Showing posts with label formal language. Show all posts
Showing posts with label formal language. Show all posts

26 Apr 2011

No Reign May Be: Infinite Unsaying in Clifford Duffy's 'around'

response to Clifford Duffy's poem, by Corry Shores
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All images are screen-shots of Duffy's work, from:
http://recalltopoetry.blogspot.com/2011/04/around.html
May I thank Clifford Duffy for sharing his creations.


No Reign May Be:
Infinite Unsaying in Clifford Duffy's 'around'



Without words, without things: so, what else would remain? Heidegger famously wrote, "where words break off, no thing may be." If we did not have the word-meaning for 'beside', would our world have things one-beside-another? If we cross-out all our word-meanings and grammatical functions (like predication, 'The tulip is a flower, is yellow,') can anything in our world remain? Not only would all the namable things be crossed-out, so too would the possibility of making designations; for, to designate something, we need to differentiate it from other things in its context or against which it would be counter-defined (we might need 'dark' to know 'light'). But without grammatical-functions like predication, negation, and prepositional relation (dark lingers before light), can we even begin to have a world of things?

How might we further this line of thinking? Regarding the sentence, "where words break off, no thing may be," let's draw our attention to no thing. We could think of it as nothing, or perhaps as a 'no-thing.' What would a no-thing be? It is not a thing, but it is not nothing, either. And from this sentence, we would assume that a no-thing appears when words 'break off'. It could be that the no-thing was there even while there were words, but we only notice it when the words break-off and leave the no-thing standing by itself.

Such observations might be of interest when we consider the mechanics of language. When driving and we see a stop signal, our feet hit the brake without a second thought. It is as if there is no need to translate the meaning of the sign: the same way that one gear need not think about another gear in order for them to turn one another, our bodies too are like gears that react automatically to the signs we see. But first recall when we initially learned how to drive. When we saw the red, we did not automatically stomp our feet without hesitation. There were so many possibilities available. Where exactly is the brake pedal? We have to hit it without looking for it? But is not the gas pedal right next to it? If we hit the gas accidentally, our situation could become unpredictably catastrophic. Our body does not yet have the word-thing 'red means stomp on brake'. Instead, we have something much more interesting. But after we learn the meaning of the sign, there is no longer anything there. We hit the brake without the red, the stomp, or the stopping entering our attention. These are all very concrete real things in our world, and they are so much a part of our world that we do not notice them. But before they existed, before we had the word-meanings for these parts of our world, they were so much more, they were infinitely more, they could have been anything! Red could very easily have meant catastrophe as much as total safety and orderliness. In this case, words broke-off, but there was not nothing, nor was there a concrete something. Rather, there was a no-thing, an infinite non-something, a something = x to the power of infinity, or we might say, a thing = x to the nth power.

Recalling again us first learning to drive, we had to learn a new language, of sorts, the language of traffic signs, both customary ones, like traffic-lights and official street signs and signals, and also contextual signs, like seeing a fluid on the road and knowing this signals to us to slow down so not slip-off the road. But when first acquiring this 'vocabulary', we learn that the language-meanings we already know do not always help. We need to learn the new language, yet we are lost at first because we try to apply the languages we already know. Perhaps we first considered those mechanical reactions we knew from the language of bike-riding, but back-peddling or gripping the steering wheel does not brake the automobile. So when first learning to brake the car, we exhaust all the 'words' we have in the 'languages' we know, running through all the things already in our world. That does not mean only nothing remains! There is an infinity at the edge of these limits!: catastrophe or safety from death, happening by so many unforeseeable ways. When we exhaust the words of our language and the things of our world, what remains is infinite no-thingness.

So let's not interpret this no-thing as an absence. It could hardly be so! How is something infinite also something absent? This means then that when Heidegger speaks of 'saying,' which is also an 'unsaying,' we need not think that language (in how we feel it immediately) has anything to do with something missing from it. When first learning how to brake a car, it is not that we face a nothing or an absolutely foreign otherness. Language becomes exhausted. All its possibilities have run-through and run-out. At last!: the birth of the full and immediate no-thing, the absolute thing = xn. Please note the difference between thing = x and thing = xn (or no-thing = xn. Also consider the formulations sign = x and no-Sign = xn.) An extensional meaning refers to the things in the set it refers to. Oedipus, recall, married his mother. So 'son of Jocasta' and 'husband of Jocasta' refer to the same thing, the same object in a set, which in this case is a set with one member, the thing we also call Oedipus. But what happens in our minds and bodies when we say to ourselves, 'The son of Jocasta is the husband of Jocasta'? This does not give us the same feeling as when we say 'Oedipus is Oedipus' or 'Oedipus is Jocasta's son', for example. This is because even though both 'son of Jocasta' and 'husband of Jocasta' extend into the same meaning-reference (Oedipus), they do not at all intend the same meaning. The words and propositions in our language have intensional meanings as well as extensional ones. It is hard to define what an intensional meaning really is, but we might think of it as the impulses and tendencies for a meaning to want to reach-out into different suggested or 'virtual' implications. It is implicit rather than explicit, and in a way, it is inexplicably implicit, because it is difficult to formalize (those with superb analytical philosophical skills can in fact formalize this, as for example the remarkable contributions Roger Vergauwen has made to formalized intensional logic and semantics). So 'thing = x' might suggest that there is an unknown thing, or an empty set; in the case of sign = x, we might at first think of it as an empty reference, or we may even regard it to indicate that at the core of all signification there is absence, in the sense that absence is always a prerequisite condition for representation (for otherwise, if the thing were present, it would be mere presentation and not representation). But this is to regard 'x' only in an extensional way. When however x is to the nth power, this means that while no extensional meaning is designated, there is an infinity of virtual intensional meaning. It is not just that the x is an empty place-marker meaning that it could be filled by anything, but rather that it virtually is tending with infinite power to mean a great variety of things, like how the no-thing/no-sign of learning to brake a car virtually implies so many meanings, ranging from utter disaster to complete safety to so much else that it may prove itself to mean.

So let's consider a Deleuzean reading of Heidegger's philosophies of language, rather than one that follows a more post-structuralist leaning (I tried that here). Perhaps we see such an approach suggested in Deleuze's essay, 'An Unrecognized Precursor to Heidegger: Alfred Jarry' (may I thank Miguel Paley for bringing this text to my attention and for offering his research on it). A closer analysis is forthcoming. For now, let's look at the final line:
The thing is the limit of language, as the sign is the language of the thing. When a language is hollowed out by its turning within language, it finally completes its mission: the Sign shows the Thing, and effectuates the nth power of language, for
"where word breaks off no thing may be." (98d)
Let's return to this for a deeper explanation another time, after we conduct a closer reading of the article. For now, I suggest we turn again to Clifford Duffy's works in our effort to understand language in a Deleuzean way, in this case, what might be a poetic unsaying to the nth power.

Duffy's poem, 'around', says to us things about rain. But consider when we watch rain falling on a puddle. It has a chaotic pattern of falling. In one way, it is completely redundant, because it's just drop drop drop, every time. But also, its chaos makes it continually unique, which can entrance us, much like how this happens when we watch the dance of fire tongues, always repeating, while always unique and singular. Rain, in a way, is continually exhausted, having done the same thing in seemingly every possible arrangement and combination, and yet also somehow always uniquely this particular fall of drops. Rain, like fire, can give us this experience of the no-thing = xn. Having seen so much rain in our lives, we have exhausted all possible words to give what we witness. And yet, that does not mean we stop noticing the raindrop patterns. They keep giving to us something infinite, lying at the limits of language.



The falling of rain is cyclonic cycling, one that is as wild as a cyclone but cyclical in its self-exhaustive repetition, like the variational rhythm of these lines above, and the rhythm of space-text temporalizations, the breaks which make our eyes fall from line-to-line, word-to-word, in a rain-like way. This is rhythm in an intensive sense, as a something creating the feeling of motion by acceleration-changes. We do not feel motion when our train or car maintains the same speed. It is only when the speed changes that we feel ourselves move forward or backward. There is no such thing as a steady rhythm, at least it would not be a rhythm with any intensity.

The no-thing is not these gaps, nor would it be a sense that something is being left-out of these poetic lines. Where the words break-off is not the empty spaces. Those are words too, don't forget! Rather, the words here are in a continual state of self-breaking, of being cracked and self-differentiated. 'rain to rain' and 'rain on rain'. It is rain raining upon itself, as different unto itself. The word/thing 'rain' is self-broken from the beginning of its utterance/appearance. Rain unsays itself with each drop. That's how it keeps being rain. Something that repeats without changing has not actually reappeared. Something can only reappear if its next appearance is self-differentiated, so that it may stand-out from itself. This is not as simple as rain at time1 being different from rain at time2. Rather, rain1 and rain2 are found in time = 0; which is to say, rain varies from itself from the beginning and before any time has passed, and it is only on account of the tension, the intensity, of an imcompossible combination of differences between a self and itself that we get the power to create the motion of time. If there were no paradoxes right now, no contradictions, no things being themselves and not themselves, then everything will have stopped! There would be no power, no tension, no force for change. 'Becoming' is not a process of something moving from being itself to not being itself; no; rather; 'Becoming' involves something already from the start not being itself, and on account of that, it has the cause, reason, ground, motivation, and power to change itself.

Duffy's 'rains' here, are becomings: each is a self-difference upon itself; it is 'rain to rain' and 'rain on rain', where each rain is itself self-differential. And it is at least partly because each moment differentiates from itself that there is intensive rhythm and the unsaying of no-things to powers n-infinite (or put more simply, true becomings).

Following that first part is something like a pause in the rain or a calm before a storm, if you will. During such moments, we do not at all feel a lack, a lack of rain and storm. In fact, these are the moments we feel the weather most acutely, when it announces itself in its infinite no-thingness, in those moments we would exhaust all words to describe it, and instead just have a direct immediate affectual connection with the weather itself.



Now consider the raining of words to come.




Our eyes rain rhythmically down the text, again in self-differential repetitions. The poem goes through words we have that might exhaust the meaning of rain. It is a god, a faucet, a reigning, a goddess, a goddess raining by reigning /and/ reigning by raining, over the earth.

Loves have secret loves. Why is it each dropping is a repetition but self-differential? Like when watching fire, we love not just what we see, but the motions that continue arriving. We did not expect them, so the next moments we did not previously know that we loved. They are secret loves of our loves, and by giving to us each time these ever newer and newer secret loves, the rain repeats in a delightfully differential way.




Clifford Duffy. 'around.'
http://recalltopoetry.blogspot.com/2011/04/around.html


Gilles Deleuze. 'An Unrecognized Precursor to Heidegger: Alfred Jarry.' in Essays Critical and Clinical. Transls. Daniel W. Smith & Michael A. Greco. Minneapolis: University of Minnesota Press, 1997.

25 Feb 2009

Mathematical Methods in Linguistics, Partee, Meulen, and Wall, Ch 8, Formal Systems Axiomatization, and Model Theory



by
Corry Shores
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Partee, Meulen, and Wall

Mathematical Methods in Linguistics

Chapter VIII: Formal Systems, Axiomatization, and Model Theory

8.1: The syntactic side of formal systems

This chapter will examine formal syntactic and semantic systems. In the following we will use recursive definitions to specify sets. With them we will make inductive proofs and form axiomatic systems. We then illustrate the connections between grammars and formals systems.

8.1.1: Recursive Definitions

We will illustrate recursive definitions.
We consider series of letters. We call them "strings."
Now to understand the next term we first ask, what would "a" look like if she looked in the mirror? She would see herself. She and her image would be standing next to each other, as if the mirror were standing between them:
aa
We notice that if we cut this string in half, we get the same thing on both sides,
a and a.
Now consider when string aba sees herself in the mirror. We get
abaaba.
Notice again we may cut this string in half to get two of the same thing:
aba and aba.
This time we see the string
aaabbaaa.
May we cut this string in half to produce mirror images? Yes:
aaab and baaa.
We call these strings "mirror-image strings."
What about
aabaa? This is not a mirror image string, because we cannot divide it into two equally sized pieces, let alone two mirror-image pieces.

Definition:
mirror-image string: a string that may be divided into halves, the right half consisting of the same sequence as the left half but in the reverse order. For example
aaaa, abba, babbab, and bbabbabb
are mirror-image strings, but
babb, aaab, and bab
are not.

We may group things into discrete sets. Then we may give the set as a whole one name. In our cases we will call the set of mirror-image strings:
set M.
We know that
aa
is included in the set of mirror-image strings M. So we will symbolize their inclusion in that set with the symbol ∈ . So to say that
aa
is included in M, we write:
aa M
Now, we also know that
bb
would be included in M. so
bb M
But since they are both included, we will use "&" to mean "and."
Together we have:
1) aa M & bb M
We will take this as the first line in our formal definition.
Now in algebra we often use variables, such as x and y. These allow us to speak of any possible term that might rightly substitute-in for that variable. In our definition, we will want to speak of all possible terms that could substitute-in for such a variable. This will allow us to show how we may build more complex terms in the set. But note that there are infinitely many possible terms that could be mirror image strings. Just think about the fact that you can keep doubling mirror image strings to produce more that fit the definition:
aabbaa
aabbaaaabbaa
aabbaaaabbaaaabbaaaabbaa
and so on.
This doubling can go on forever. So we cannot actually name every possible term in the set. However, we can still use a "recursive definition" to show how any possible member of the set could be constructed.
So again, we want to talk about variables. And we want to say, "for all possible substitutions for x." Then we would say something about all such substitutions.
We will symbolize "for all possible substitutions for x" as
∀x.
Now finally, we will want to use "if ... , then ..." statements. We wanted to say, "If A, then B" we would use the arrow symbol → and write: "A → B".

Now, returning to our recursive definition, we know that because these are mirror images, the beginning and the end of each have to both be the same letter. So if the string begins with a, it must end with a. And the same for b.

Now, lets consider the variable x. And let's place it between aa and bb. What we get is
axa
and
bxb.
So let's determine the simplest substitutions for x.
Can x be either a or b?
That would produce:
aaa
aba
bab
bbb
All these are odd. So they do not divide evenly into two parts. So instead we need an even number of letters. The smallest even number is 2, so let's look at our options for 2 variables.
aa
ab
ba
bb.
Now let's put them between aa and bb by substituting them for the x in axa and bxb.
aaaa
aaba
abaa
abba
and
baab
babb
bbab
bbbb.
We see that for both axa and bxb, the only two letter substitutions for x that produced mirror images were aa and bb.
These of course are the two we started-with. So the simplest mirror-image strings are aa and bb. And the next simplest are
aaaa
abba
baab
bbbb.
We know that we cannot substitute x with 3 letters, because that would create an odd number, which does not divide equally into two. So what are the possible 6-letter mirror-image strings?
aaaaaa
aabbaa
abbbba
abaaba
baaaab
bbaabb
babbab
bbbbbb
We see then that we still may fit the axa and bxb format. The possible substitutions for x would be:
aaaa
abba
baab
bbbb.
Now in the first case of substitutions for x, we used no more than what we started with: aa and bb. Then we obtained
aaaa
abba
baab
bbbb.
Then in the second case of substitutions for x, we used no more than the above list. We will find that this pattern continues, where to continue to produce new strings, we need only replace xin axa and bxb with whatever we have previously found or constructed in the set.

We also notice in the case of a constructed sequence such as abba, we know that the x for the axa that we used to construct this, which here is bb, also is in the set. And because each internal sequence will have been build from some already in the set, we can say that for sequence in the set, the middle parts (all but the end terms) will also be in the set. For, they are the "building" blocks we used to make that longer sequence.
So we want to define what possible middle parts x belong in set M. So we will say that if middle part x is in set M, then axa and bxb will also be in that set. For this is how we build the sets. We want to say specifically, "First consider all possible substitutions for x. If that substitution for x is in the set of mirror-image strings M, then so too will its next most complex forms, axa andbxb be included in the set." The formal way of writing this is:
2) (∀x) (x M → (axa M & bxb M))
So this will be our second line in our formal definition. The first one was:
1) aa M & bb M
This established our "building blocks." These blocks we used to form the basic strings. And, these blocks recur throughout any other possible mirror-image string. So we call this first line the "base" of the recursive definition.
In the second line:
2) (∀x) (x M → (axa M & bxb M))
we showed how we may recurrently instantiate these building blocks so to create more complex strings. So this second step we call the "recursion step" or just the "recursion." It says that "for any string x if x M is true, then it is also true of the strings formed from x by concatenating an a at both ends or a b at both ends." (182b)
Now, if we only had these two steps, we open possibilities we do not want. For example, we might obtain acca or bllb. But we only want a's and b's. But of course, if we just follow the two steps, we will not have reason to begin to include other letters. Roger Vergauwen explains that really this is a problem in rare cases in mathematics that involve infinite sets. But in the very least, we can say that we want to indicate that there are no more rules that will influence the recursion. This way there can be no doubts.
So we will write a line that restricts these alternate unwanted possibilities. We call it the "restriction":
3. M contains nothing but those members it has by virtue of lines 1 and 2.

All together, we have:
1) aa M & bb M
2) (∀x) (x M → (axa M & bxb M))
3) M contains nothingbut those members it has by virtue of lines 1 and 2.

This was enough to give a recursive definition. The basic steps are:
1) base
2) recursion
3) restriction

We find that most recursive steps say that the second clause can only be true under the condition that the first one is true. We calls these if ..., then ... sorts of statements: conditionals. In recursive steps, usually these conditions use the terms being defined in both the antecedent if clause and the consequent then clause. Yet, this makes the definitions seem circular. So consider for example a possible definition for the term "subset":
For any sets A and B, A is a subset of B if and only if every subset of A is also a subset of B.
Now, that is basically what our second line does:
2) (∀x) (x M → (axa M & bxb M))
It says that all the possible substitutions for x belong to M so long as they already belong to M.
However, we avoid this circularity with the base step. The base step provides the conditions for the conditional statement, which makes it no longer conditional in a sense. We already know that the conditions will be fulfilled, and those conditions are already provided. The conditional sentence form just allows us to make more complex forms to re-input back into the conditional sentence, so to produce even more complex sequences.

We know that this definition is not circular, because we can draw inferences from it. So say we want to prove that abaaba and bbaabb are included in the set of mirror-image strings. Let's follow this proof step by step.
We begin with the base step:
1) aa M & bb M
Then the recursive step:
2) (∀x) (x M → (axa M & bxb M))
Now, because in step 1 we know that both aa and bb are included in the set of mirror-image strings, we know that individually just aa is included in that set. This is a simplification of step 1 (abbreviated Simp.). So:
3) aa M 1, Simp
The "1, Simp" means a simplification of line 1.
Now, we will consider aa as a substitution for x in line two, which was:
2) (∀x) (x M → (axa M & bxb M))
So let's plug aa in for x: We can do that, because our second step speaks of "all x's". So this proposition holds universally for any possible instantiation for the variable x. Hence we call the principle allowing us to make this substitution: Universal Instantiation (U.I.)
4) aa M → (aaaa M & baab M) 2, U.I.
The '2, U.I.' means a universal instantiation of line 2.
Now, consider this argument based on a conditional statement.
If it is cloudy, then the sun won't shine.
It is cloudy.
Therefore, the sun won't shine.
When we affirm the if clause antecedent, we thereby affirm the then clause consequent. We call the principle that allows us to make this inference: Modus Ponens (M.P.). So by affirming the antecedent, we obtain no more than the consequent. Thereby we get:
5) aaaa M & baab M 3,4 (M.P.)
The "3,4 (M.P.)" means that we used line 3 to affirm the antecedent in line 4, and by Modus Ponens obtained just the consequent.
We again have a conjunction by '&.' So we again like in line 3 simplify line 5 to get:
6) baab M 5, Simp.
Now, just like in line 4 where we plugged aa into the recursive conditional sentence, we will do so again for baab to obtain:
7) baab M → (abaaba M & bbaabb M) 2, U.I.
Recall that step 4 was
4) aa M → (aaaa M & baab M)
We notice that we took one part of the statement, the baab, and we plugged it back into that very recursive conditional again. So the same step recurred. Hence the recursivity of this definition.
Now, returning to step 7, which was
7) baab M → (abaaba M & bbaabb M) 2, U.I.
We can affirm the antecedent with step 6, this allows us to derive:
8) abaaba M & bbaabb M 6,7 M.P
We see that abaaba and bbaabb did not appear in the original definition. However, they resulted from the definition. So we can know that this definition is not circular.
But we need both the base step and the recursive step to make this definition productive.

From:
Partee, Meulen, and Wall. Mathematical Methods in Linguistics. Springer, 1990.
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