Notes Toward a Formal Typology of Argument V

In the first post I laid out in a rough way the notation for the typology.
In the second post I introduced the notion of attenuation and used it to establish hierarchies of arguments.
In the third post I introduced the notion of preclusion and used it to show how distinct hierarchies of arguments are interrelated.
In the fourth post I introduced the notion of candidacy in order to show one way the typology could be extended (and corrected an inaccuracy in earlier posts).

Here I'd just like to give an example. But first it may be useful to note a few points that help when applying the typology to real-world examples.

The first is that the only difference in bases that is important is a difference that affects the structure of the argument. Thus, one can have as one's bases a thousand separate pieces of evidence, and if they all together only form one reason for a conclusion, they still can be treated as a simple, one-base type of argument. It's only when one base is put forward as a reason for thinking that another base is a reason for drawing a conclusion that we get into multiple-base hierarchies.

The second is that in a one-base case, multiplying R's doesn't really affect anything. a : R_a(R_a(XT)T) is just a more verbose a : R_a(XT).

In Hume's Enquiry Concerning Human Understanding there is a famous passage in which Hume presents a counterexample to his own position, the missing shade of blue:

There is, however, one contradictory phenomenon, which may prove that it is not absolutely impossible for ideas to arise, independent of their correspondent impressions. I believe it will readily be allowed, that the several distinct ideas of colour, which enter by the eye, or those of sound, which are conveyed by the ear, are really different from each other; though, at the same time, resembling. Now if this be true of different colours, it must be no less so of the different shades of the same colour; and each shade produces a distinct idea, independent of the rest. For if this should be denied, it is possible, by the continual gradation of shades, to run a colour insensibly into what is most remote from it; and if you will not allow any of the means to be different, you cannot, without absurdity, deny the extremes to be the same. Suppose, therefore, a person to have enjoyed his sight for thirty years, and to have become perfectly acquainted with colours of all kinds except one particular shade of blue, for instance, which it never has been his fortune to meet with. Let all the different shades of that colour, except that single one, be placed before him, descending gradually from the deepest to the lightest; it is plain that he will perceive a blank, where that shade is wanting, and will be sensible that there is a greater distance in that place between the contiguous colour than in any other. Now I ask, whether it be possible for him, from his own imagination, to supply this deficiency, and raise up to himself the idea of that particular shade, though it had never been conveyed to him by his senses? I believe there are few but will be of opinion that he can: and this may serve as a proof that the simple ideas are not always, in every instance, derived from the correspondent impressions; though this instance is so singular, that it is scarcely worth our observing, and does not merit that for it alone we should alter our general maxim.

So let's take the background argument, which is a set of reasons for the claim that every simple idea has a corresponding impression whence it derives. This is an argument of type a : R_a(IT), where I is the claim about the relation between ideas and impressions. He then presents a contradictory phenomenon, the fact that we can conceive of the scenario of the missing shade of blue (mb), which "may prove that it is not absolutely impossible for ideas to arise, independent of their correspondent impressions," in other words, that it is possible for I to be false. There is some ambiguity here about whether the argument is supposed to be of type mb : R_mb(IMF) or the slightly weaker mb : R_mbM(IF). The phrase "not absolutely impossible" suggests to me that the intent is the weaker one; the idea appears to be that it is not in the strictest sense impossible for simple ideas to arise without simple impressions, not that it is possible given actual conditions. The type would then be mb : R_mbM(IF). But this is controversial; one might choose to emphasize the terms in which Hume ends his description of the scenario: "this may serve as a proof that the simple ideas are not always, in every instance, derived from the correspondent impression". This might well be taken to suggest that the argument is of type mb : R_mb(IMF). However, if we might also wish to emphasize the 'may' in "this may serve", in which case we have a different sort of argument altogether, namely, mb : MR_mb(IF). In this interpretation Hume is only admitting to the scenario's possibly being a reason for thinking that I is false. Thus we have three superficially similar but in fact rather different interpretations of the missing shade of blue:

mb : R_mbM(IF)
mb : R_mb(IMF)
mb : MR_mb(IF)

This sort of ambiguity is, in fact, very common in arguments made in modal terms that are not made very, very carefully. If the argument is of the first type, it is an argument that the missing shade of blue suggests that it is possible for I to be false (under some circumstances or other, which may or may not actually occur). If of the second, it says that I can be false under some conditions that might actually occur. And if of the third, it says that the missing shade of blue might be a reason for thinking that I is false. Which one we choose affects how we understand Hume's counter-reasoning, but it will be convenient to use the term B_mb to mean the exclusive disjunction: either the first, or the second, or the third.

After giving the scenario, and saying that it may be taken as proof that I is false, Hume then goes on to say the words that have so puzzled commentators, namely, "this instance is so singular, that it is scarcely worth our observing, and does not merit that for it alone we should alter our general maxim." This reasoning is, at least at first glance, of type s, mb : R_s(~B_mb(IF)T). That is, it's an argument that s is a reason for thinking the missing shade of blue is not a reason for thinking I false. Now, how does our understanding of B_mb affect our understanding of this line of reasoning?

If B_mb should be understood as mb : R_mbM(IF), then the idea would appear to be this: the missing shade of blue really is a legitimate reason for thinking that it is possible for I to be false. But the singularity of the case could mean, among other things, that it is unlikely, and perhaps not even possible, for I to be false under conditions that would actually obtain in the real world. Thus I can be taken to be true as a "general maxim" on the original argument Hume made.

If B_mb should be understood as mb : R_mb(IMF), then the point of mentioning the singularity of the case is not to say that it's so unusual that it might not ever occur, but that it's so unusual that even if it does occur it's not a good reason for taking I as Hume actually understands it to be false. For instance, the idea might be that Hume is only interested in what generally occurs "in the wild" (in a restricted application of the principle), not under highly artificial circumstances like that which the missing shade of blue would have to be.

If B_mb should be understood as mb : MR_mb(IF), then the singularity of the case would indicate that the scenario is not even a good candidate for an argument that I is false. This would seem to be contradicted by what Hume says immediately before this ("this may serve as a proof"), although, of course, that carries a typical Humean ambiguity: it may serve as a proof, but does Hume think it really does serve as one, or is he just observing that other reasonable persons might think it does? But it does seem less likely, so we can perhaps set aside the mb : MR_mb(IF) as an unlikely possibility.

Now, Hume is very often criticized for his response, but it is clear that many commentators take the argument to be of type s, mb : R_s(~R_mb(IMF)T) -- that is, they take it as a straightforward denial that the missing shade of blue is a counterexample to I, where it is taken as actually being able to occur. But it's important to note that we can take the conclusion of the argument to be not (IMF) but M(IF), in which case almost all the criticisms of Hume's treatment of the missing shade of blue miss the point, for they take it as saying that it can really occur, whereas Hume might not even be committing himself to it's being able to occur at all -- he might just be saying that there is some conceivable scenario (and for him conceivability is possibility) in which it might, without committing himself to the claim that this scenario is at all consistent with other truths about the world. That is, all he would be conceding is that I is not a necessary truth; and he is right that singularity can show this to be irrelevant to actually taking I to be true, i.e., can be an undercutting defeater for I.

This is all very crude, but I think you can get the idea. As I said in the first post, this whole thing is just an idea that came to me one Monday in late July and was worked out over a few days of sporadic thought. It's likely to need fine-tuning in a great many spots. Comments are very welcome.

Logical Epochs

I find it interesting that philosophers often have a tendency to speak of logic as going through two epochs: the traditional, or Aristotelian, in which syllogistic reigns, and the modern, governed by the predicate calculus. Of course, there are many more epochs than this, but it's easy for people to do it. Thus when we get into other epochs in logic, like the splendid era started by Boole and De Morgan, we find that there's often an attempt to assimilate it to one or the other. Englebretsen, working in a modern extension of the traditional term logic, regards Carroll as the last great traditional logician; it's not uncommon for people used to the predicate calculus to treat Boole et al. as a sort of prolegomena to Frege.

Of course, the great algebraic logicians -- Boole, De Morgan, Jevons, John Neville Keynes, Lewis Carroll, John Venn -- are not so easily shunted to one side or another. They certainly saw themselves as making a sharp break with traditional logic -- which they would have understood in terms of the old staple of Aldrich's Artis logicae compendium, published in 1691 but still the standard reference for logic up until Whately published his Elements of Logic in 1826 (and in many places in England for yet longer). (The reader of George Eliot might recall poor Maggie trying to read Tom's school edition of Aldrich, and not being able to divine how the work related to the living world.) No one can read Venn's Symbolic Logic without recognizing that this is in his mind a radically different approach than anything traditional logic has to offer. And there is something to it. He regards the traditional emphasis on particular propositions (I and O) to be largely misguided -- one can put the traditional syllogistic, with I and O so prominent, in Boolean terms, but that is merely one permutation, and not an essential feature at all; logicians only work with universal propositions, and only deign to bother with particular propositions insofar as these can be considered incomplete universal propositions. One also finds that he reviewed Frege's work; it was a scathingly contemptuous review. He regarded Frege's notation as clumsy and generally not well-suited to application (which everyone agrees with) and his logical work to be far inferior to the algebraic logic being done in the wake of Boole (which is rather more controversial).

It's also true, I think, that there tends to be a complete difference in logical approach. It's dangerous to lump all the pre-Boolean work into the category 'traditional logic'; that ignores the fact that, for instance, the Ramists or the Cartesians tried to incite revolutions against what they perceived as the Aristotelian oppressor. And the sort of traditional logic the algebraic logicians would have known would have been a mix of simplified school manuals and Whately's adulterated revival version. But we can perhaps say to some extent that whereas logicians in the Aristotelian mode are intently focused on demonstration, the Booleans were not. They are very helpful in telling us what they are obsessed with, and it is not demonstration at all. Jevons puts it succinctly in his Philosophical Transactions: "Boole first put forth the problem of Logical Science in its complete generality: Given certain logical premisses or conditions, to determine the description of any class of objects under those conditions." As Keynes puts it in a passage in Studies and Exercises in Logic, where he quotes Jevons on this point:

Given any number of universal propositions involving any number of terms, to determine what is all the information they jointly afford with regard to any given term or combination of terms.

This is closely related to another obsession of algebraic logicians, the finding and eliminating of superfluous premises. It is not something we generally do much any more, at least as anything more than an occasional technical exercise; but for them the whole point of logical analysis is to determine precisely the most the premises give you and to determine precisely how you can get your conclusion without going over the same ground twice. But the predicate calculus is not used this way; I imagine that if there's an obsession it has brought about, it is identified by Quine in that passage in which he says that the whole point of logical grammar is to facilitate the tracing of truth conditions. One sees this difference in a number of ways they approach things. The algebraic logicians are generally dismissive of any attempt to reason about objects without specifying a domain of discourse; any symbolic system, for instance, that does not clearly specify a universe of discourse is regarded, by that very fact, as deficient. This does, in fact, tie in with their interest in the information afforded by terms in universal propositions. It is often difficult to get people trained in predicate calculus even to take universes of discourse seriously.

Logic, then, at least to a limited extent is not merely a set of formal considerations; it involves an approach, and what you are trying to do plays a key role in how the whole operation works. What counts as success, what counts as failure, what counts as even important, shifts depending on your view of the ends of logical thinking.

Another Poem Draft

Starmaker

The Starmaker on an everlasting throne
whirls the sky, orders stars with law;
the moon, now full, sunfire-dressed,
in borrowed light makes stars dim in awe;
now she pales, her light dispersed,
the evening star shines coldly;
eve turns to morrow, the star bows out,
the sun leaps up, shines boldly.

Juvenilia

Going through some old papers I came across a good number from my high school and undergraduate days -- jottings about philosophy and theology, and a truly enormous number of extraordinarily bad poems. For most of them the decent thing to do is just to shred them. But some of them are OK enough that I thought it would be interesting to put them up. At least, I find it interesting to see my strange younger self on some of his better days, managing to be insightful and to miss the point in the same sentence, or writing some lines of poetry that are in some ways decent enough even if clumsy and flawed. It's also a bit uncomfortable to see that my poetic sensibilities have not greatly improved since then; although I have certainly become more consistent. ("Visuddha," for instance, is so exactly the sort of poem I tend to write even now, that it's startling to come across it; I would make different poetic choices now, but it's still a very Brandon sort of poem.) A sign of the limits of whatever talent I have for poetry, I suppose. I have also become less venomous in my bite, or else am just less likely to bite. In any case, here are a few of the poems; "Blue" seems to have been one that I particularly liked at the time, because instead of being scrawled it is carefully written in my best hand.

Protest

This thing called love is some strange chemical,
That, by electric impulse in the brain
Is given by some gland through running blood
To every part of me, that is, my flesh;
And it is nothing more, tho' songs protest.
And this strange process, if even strange it be,
Is but the brain's reaction to a scent,
That, subtle on the nose, does sit on us,
Or else to some eye's vision which we today
Are taught by conditioned process to conceive
To be attractive, and so we change.
And true, who this will feel, shall feel with speed,
And this, the hormone's kiss, is like a drug,
But this is merely accidental work.
And if you ask me why we have this change,
I say it is a consequence of time
Wherein is added randomness and codes
Of certain elements whereby we're made.
This is all love is, and nothing more.
This code has settled down in such a way
Because it makes the species to survive,
Thus sex is all for children, and all love
Is but a code for sex, and this is how
The species does go on, and whence derives.
And so you you see, although I love, I know
That this is all a chemistry inside
That grew from events most physical
According to a process wholly known,
And love is nothing more than this,
And I know it all; this is all it is.

The Juggernaut's Child

All space is an ocean with currents of iron
That whirl in their pathways and roll through the void,
Bearing in procession the mad adorations
of evil, of devils, of fools, of the tower.
Lightning from heaven the balance deposing,
Black syncopations like coils of snakes
Twist through the systems and through human lives,
Major misfortune with minor collides.

The stars in their courses are battling madly,
Insane in their flowings and dark in their light.
Where is the widow who once was a lady?
Here in the darkness, in the depths of the night.

And wild stones rolling, ever faster and faster,
Slide down the slope at a frightening speed.
We are the stones in avalanche falling,
Rushing headlong, sans hope, without heed.

Oceans are heavy, they crush all the light,
But they are as nothing to fate on its throne,
Who is a mad fool on the day of dissension,
Broken and breaking, almost random in being,
Giving out rings to those undeserving,
And here, in the shadow of death and decay,
Children are born and this is their life.
We are those children, as are our children.
Our world is the birthing of the Juggernaut's child.

Reflection on Paddington Station

In life, the course of which we never quite foresee,
There yet are mighty junctions where we choose our next divide
And shift from train to train to take another ride.
And if we have no willing, then other folk may be
So kind as to take us in, and kind enough to guide our way,
That we shall get to somewhere by the ending break of day.
But one small question rises, teh problem of our aim,
For the riding on life's underground is more than simple game,
And 'tis no use going one way when you need the other side,
And not every point is best to be when we have somewhere died.

Scholared Academics

Let us analyze our illness and calculate our death,
Let us talk in pure convention until we lose our breath.
Let us bow in adoration to an image on the shelf,
Which we in all our learned madness crafted from ourself.
Let us take away the truth and live a life on dreams
That, not even showing truth, deceive by all that seems,
And if we do not like the real, cast it to the night,
For we are scholared academics who spread the subtle light,
And our opinions in this era outrule the endless sages
Who drew from life in endless pattern the wisdom of the ages,
But if they happen with ourselves most sweetly to agree,
Let us give them credit for such wise felicity;
While if they contradict ourselves in prose, in speech, in song,
Then we castigate their minds, because they are then wrong.
And never shall we acknowledge that we may have a flaw
For we divinity itself in our own image draw.
All these have no education who with us disagree
And every student who has learned will with our own eyes see.
You ask, is this not all against all for which we stand?
This shows that you have missed the point, and do not understand.
We are scholared academics: 'tis we who make the real
And documents in ink on paper are our shield and steel.

Trajan by Baptism

The rule of the lion is rule over pride,
where killed is the evil which gnaws one inside,
where found is the godhood which in one has died,
where blood washes all, and one's death is new life.

The wolves in their howling long for their dream
in packs where they bind sweet alliance's seam,
where wonder is lost in an infinite stream,
where blood washes all, and one's death is new life.

For never is given to man's mortal eye
by wings of his own to sunny vales fly
or else he will fall with Icarus' cry--
but blood washes all, and one's death is new life.

And only by others is blood gladly shed,
And only by one who most painfully bled
can ever arise the soul of the dead,
when blood washes all, and one's death is new life.

In the Street

The clowns in the street begging for bread
are but the shrouds that cover the dead,
or else the sheets on which sick are lain,
and so without fault are covered with stain;
and all around them a city does fall
with crumbling decay like a ruining hall.

Blue

Like running currents swiftly flowing
by the sunset early breaking
on the hills of far away,
I passed onward in my journey
through the box they put me in
to where the wind in gusty blowing
nudges all the leaves to waking
in the cool of singing day,
which made the seer merely be
in ways the mortal has not been.

With darkness in the hindways falling,
light before me, now I stride,
and bless the rising of the coolness
(which the guessing thinkers lose),
freed from chambers dark, imposing,
to walk with angels in their calling,
purged of envy and of pride,
crushed with realness of all trueness,
in existential paths I choose --
and choosing, all the wide ways closing.

Yet doors that open, open inward,
and the laugh of wing-blessed singer
is like rose's scentings dreaming
every day new paths to yield,
or else like the kisses of the stream,
and though I seek not to discard
all my hope, it does not linger:
is it all a pleasant seeming
or do I realms and kingdoms build --
as this is real and yet a dream?

Visuddha

Destruction and burning, wherein there is life,
Slashing and slaying, perpetual strife,
Resolved in a moment to infinite peace,
Might from the burning, and joy without cease.

Slashing and slaying in fountains of blood,
The riving of heaven unto the good,
Leaps up like a salmon to death in the night --
The deepness, the darkness, of life and of light.

Visuddha: the purging, the drawing in dust
With blood from bright swords long put to rust--
This is our living, the hope of our eye,
For no god shall live unless it first die.

Bleg

Does anyone know of any good, accessible (for undergrads), short, and (preferably) recent essays on feminist theory or, indeed, any aspects of feminist thought? I have one or two in mind, but I wanted to find out if any of my readers had come across some recently that I might not have.

PC52

The Philosophers' Carnival LII is up at Philosophy, et cetera, where it began on August 23rd three years ago. It's been three years, over fifty posts, and quite a splendid set of links. Here's wishing many more!

A Poem Draft

Ripples

The moon is full tonight,
as is my mind:
memories of another life,
another time.

On the river-water
boats ripple as they glide;
each ripple is half-remembered
as it rises up, then dies.

Posting is likely to be light this week. In the pipeline: the final posts of a number of series (on formal typology of argument, on Hume's Dialogues, etc.), a discussion of the tractability of moral disagreement (you read it right), and a 'Philosophy 101' series of posts, some juvenilia, and more.