restoration with sublation as the unified response of healthy tradition to deformations of tradition (redemption and transfiguration in sacred Tradition)
key principles of formalized iconography
(1) indentifiability by intellect
(2) definite representation
(3) purified form (removal of anything inappropriate)
- spontaneous devotional iconography may violate strict application of principles, sticking to the spirit rather than the letter
- causation : definite representation :: remotion :purified form :: eminence : intellectual identification
the principle of progressive solemnity applied to iconography
Part of the point of argument is to build shared reasoning.
the withwardness of reason
A dangerous feature of modern life that infects the 'liturgy wars' is the temptation of thinking that if you just find the Magical Method, all things will automatically fall into place; for all sides in the 'liturgy wars' tend often to treat liturgy as method.
incoherent plausibilities as a contributor to discovery (development of aporia)
universalism // indifferentism
The Church is as it already was, and it is what it already was. It is its past, not only in that its past pushes along behind it, and that the Church possesses what is past as something present-at-hand and effective, but also that hte Church has grown up in a traditional way of interpreting and understanding itself, and by this understanding its possibilities are disclosed and regulated.
It is an error to treat tradition and its primordial sources as opposed.
signs as gear (Heidegger)
Vestments are both signs and gear, but they involve the notion of carrying so as to express a being-related to the world.
the traditionary character of clothing practice
the analogy between vestment and office or role (complicated by vestment also, and because of this, being an instrument of office -- role gear)
- uniforms are standardized role-gear
the aroundness of clothing vs the aroundness of environs
As role-gear, clothing differs from characteristic tools by being not merely carried (like a hammer in the hand) but in some sense carrying itself in being carried. We carry our clothes in wearing them by arranging them so that they do the carrying (hanging, buttoning and fastening, tying, conforming).
the protecting and displaying functions of garments
- the displaying function has an interesting relation to the hiding subfunction of protecting
- the protecting function may be moral as well as physical
the analogies between garment and home (note turtles and snails and hermit crabs here)
The being of the Church Militant is the Passion of Christ; its traditionary sojourn through the world is its fulfillment by the Passion.
Kant's refutation of idealism as a sort of ontological argument (Heidegger)
analogy between knowing causal power | knowing external world
sensation | sensation
one's own volition | analogy to violation
endeavor | resistance
volition of God | occasion
regularity | coherence
readiness to act | readiness to appear
Sacramental vocation is the vocation of Christ's Passion.
sacramental vocation as anticipatory martyrdom
the Church's mode of unwellness (uneasiness in the world, queasiness at the world)
the mood of drudging
The fortitude of the Church: its founded-on-the-Rock-ness; the prudence: its helped-by-the-Spirit-of-Truth-ness; the justice: its many-members-in-one-Body-ness; the temperance: its chasteness-as-the-Bride-of-the-Lamb-ness.
The cultural goods of the Church as material for the disclosure of the Church as it has been.
The Church is based on Tradition, and its calendarizing of its worship is a way of making accessible the Tradition within which it works. but the Church works with many calendars, as it were spreading out and shifting around within this Tradition.
the calendar of saints as providing a rough sketch of the content of Tradition
hylomorphism of ideas and means of production
Imagination is naturally eclectic.
Even a stupid man's traditions have a sort of dignity.
methods of seeking the right vantage for problem-solving.
forms of promulgation: creation, command, shared practice
Peirce's 3 conditions for the existence of a sign (MS 221)
(1) characters distinguishing it from other objects
(2) must be causally linked to its object in some way
(3) address to the mind such as to relate the mind to the object
distinction, principiation, manifestation
nouns as less sparse pronouns
as-if-it-were as a modal operator
Our world is such that it works in several ways as if it were a world in which a detective named Sherlock Holmes lived on Baker Street and vanquished Moriarty at Reichenbach Falls. (Our world is such that it also works in several ways as if it were a world in which a general named Julius Caesar crossed the Rubicon, and was slain by Brutus and others.) It isalso such that it works in several ways as if there were no such detective at all. This has to be stronger than mere consistency, but of course much weaker than requirement, or even apparent requirement.
Our capacity to invent fictions arises naturally from our ability to inquire into things.
inquiry as a process of story-making
fictional characters // geometrical diagrams // arithmetical calculations // sentence formations
primary reference, collateral reference, qua-reference
objects of thought as proximate ends of means to the remote ends of truth
The orthopraxy relevant to the liturgy consists of acts of faith, hope, and love.
goodness of inquiry in terms of results, of coherence of inquiry, of contribution to understanding
the maximizing of what tends to solve intellectual problems
∃xFx Allow F's
∀xFx Do not allow non-F's (Allow only F's)
cast from Eden -> long period of rebellious wandering -> judgment -> covenant
liberation from slavery in Egypt -> long period of rebellious wandering with judgment -> covenant
"When we speak of Christ's priesthood, what else do we mean than the incarnation?" Fulgentius
"For it is in him that our human nature becomes a redemptive offering."
convalidation of reference
black holes as supertask-like
aesthetics and the splendor of ethics
the principle of overflow (redundantia) in fine arts
being with a work of art
works of art as presenting th eworld to us
works of art and the salience of space and time
the quasi-vocative characters of certain works of art
From the craving of the world springs impatience and cruelty, jealousy and conceit and self-exaltation, boorishness and selfishness and irritability, retaliation and self-indulgence and lie, despair, and, in the end, death.
Patience is required for the full fructification of love.
causation, remotion, and eminence in interpretation of Scripture
one : prudence :: holy : justice :: catholic : temperance :: apostolic : fortitude
"Our pilgrimage on earth cannot be exempt from trial. We progress by means of trial. No one knows himself except through trial, or receives a crown except after victory, or strives except against an enemy or temptation." Augustine
The body of the Church is a picture of the Spirit.
'straw man' requires that are determine
(1) the structure of A
(2) the structure of B
(3) an end in light of which A is not adequate to represent B
Straw man and idealization are both based on one argument/position representing or modeling another.
straw man as a case of defective type (relative to archetype)
fallacies and defective causes in inquiry
examination of inquiry // examination of conscience
- inquiry : true :: conscience : good
- things fatal to inquiry:
distraction from inquiry
despair of inquiry
distortion of inquiry (wrath, envy, pride)
the moral discipline implicit in inquiry
Saturday, June 10, 2017
Poem a Day 10
Lovers' Boasts
A lover's boast is swiftly made;
the heart is not so firm and sure.
Too often trust is ill-repaid,
too often love does not endure.
But hope still bubbles at the source,
still flows from forth its ancient fount,
becomes a river in its course,
thus all the laws of time to flount.
And, strangely, things that lovers feel,
which swift and sudden passions form,
sometimes seal as strong as steel
and weather even raging storm.
At times a passing word will last,
a breath of promise shape the age,
a vow in day, in night will last,
flirtation grace undying page.
A lover's boast is swiftly made;
no human word is guarantee.
Too often promises will fade,
too often lovers' hopes will flee.
But passions sometimes seed a faith
that stands so pure and oaken-strong,
all else compared is but a wraith,
for it is real and ages long.
A lover's boast is swiftly made;
the heart is not so firm and sure.
Too often trust is ill-repaid,
too often love does not endure.
But hope still bubbles at the source,
still flows from forth its ancient fount,
becomes a river in its course,
thus all the laws of time to flount.
And, strangely, things that lovers feel,
which swift and sudden passions form,
sometimes seal as strong as steel
and weather even raging storm.
At times a passing word will last,
a breath of promise shape the age,
a vow in day, in night will last,
flirtation grace undying page.
A lover's boast is swiftly made;
no human word is guarantee.
Too often promises will fade,
too often lovers' hopes will flee.
But passions sometimes seed a faith
that stands so pure and oaken-strong,
all else compared is but a wraith,
for it is real and ages long.
Friday, June 09, 2017
Evening Note for Friday, June 9
Thought for the Evening: Invented Traditions
I have recently been reading The Invention of Tradition, edited by Eric Hobsbawm and Terence Ranger, which I picked up at Half-Price Books a while back; it is quite interesting. Hobsbawm defines 'invented tradition' in the following way:
A number of examples are given throughout the book by the different authors. For instance:
* The kilt, which is practically a distinguishing feature of Scotland today, was invented in the eighteenth century by an Englishman, Thomas Rawlinson, who owned a business that hired a lot of Scotsman for dangerous jobs like lumberjacking. The traditional Highland costume was a belted long tunic that had to be hitched up for hard work, and Rawlinson designed the kilt to be a vaguely similar garment that would be more practical. His design was indeed much more practical for difficult work, so it spread like wildfire through the Highlands among Highlanders, who often were involved in various kinds of manual labor jobs. It then began to be replaced by trousers, but the Highland regiments in the military continued to have it as part of their military uniform, so when there was a revival of Scottish nationalism in the nineteenth century, it was in people's minds as a distinctively Scottish kind of dress. Likewise, association of tartans with particular clans arose only in the nineteenth century, mostly as a marketing gimmick. (Hugh Trevor-Roper, "The Invention of Tradition: The Highland Tradition of Scotland")
* Most of the public rituals and traditions we associate with the British Monarchy are fairly recent -- through most of the nineteenth century the public face of the Monarchy was notoriously shabby and unimpressive, with a poor sense of ceremony. This began to change in the 1870s, and as the Crown exercised less and less power, it began to be put forward more and more as a purely symbolic representation of British unity. (David Cannadine, "The Context, Performance and Meaning of Ritual: The British Monarchy and the 'Invention of Tradition', c. 1820-1977")
It quickly becomes clear that what the authors call 'invented traditions' are in fact one way in which traditio, handing down, is standardly done. The forest of old ways tends to dry up and burn out as people lose a sense of their purposes and meanings, or as invasive species take root and steal away their original nutrition; something must fill the void left. Either those old ways will be entirely replaced, or there will be some revitalization. If they are replaced, the new ways become the new forest, and undergo the same cycle. Residues and remnants of old ways, sometimes scholarly, sometimes distorted, sometimes only speculated, sometimes entirely imaginary, often become the seeds for either a return of something approximately like the old forest, or, more often, a compromise forest between the old and the new. The return of the old ways is often merely approximate, or by analogy, or sometimes more as a symbolic aspiration than an actual return. And the cycle begins again.
The actual processes involved are, of course, various, and, despite the name 'invented traditions' are not necessarily invented in the ordinary sense of the term -- they may just be a shift from a literal understanding to a symbolic one, or they may just be natural responses, consistent with prior traditions, to new situations that become stable precedents. They may -- indeed, in the Romantic period often were -- scholarly reconstructions, or, even more commonly, popular presentations of scholarly reconstructions that become part of people's folkloric self-understanding.
While it wouldn't be considered an 'invented tradition' in the above sense, an interesting analogy to some of these situations can be seen in Ivar Aasen's Nynorsk, an attempt to find the more purely Norwegian framework in the heavily Danish-overladen Norwegian language. He did this by working out what a purely Norse-based Modern Norwegian might be like. Thus we have the situation, which is remarkably common, of an actually traditional practice -- in this case, Dano-Norwegian -- in a struggle with a reforming purist-traditionalist practice -- in this case, Nynorsk -- both putting themselves forward as the appropriate tradition. In the case of the Norwegian language, this became tangled up with political disputes, leading to a considerable number of artificial interventions, none of which succeeded; and the result is that the Norwegian language today is quite an extraordinary mess, with no organic solution to the struggle yet found or, for that matter, in sight. In the end, only a long stretch of time and a lot of ordinary interaction will heal a muddle of tradition created by politics.
Various Links of Interest
* G. E. M. Anscombe, On Transubstantiation: "It is easiest to tell what transubstantiation is by saying this: little children should be taught about it as early as possible."
* Hume as Historian at "Incudi Reddere"
* Thomas Storck, The Sin of Usury
* Daniel J. Lasker, Translations of Rabbi Judah Halevi's Kuzari
* I'm thinking of doing some Unamuno for a fortnightly book this summer, so this discussion of his quijotismo by Mariana Alessandri is timely.
* Martha Bolton, Mary Shepherd, at the SEP
* James V. Schall reviews John Safranek's The Myth of Liberalism
Currently Reading
Agatha Christie, Murder on the Orient Express
Vladimir Soloviev, The Justification of the Good
St. Romanos the Melodist, On the Life of Christ: Kontakia
Edward Feser and Joseph Bessette, By Man Shall His Blood Be Shed
John of St. Thomas, The Gifts of the Holy Spirit
I have recently been reading The Invention of Tradition, edited by Eric Hobsbawm and Terence Ranger, which I picked up at Half-Price Books a while back; it is quite interesting. Hobsbawm defines 'invented tradition' in the following way:
It includes both 'traditions' actually invented, constructed and formally instituted and those emerging in a less easily traceable manner within a brief and dateable period-- a matter of a few years perhaps -- and establishing them with great rapidity....'Invented tradition' is taken to mean a set of practices, normally governed by overtly or tacitly accepted rules and of a ritual or symbolic nature, which seek to inculcate certain values and norms of behaviour by repetition, which automatically implies continuity with the past. (p. 1)
A number of examples are given throughout the book by the different authors. For instance:
* The kilt, which is practically a distinguishing feature of Scotland today, was invented in the eighteenth century by an Englishman, Thomas Rawlinson, who owned a business that hired a lot of Scotsman for dangerous jobs like lumberjacking. The traditional Highland costume was a belted long tunic that had to be hitched up for hard work, and Rawlinson designed the kilt to be a vaguely similar garment that would be more practical. His design was indeed much more practical for difficult work, so it spread like wildfire through the Highlands among Highlanders, who often were involved in various kinds of manual labor jobs. It then began to be replaced by trousers, but the Highland regiments in the military continued to have it as part of their military uniform, so when there was a revival of Scottish nationalism in the nineteenth century, it was in people's minds as a distinctively Scottish kind of dress. Likewise, association of tartans with particular clans arose only in the nineteenth century, mostly as a marketing gimmick. (Hugh Trevor-Roper, "The Invention of Tradition: The Highland Tradition of Scotland")
* Most of the public rituals and traditions we associate with the British Monarchy are fairly recent -- through most of the nineteenth century the public face of the Monarchy was notoriously shabby and unimpressive, with a poor sense of ceremony. This began to change in the 1870s, and as the Crown exercised less and less power, it began to be put forward more and more as a purely symbolic representation of British unity. (David Cannadine, "The Context, Performance and Meaning of Ritual: The British Monarchy and the 'Invention of Tradition', c. 1820-1977")
It quickly becomes clear that what the authors call 'invented traditions' are in fact one way in which traditio, handing down, is standardly done. The forest of old ways tends to dry up and burn out as people lose a sense of their purposes and meanings, or as invasive species take root and steal away their original nutrition; something must fill the void left. Either those old ways will be entirely replaced, or there will be some revitalization. If they are replaced, the new ways become the new forest, and undergo the same cycle. Residues and remnants of old ways, sometimes scholarly, sometimes distorted, sometimes only speculated, sometimes entirely imaginary, often become the seeds for either a return of something approximately like the old forest, or, more often, a compromise forest between the old and the new. The return of the old ways is often merely approximate, or by analogy, or sometimes more as a symbolic aspiration than an actual return. And the cycle begins again.
The actual processes involved are, of course, various, and, despite the name 'invented traditions' are not necessarily invented in the ordinary sense of the term -- they may just be a shift from a literal understanding to a symbolic one, or they may just be natural responses, consistent with prior traditions, to new situations that become stable precedents. They may -- indeed, in the Romantic period often were -- scholarly reconstructions, or, even more commonly, popular presentations of scholarly reconstructions that become part of people's folkloric self-understanding.
While it wouldn't be considered an 'invented tradition' in the above sense, an interesting analogy to some of these situations can be seen in Ivar Aasen's Nynorsk, an attempt to find the more purely Norwegian framework in the heavily Danish-overladen Norwegian language. He did this by working out what a purely Norse-based Modern Norwegian might be like. Thus we have the situation, which is remarkably common, of an actually traditional practice -- in this case, Dano-Norwegian -- in a struggle with a reforming purist-traditionalist practice -- in this case, Nynorsk -- both putting themselves forward as the appropriate tradition. In the case of the Norwegian language, this became tangled up with political disputes, leading to a considerable number of artificial interventions, none of which succeeded; and the result is that the Norwegian language today is quite an extraordinary mess, with no organic solution to the struggle yet found or, for that matter, in sight. In the end, only a long stretch of time and a lot of ordinary interaction will heal a muddle of tradition created by politics.
Various Links of Interest
* G. E. M. Anscombe, On Transubstantiation: "It is easiest to tell what transubstantiation is by saying this: little children should be taught about it as early as possible."
* Hume as Historian at "Incudi Reddere"
* Thomas Storck, The Sin of Usury
* Daniel J. Lasker, Translations of Rabbi Judah Halevi's Kuzari
* I'm thinking of doing some Unamuno for a fortnightly book this summer, so this discussion of his quijotismo by Mariana Alessandri is timely.
* Martha Bolton, Mary Shepherd, at the SEP
* James V. Schall reviews John Safranek's The Myth of Liberalism
Currently Reading
Agatha Christie, Murder on the Orient Express
Vladimir Soloviev, The Justification of the Good
St. Romanos the Melodist, On the Life of Christ: Kontakia
Edward Feser and Joseph Bessette, By Man Shall His Blood Be Shed
John of St. Thomas, The Gifts of the Holy Spirit
Poem a Day 9
Friendship
In hope and faith is friendship formed.
By hope we hold our heart's desire;
walking freely, foot put forward,
with step on step, we stride the way.
By faith we pledge, we form a pact,
we seal by sign two different souls,
and I to we thus weld as one.
In hope and faith is friendship formed.
By hope we hold our heart's desire;
walking freely, foot put forward,
with step on step, we stride the way.
By faith we pledge, we form a pact,
we seal by sign two different souls,
and I to we thus weld as one.
Thursday, June 08, 2017
Social Life and Personality
Deprive a concrete human personality of all that is in any way due to its relations with social and collective wholes, and the only thing left will be an animal entity containing only the pure possibility or empty form of man--that is, something that does not really exist at all....
Social life is not a condition superadded to the individual life, but is contained in the very definition of personality which is essentially a rationally-knowing and a morally-active force--both knowing and acting being only possible in the life of a community. Rational knowledge on its formal side is conditioned by general notions which express a unity of meaning in an endless multiplicity of events; real and objective universality (the general meaning) of notions manifests itself in language as a means of communication, without which rational activity cannot develop, and, for lack of realization, gradually disappears altogether or becomes merely potential.
[Vladimir Soloviev, The Justification of the Good, von Peters, ed. Catholic Resources (Chattanooga, TN: 2015), p. 221.] This clearly is influenced by the arguments of nineteenth-century traditionalists, but it looks like Soloviev has generalized some of the ideas slightly.
Poem a Day 8
Caritas
Birds may be trained and tamed,
learn to love their bars and bonds,
eat from the hand and turn a phrase,
but perhaps one day
the latch on the door is unlocked,
and, flying to freedom,
from bough to bough flits the bird.
The captor may call, but in vain;
bird to bird sings a song,
flying, floating, fleeing wide,
a righteous sun above its head,
the breath of God beneath its wing.
Birds may be trained and tamed,
learn to love their bars and bonds,
eat from the hand and turn a phrase,
but perhaps one day
the latch on the door is unlocked,
and, flying to freedom,
from bough to bough flits the bird.
The captor may call, but in vain;
bird to bird sings a song,
flying, floating, fleeing wide,
a righteous sun above its head,
the breath of God beneath its wing.
Wednesday, June 07, 2017
Elements of Modal Logic, Part IX
Part VIII
Let us take some examples to see how we can use what we have learned so far. Suppose that I, being a wizard with a green thumb, am surveying my new garden of weird flowers. I have some things that I have in every quarter of my garden. That's a Box. I have other things that are in at least one quarter. And if something is in every quarter of my garden, it is in at least one quarter, and that means Box includes Diamond. So we want at least 1234D.
Now, since Box here means 'in every quarter', Not-Box means 'not in every quarter (i.e., not in some quarters)', Box-Not means 'in every quarter not (i.e., in no quarter)', and Not-Box-Not means 'not in every quarter not (i.e., in some quarters)'. From this it is easy to see how the corresponding Diamonds work, using Rules (3) and (4). Our square of opposition is (again, I only show Box, but each corner can be translated into a Diamond version):
If we take an item in our reference table, like Box (Man-eating Dandelions), then we can use the square to say what's consistent and inconsistent with this item. For instance, Box (Man-eating Dandelions) is contrary to Box-Not (Man-eating Dandelions), and it is contradictory to Not-Box (Man-eating Dandelions); so Box (Man-eating Dandelions) rules both of these out. On the other hand, it requires Not-Box-Not (Man-eating Dandelions) -- Rule (D) tells us that Box includes Diamond. And Rule (4) tells us that Not-Box-Not (Man-Eating Dandelions) is interchangeable with Diamond (Man-Eating Dandelions), so that's required, too. We can put our reasoning in a simple form:
(i) □ (Man-Eating Dandelions)
Therefore:
(ii) ◇ (Man-Eating Dandelions) -- [from Rule (D)]
Which is the same as:
(iii) ~□~ (Man-Eating Dandelions) -- [from Rule (4)]
If we look at Diamond-Not (Venus Fly-Traps), this is the same corner as Not-Box (Venus Fly-Traps); it is inconsistent with both Not-Diamond-Not (Venus Fly-Traps) and Box (Venus Fly-Traps), since those both are the same. However, it doesn't tell us anything about the other two corners -- it's consistent with Not-Diamond (Venus Fly-Traps) and Diamond (Venus Fly-Traps).
It's worth taking some time to explore how each of the corners of the 1234D square, both with Box and with Diamond, relate to the others, because it is just so very common, and if you know this very well, you know a huge amount of modal logic, because it's how you fully understand what a given Reference Table means.
So let's think about what our other tables have to be, given our Reference Table.
If you are only using 1234, you should always do Diamonds first, but we are using 1234D, and both Rule (D) and our square of opposition tell us that Box includes Diamond. Note that by our rules, Not-Diamond (Tulips) [= There are no quarters with tulips] means the same as Box-Not (Tulips) [= every quarter has no tulips]. If I were just using 1234, I might not have any quarters in my garden (maybe I haven't planted any yet). But (D) tells me that if I have any Boxes in my Reference Table, I have at least one quarter in my garden that has them (which is Diamond), so I know I have at least one quarter in my garden, and since Boxes are true of every table that we have, all our Boxes tell us about that quarter:
I put 'No Daisies' because we know that there are no daisies anywhere (Box-Not). And I put 'No Tulips' because Not-Diamond means the same as Box-Not.
With the flowers that are only Diamond, though, we have to be much more careful, because while we know they each are on at least one table, we don't know which or how many! Maybe Jam-and-Butter-Cups are only in one quarter. Maybe they are in two. Maybe they are in three. It could even be that they are in all four. Our Reference Table doesn't tell us. And it's even trickier, because we have more than one Diamond, and we don't know if they are talking about the same quarters or different quarters. There are lots of possibilities. So when I put them on my table, I can't assume that every table is a different quarter. Maybe I am accidentally giving two incomplete descriptions of the same table, and they really should be on the same table! But I can't just put them on the same table, either, because maybe they are all on different tables! Since the Reference Table doesn't tell us, we have to be careful to remember that we don't know these things. Not assuming that you know something you don't is often the single most important thing in logic.
What we do know from the Diamonds, is that (whether or not they are overlapping or separate) the following things are true:
(1) I have at least one quarter with Jam-and-Butter-Cups.
(2) I have at least one quarter with Screaming Mandrakes.
(3) I have at least one quarter without Venus Fly-Traps.
Again, I don't know if these quarters are the same or different. If I put them all on one table, I might be wrong. If I put them on different tables, though, then as long as I remember that any of my tables might be incomplete, and that any two tables might be giving incomplete descriptions of the same quarter, I will be just fine.
Or, if we prefer to have it in a slightly more diagrammatic form:
Part X
Let us take some examples to see how we can use what we have learned so far. Suppose that I, being a wizard with a green thumb, am surveying my new garden of weird flowers. I have some things that I have in every quarter of my garden. That's a Box. I have other things that are in at least one quarter. And if something is in every quarter of my garden, it is in at least one quarter, and that means Box includes Diamond. So we want at least 1234D.
| [1234D] Reference Table (Weird Flowers in Quarters of the New Garden): |
|---|
| Box (Man-eating Dandelions) |
| Box (Fire-breathing Snapdragons) |
| Diamond (Jam-and-Butter-Cups) |
| Box (Really Red Roses) |
| Diamond (Screaming Mandrakes) |
| Box (More-than-deadly Nightshades) |
| Box-Not (Daisies) |
| Not-Diamond (Tulips) |
| Diamond-Not (Venus Fly-Traps) |
Now, since Box here means 'in every quarter', Not-Box means 'not in every quarter (i.e., not in some quarters)', Box-Not means 'in every quarter not (i.e., in no quarter)', and Not-Box-Not means 'not in every quarter not (i.e., in some quarters)'. From this it is easy to see how the corresponding Diamonds work, using Rules (3) and (4). Our square of opposition is (again, I only show Box, but each corner can be translated into a Diamond version):
If we take an item in our reference table, like Box (Man-eating Dandelions), then we can use the square to say what's consistent and inconsistent with this item. For instance, Box (Man-eating Dandelions) is contrary to Box-Not (Man-eating Dandelions), and it is contradictory to Not-Box (Man-eating Dandelions); so Box (Man-eating Dandelions) rules both of these out. On the other hand, it requires Not-Box-Not (Man-eating Dandelions) -- Rule (D) tells us that Box includes Diamond. And Rule (4) tells us that Not-Box-Not (Man-Eating Dandelions) is interchangeable with Diamond (Man-Eating Dandelions), so that's required, too. We can put our reasoning in a simple form:
(i) □ (Man-Eating Dandelions)
Therefore:
(ii) ◇ (Man-Eating Dandelions) -- [from Rule (D)]
Which is the same as:
(iii) ~□~ (Man-Eating Dandelions) -- [from Rule (4)]
If we look at Diamond-Not (Venus Fly-Traps), this is the same corner as Not-Box (Venus Fly-Traps); it is inconsistent with both Not-Diamond-Not (Venus Fly-Traps) and Box (Venus Fly-Traps), since those both are the same. However, it doesn't tell us anything about the other two corners -- it's consistent with Not-Diamond (Venus Fly-Traps) and Diamond (Venus Fly-Traps).
It's worth taking some time to explore how each of the corners of the 1234D square, both with Box and with Diamond, relate to the others, because it is just so very common, and if you know this very well, you know a huge amount of modal logic, because it's how you fully understand what a given Reference Table means.
So let's think about what our other tables have to be, given our Reference Table.
If you are only using 1234, you should always do Diamonds first, but we are using 1234D, and both Rule (D) and our square of opposition tell us that Box includes Diamond. Note that by our rules, Not-Diamond (Tulips) [= There are no quarters with tulips] means the same as Box-Not (Tulips) [= every quarter has no tulips]. If I were just using 1234, I might not have any quarters in my garden (maybe I haven't planted any yet). But (D) tells me that if I have any Boxes in my Reference Table, I have at least one quarter in my garden that has them (which is Diamond), so I know I have at least one quarter in my garden, and since Boxes are true of every table that we have, all our Boxes tell us about that quarter:
| TABLE 1: Some Quarter in the New Garden |
|---|
| Man-eating Dandelions |
| Fire-breathing Snapdragons |
| Really Red Roses |
| More-than-deadly Nightshades |
| No Daisies! |
| No Tulips! |
I put 'No Daisies' because we know that there are no daisies anywhere (Box-Not). And I put 'No Tulips' because Not-Diamond means the same as Box-Not.
With the flowers that are only Diamond, though, we have to be much more careful, because while we know they each are on at least one table, we don't know which or how many! Maybe Jam-and-Butter-Cups are only in one quarter. Maybe they are in two. Maybe they are in three. It could even be that they are in all four. Our Reference Table doesn't tell us. And it's even trickier, because we have more than one Diamond, and we don't know if they are talking about the same quarters or different quarters. There are lots of possibilities. So when I put them on my table, I can't assume that every table is a different quarter. Maybe I am accidentally giving two incomplete descriptions of the same table, and they really should be on the same table! But I can't just put them on the same table, either, because maybe they are all on different tables! Since the Reference Table doesn't tell us, we have to be careful to remember that we don't know these things. Not assuming that you know something you don't is often the single most important thing in logic.
What we do know from the Diamonds, is that (whether or not they are overlapping or separate) the following things are true:
(1) I have at least one quarter with Jam-and-Butter-Cups.
(2) I have at least one quarter with Screaming Mandrakes.
(3) I have at least one quarter without Venus Fly-Traps.
Again, I don't know if these quarters are the same or different. If I put them all on one table, I might be wrong. If I put them on different tables, though, then as long as I remember that any of my tables might be incomplete, and that any two tables might be giving incomplete descriptions of the same quarter, I will be just fine.
| TABLE 1: Some Quarter in the New Garden (Don't Know Which) | TABLE 2: Some Quarter in the New Garden (Don't Know Which) | TABLE 3: Some Quarter in the New Garden (Don't Know Which) |
|---|---|---|
| Man-eating Dandelions | Man-eating Dandelions | Man-eating Dandelions |
| Fire-breathing Snapdragons | Fire-breathing Snapdragons | Fire-breathing Snapdragons |
| Really Red Roses | Really Red Roses | Really Red Roses |
| More-than-deadly Nightshades | More-than-deadly Nightshades | More-than-deadly Nightshades |
| No Daisies! | No Daisies! | No Daisies! |
| No Tulips! | No Tulips! | No Tulips! |
| Jam-and-Butter-Cups | Screaming Mandrakes | No Venus Fly-Traps! |
Or, if we prefer to have it in a slightly more diagrammatic form:
Part X
Subscribe to:
Posts (Atom)