sources for seeing how complicated arguments are put together, and any decent argument for the claim that Descartes was poisoned would have to be a complicated one, since all the straightforward evidence suggests pneumonia.

Lightning from the Tops of heaven

From "A Curse for a Nation"
by Elizabeth Barrett Browning

I heard an angel speak last night,
And he said 'Write!
Write a Nation's curse for me,
And send it over the Western Sea.'

I faltered, taking up the word:
'Not so, my lord!
If curses must be, choose another
To send thy curse against my brother.

'For I am bound by gratitude,
By love and blood,
To brothers of mine across the sea,
Who stretch out kindly hands to me.'

'Therefore,' the voice said, 'shalt thou write
My curse to-night.
From the summits of love a curse is driven,
As lightning is from the tops of heaven.'

You can read the rest of it, a sharp denunciation of American slavery, here.

Logical Obverse

There was a flap recently about Cherie Booth, the wife of Tony Blair, who had given a lenient sentence to a man, saying, "I am going to suspend this sentence for the period of two years based on the fact you are a religious person and have not been in trouble before. You caused a mild fracture to the jaw of a member of the public standing in a queue at Lloyds Bank. You are a religious man and you know this is not acceptable behaviour." I hadn't paid much attention to it, but wandering about I happened to come upon an essay and discovered that A.C. Grayling needs a little assistance with the application of logic.

Grayling argues in response to Booth's claim:

As a barrister Mrs. Blair should be able to see the inadmissible corollary of passing lenient sentences on believers because they are believers; namely, that non-believers should receive less lenient sentences. If she had said – and said twice – in passing judgment on a person she knew to be non-religions, ‘I am going to apply the full penalty of the law based on the fact that you are not a religious person,’ she would not have merited any less of an outcry than she has caused, for the very good reason that this is the logical obverse of what she in fact said, and would be as unacceptable.

Unfortunately, this is not the logical obverse of what she in fact said. What she in fact said (and Grayling had just quoted her as saying it a few sentences before) was that she was giving a lenient sentence because Shamso Miah was a religious man and had not been in trouble before. From this it does not follow that not being a religious person is a reason for not being lenient: there is a conjunctive term here, and one conjunct of it has wandered out of Grayling's analysis. Moreover, Grayling clearly gets the whole thing backwards. We can see this by simplifying a bit and recognizing that to get the 'logical obverse' Grayling wants, we have to interpret Booth as assuming something like

All people who should get lenient sentences are religious people.

The logical obverse of this is that no people who should get lenient sentences are non-religious people; which is equivalent by conversion to the claim that no non-religious people should get lenient sentences, which Grayling seems to have in mind. But even if we ignored the conjunction, and even if we take Booth to be making a strong assumption, the actual assumption attributed to Booth to make her comment make any sense would have to be something more like:

All religious people should get lenient sentences.

And the logical obverse of this is that no religious people are not people who should get lenient sentences; if we also contrapose, then we can get the claim that all people who should not get lenient sentences are non-religious people, but it does not follow from this that if you are non-religious you don't get a lenient sentence; the assumption is consistent with some people who are non-religious getting a lenient sentence. You can only get to something like Grayling's 'logical obverse' by an illicit conversion.

Such an assumption is too strong in any case; it is consistent with the claim that being religious is sometimes a reason for leniency, and that only some religious people should get leniency because of being religious -- due to the way in which they are religious. And, indeed, the conjunctive term requires a weaker assumption. The strongest assumption Booth can be possibly committed to on the basis of her words is the following:

All religious people who have not been in trouble before should get lenient sentences.

This just tells us that some religious people should get lenient sentences. (The logical obverse of this, in case you are wondering is, with the conjunctive term, that no religious people who have not been in trouble before are not people who should get lenient sentences; and in the I-form without the conjunctive term, it just tells us that some religious people are not people who should not get lenient sentences.) There is no way to get from this to anything like Grayling's 'logical obverse'.

But one could also plausibly suggest that the claim is consistent with an even weaker assumption, because she doesn't have to be making a universal assumption; her assumption can be particular:

Some religious people who have not been in trouble before should get lenient sentences.

All that is required to apply such an assumption in this case is to hold that there is reason to think that this is in fact one of those cases. And this indeed is much closer to the claim that Booth seems to be making, namely, that some people who should get lenient sentences should get them because they are religious people who have not been in trouble before.

From all this we can see that pretty much nothing follows about non-religious people from Booth's claim. From the fact that religion, even all religion, is a reason for leniency we cannot conclude that lacking a religion is a reason for non-leniency; it may not be a reason one way or another. This is true a fortiori if we only admit that some things that are called religion -- for instance, things relevant as a testimony to character, which I take it is why Booth brought it up in the first place -- are reasons for leniency. For instance, the fact that someone is a philosophy professor is a reason for concluding that they can handle the logic of an argument; but it does not follow from this that we are therefore saying that the fact that someone is not a philosophy professor is a reason for concluding that they can't handle the logic of an argument. There is no way to get the one from the other.

I suspect that Grayling has confused himself with the word 'because'. In this context it works as reduplicative that qualifies how the term in the predicate applies to the subject. Saying that X should get Y because X is Z, this is the same as to say that X should get Y on account of being Z. If I say that religious people are people who should get lenient sentences on account of being religious, this has as its corollary that non-religious are not people who should get lenient sentences on account of being religious. This is not only obviously true, it is not the same as the corollary claimed by Grayling, that non-religious people should not get lenient sentences on account of being non-religious. That would involve an illicit shift in the reduplicative: since the reduplicative limits what about the term is being applied to the subject, changing the reduplicative changes the whole predicate and introduces an equivocation.

There are any number of other problems with Grayling's argument that follow from his failures here. For instance, he says,

So she must be assuming – leaving all history and the contemporary world aside as providing too many troubling counter-examples – that there is a tendency for religious people to be of good character because they are religious (and not, say, that they have a tendency to be of good character because they are people).

But this can only be the conclusion if you leave out the fact that the reason given by Booth was conjunctive: it was the conjunction of the fact that he was a religious person and that he hadn't been in trouble before. So all Booth must be assuming is that people who are religious and have not been in trouble before tend to be of the sort of good character that makes it reasonable to be lenient in the context of a court. It does not require the assumption that all religious people tend to be of good character; it is consistent, for instance, with religion being thrown out as a legitimate reason for thinking a person of good character if they have been in trouble before. Moreover, it does not even require (contrary to the course Grayling's argument later takes) that the 'good character' involved be fully moral good character rather than good character in the limited sense that they are social enough that they can be expected to behave themselves better in the future. But it's clear that when courts in considering leniency look at whether a person has good character they are really only doing the latter: they are simply looking for indications that a person is not a complete social misfit, but someone who participates in the community, who has shown that they are capable of participating in the community without misbehaving, and who has been shown to have social incentive to be more careful in the future.

It is very extraordinary how much of a tempest in a teapot this whole thing is: so much sleight of hand to make the whole thing seem more severe than it is. One can reasonably argue that religion should not be a reason considered when determining whether to be lenient or not. However, one cannot reasonably argue that considering religion as one reason for leniency commits anyone to the claim that people who are non-religious should not be treated leniently on account of the fact that they are non-religious.

Knots and Problems

Now in loosening a physical knot it is evident that one who is unacquainted with this knot cannot loosen it. But a difficulty about some subject is related to the mind as a physical knot is to the body, and manifests the same effect. For insofar as the mind is puzzled about some subject, it experiences something similar to those who are tightly bound. For just as one whose feet are tied cannot move forward on an earthly road, in a similar way one who is puzzled, and whose mind is bound, as it were, cannot move forward on the road of speculative knowledge. Therefore, just as one who wishes to loosen a physical knot must first of all inspect the knot and the way in which it is tied, in a similar way one who wants to solve a problem must first survey all the difficulties and the reasons for them.

[Thomas Aquinas, Commentary on Aristotle's Metaphysics, Rowan, tr. Dumb Ox (Notre Dame, IN: 1995) p. 128.]

Recognizing this requires recognizing that the 'history' and 'problems' sides of philosophy, often treated as distinct approaches, need to be integrally related. I've previously called this the problem of the philosophical problem (and here); it's still one of the key issues in the study of the history of philosophy. Philosophical problems are knots, knots must be tied, and the tying of the knot is the history of it; without seeing how the knot was tied one simply does not know what kind of knot it really is.

Sommers Notation, Part V

(Part IV)

Basics of Propositional Logic

Here I will diverge from both Sommers and Englebretsen a bit. In their scheme, we can represent conditionals (if p then q) as:

-[p]+[q]

Conjunctions are represented as:

+[p]+[q]

And disjunctions as:

--[p]--[q]

This works, and it ties in with the history of logic; for instance, the similarities between conditionals and A propositions have been recognized for centuries now.

However, I would suggest that it is a mistake to apply quantity directly to nominalized propositions; the minuses and pluses associated with them work much more like quality than quantity. We should perhaps see nominalized propositions as predicates that are predicated of the domain. Thus, “p and q” is predicated of some domain, e.g., T for the domain of truths:

±T+(+[p]+[q])

Then disjunction is related to conjunction as negative quality to affirmative quality.

±T-(-[p]-[q])

That is, “p or q” tells us that the one case that can’t occur is the one in which –p and –q are both true. And the material conditional, “if p then q,” is (as it is in other forms of logic) a kind of disjunction:

±T-(+[p]-[q])

All the standard rules of propositional logic, then, follow from rules for conjunctive and disjunctive predicates. For instance, conjunction simplification follows directly from CPSimp.

The subject term here does much the same work as an assertion sign here. We also see how judgment and quality are related, since they turn out to be the same kind of thing, just at different levels.
Likewise, we can handle metapropositions easily:

± T-[p] = [p] is false (not true) = It is not the case that [p]

And so forth.

Using propositional nominalization we can easily prove that modus tollens is valid:

1. ±T-(+[p]-[q]) premise
2. ±T-[q] premise
3. ±T-[p] MI from 1,2

And likewise with modus ponens:

1. ±T-(+[p]-[q]) premise
2. ±T+[p] premise
3. ±T+[q] MI from 1,2

And so with disjunctive syllogism:

1. ±T-(-[p]-[q]) premise
2. ±T-[p] premise
3. ±T+[q] MI from 1,2

We could continue indefinitely, but we won’t here. It would, however, be useful to have a rule linking conjunctive and disjunctive predicates to conjunctions and disjunctions of propositions. We can call it a Rule of Predicate Distribution; for conjunctive predicates it is (CPD):

-X+(+Y+Z) ∴±T+(+[-X+Y]+[-X+Z]

And for disjunctive predicates (DPD):

+X-(+Y+Z) ∴ ±T-(-[+X+Y]-[+X+Z]

We could derive these rules from what we already know about Sommers notation, plus some basic knowledge of what a domain is; but the point should be clear enough without the derivation.

Because the domain is always the subject, and because in general the domain has to be the same through an entire argument (otherwise one equivocates), we can usually just leave the domain implicit and focus on the predicates alone. It is interesting, however, to consider that by reasoning in terms of different domains one could probably also handle a number of paraconsistent and modal reasonings. We will not look into the question of paraconsistent reasoning, which is almost totally unexplored, but instead look at modal reasoning, which has been explored, but only partially.

(Part VI)

Hinton's Impossible Syllogism

In his book The Fourth Dimension, C. H. Hinton argued for the existence of an undiscovered valid syllogism, IEO-4: that is, a syllogism with all of the following characteristics:

(1) The major premise was an I proposition.
(2) The minor premise was an E proposition.
(3) The conclusion was an O proposition.
(4) The syllogism was in the fourth figure.

Hinton argued for this on the basis of the fact that you could arrange the valid syllogisms in such a way that they exhibited a perfect symmetry except for one missing space, that represented by IEO-4. I will not go into his reasoning, which is rather complex, but he is right: if there were an IEO-4, this would increase the number of symmetries in the valid syllogisms. Hinton even gave an example of the form:

Some Americans (P) are of African stock (M).
No one of African stock (M) is Aryan (S).
Therefore Aryans (S) do not include all Americans (P).

Actually, Hinton gives this premise as "No Aryan is of African stock," but this would make the premises of the argument suitable for the second figure, not fourth. However, since E propositions convert, Hinton's premise is equivalent to the above premise, which gives the right premises for a fourth figure syllogism. So it seems we can take "Some P is M," add it to "No M is S," and get a valid conclusion in mood O.

This would make a beautiful test question for a logic class, because all of the following are true (and it is in fact fairly easy to show it):

(1) Hinton's example argument is a valid argument.
(2) Hinton is right that it adds an I premise to an E premise to get an O conclusion.
(3) Hinton's proposed valid mood is logically impossible.

So here's a logic challenge for you to puzzle over: show how Hinton went wrong by showing that (1), (2), and (3) are all correct.

Notes and Links

* C. S. Peirce and micrometry.

* An essay on how academia is failing graduate students; and, unfortunately, it's largely right, although some fields are worse than others. It's difficult to see that there's any way out, too. Alasdair MacIntyre talks about how institutions that are created to be instruments in the pursuit of the internal goods of practices -- like schools are instruments in pursuing the goods of education -- have a tendency to begin to take on a life of their own because they respond not only to goods internal to practices but goods external to them, as well. And it's very much the case that most of the basic institutions of the academic world have been running on auto-pilot for quite some time: practically any academic can give you a laundry list of institutional behaviors that impede the pursuit of the goods of study and research. But nobody knows what to do about it: the mine train is slowly beginning to accelerate and the brakes aren't working.

The solution is that intellectual life requires new institutions; the problem is that it is easier to recognize that new institutions are necessary than to determine what they should -- or even could -- be.

* All past episodes of Philosophers' Zone, the world's foremost philosophy-themed radio show, are available. (ht: smally)

* Jan van Eijck, Syllogistics = Montonicity + Symmetry + Existential Import (PDF)
Klaus Glashoff, Aristotelian Syntax from a Computational-Combinatorial Point of View (PDF)

* Apparently there is a new translation of The Second Sex -- and about time, since it has been known for decades that, despite its readability, there are extraordinary flaws in the Parshley translation. But apparently it's not very good.

* I am mystified by the fact that some people are surprised that people get in an uproar over the prices of ebooks; whatever reasons publishers might have, there is simply no reason obvious to consumers why a digital file should cost the same as an actual book whose price can be seen obviously to include the cost of real paper and real ink. We've been through this with music already: if the cost of the electronic version gets even too close to that of a physical version, consumers will begin to protest either with anger or with ingenious attempts to circumvent the price. In order to sell electronic versions, you have either to make them clearly cheaper than the physical version, or to provide some clear value beyond the basic product itself that consumers can see as justifying the expense. This is not based on a sense of entitlement: it's based on the anger customers feel when they begin to suspect that someone is trying to gouge or scam them. And there are plenty of people who will get the feeling that you are trying to do both when it costs fifteen dollars to click their mouse and download a file they are sure you can provide much more cheaply.

I actually wonder if a tip-jar and discount system might work better than what publishers are currently trying to do: that is, provide the electronic version fairly cheaply, but allow people to tip on books they appreciate (with the tips contractually divided between publisher and author like any other income on the book), and provide discounts on the paper version for those who have already bought electronic versions and liked them. But here again we are really dealing with newer territory than anyone likes to admit: we don't know what will work best.