On Limits

Day 18

The following is adapted from a paper I wrote for a school class once. I've had further thoughts since then, but I wanted to cover the main basis and original thought before getting into that. Normal writing is original writing or adapted from the paper, while writing in parenthesis are some of my current comments.

In a hole in the ground there lived a hobbit, ver. 2

“In a hole in the ground there lived a hobbit.”(referred to as“Iahitgtlah” henceforth), is certainly an assertion. It is the first sentence in the story and book The Hobbit, by J.R.R. Tolkien, and as such is an assertion in a unique position. I intend to speak about “Iahitgtlah” as it is for someone reading the story, not as it is for someone, for example, talking about their pet cat “hobbit” to a friend. When we read a story we tend to read the first sentence then the second, third, and so on. The info contained in the sentences of the story does not follow commutative rules of logic; Reading The Hobbit front-to-back is a different story than reading it back-to-front. Passing someone a bundle of information about a cat is different from telling them a story about that cat. A story is more than just the sentences in it, but also the order in which events take place. It is my hope that “Iahitgtlah” being the first sentence in the story will allow some activities about assertions and conversations to become clearer. (The logic in this paper is time-linear, as it is mostly about or experiences, and our experiences are mostly in time. There are times where we are confused about which event took place first and which second, in a time-linear ordering, but this is not our common experience, and I suspect that only certain types of those events will have an effect upon lim ordering.)

... and also understand that to assert something is to state some information about the world. Assertions are not empty, they always say something, if the listener understand the words. However, if we are to agree with this, we then have a question to answer: Which world? That is, which world is being represented as being in a certain way? One answer to this is the obvious, that all assertions are about the actual world, or at least what we understand to be the actual world. In this sense, “Iahitgtlah” is about the world in which someone is reading The Hobbit. I believe this to be an incomplete description of the sentence “Ithitgtlah” however, and argue that “Iahitgtlah” is about the world of the story of The Hobbit. What “Iahitgtlah” informs us about is what happens in the story. The occurrences of the characters, and the setting of the events. It may be the case that I am reading The Hobbit when I happen across “Iahitgtlah”, but someone else may be reading it as well. Someone has read it in the past, the future, in spanish, or in Braille. For each of these cases, what “Iahitgtlah” informs the reader about is not themselves or anything about the external world, but only about what happens in the story. What language the sentence is in doesn’t change what happens in The Hobbit. Whatever happens to the reader, on earth, or on the moon, does not change the informational content of “Iahitgtlah”. It is the case that “Iahitgtlah” may have been different if history was different, but then so would the story of The Hobbit. Different readers may understand the words differently, but the content is directed at the story itself, not at the world outside the story. More formally, if W is a real world for any case where “Ithitgtlah” and the preceding sentences of The Hobbit exist,  W’ is another real world with a different history, and H* is the world of The Hobbit, then if W={...x...W*...y...}, W’={...x’...W*...y’...}, if H*={“Ithitgtlah”...j...k…}, then “Iahitgtlah” informs you about H* in either W or W’, regardless of the status of x or x’, or whether the actual world is W or W’.  Even if aliens have destroyed New York, if the story of The Hobbit exists in that world, then the unique and informative parts of the sentences that make it up do as well, and vice-versa. They are co-existential. (When I wrote this paper, it was about an interpretation of a paper called “Assertion” by Robert Stalknaker. What I want ot try to argure here is that a sentence in the hobbit stands in the form of an assertions about the world. The sentence in the book asertes that in the world of the hobit, such and such a thing is the case. This becomes more complicated when we get into points of view within the story, but the major paoint of view we are concerned with now is the point of view of the reader of the story. We can certainly talk about something like the world for Bilbo within the world of the story of the hobbit for the reader, but that seems to me to be more of a concern about other minds then it does about limits. It is still relevant, especially because the reader is reading the story and so the world of the hobbit is already inside, so to speak.)

Why is this an important claim? It is precisely because “Iahitgtlah” is the first sentence of the story. H* before “Iahitgtlah” is empty, or at least as empty as any world that is accessible may be. To read a fantasy story for the first time is to enter into a realm of mystery, to not know what the story is about, what words will be used, and what laws will be followed. In normal conversation one can often safely assume that the topic will be relevant, surprises will be few, and the meaning of most words will be known and obvious. It is possible for none of that to be true in a fantasy story. This is made obvious when reading The Hobbit for the first time, as the word “hobbit” is one that is made up. It has no historical basis, and references an imaginary species in an imaginary world. There is a certain lack of background, an absence of presupposition precisely because the author has used a word the reader is presumed to not know. (Empty worlds are very interesting things. While I am not sure how an empty world works or what it looks like, since to exist in the empty world is to have it be not empty and I'm not sure how to describe a world viewed from outside, viewed from reality, I do want to try and capture some of the sense of an empty world. When we look at a book and we do not know what the story is about, we are looking at an empty world, or at least as empty a world as I have ever encountered. It is true that we can assume that the book has paper and ink in it and tells us about something, but we do not know what that something is. An empty page we are writing on is as full of possibility as anything. It has been said that a number of monkeys typing forever on typewriters may end up with every possibe story in existence being written down, but we are more than monkeys. We can come up with our own letters, and our own words for things.)

According to Stalnaker, “Presuppositions are what is taken by the speaker to be the common ground of the participants in the conversation, what is treated as their common knowledge or mutual knowledge”.(321) How can this work for “Iahitgtlah”? How can there be a conversation, or the telling of a story, when both participants do not know the meaning of a word? It seems we can say that the writer and reader have agreed to allow ‘hobbit’ to be undefined for the present time. Stalnaker offers a way to explain this when he says that “It is propositions that are presupposed...but the more fundamental way of representing the speaker’s presuppositions is not as a set of propositions, but rather as a set of possible worlds…”(321) We can understand that the reader may accept that the world of the story has something ‘hobbit’ that ‘lives in a hole’. The exact referent of the word ‘hobbit’ is still unclear here, but the sentence has placed a limit on the acceptable worlds that comprise the story. H* does not contain worlds where no hobbits exist, even if we don’t know what a ‘hobbit’ is or refers to. This is still not the end of things as presumably hobbits could go extinct in the tale, but we shall return to that possibility later. The author eventually defines what a ‘hobbit’ is, but interestingly only after first defining the well known word ‘hole’ in the second sentence of the story: “Not a nasty, dirty, wet hole, filled with the ends of worms….”(Tolkien 1) In fact, the author spends an appreciable amount of time, three pages in my book, on ‘hole’ until he gets to really talking about ‘hobbit’. Why is this? (It seems very important to me that 'hobbit' can be undefined, yet we still recognize that it has a meaning special to itself, like 'successor'. I'll have another piece on this later, but I don't think that using negative definitions helps us much to understand what is going on. Rather, 'hobbit' as written down is something like a shard. Its not a gem until we know that it lives in a hole, and so it must be a shard. It is something that we see, but do not understand. It is a word that we do not know how to use correctly, but that we can still use. It might not be yet part of the English language, but it is already a part of the world.)

I believe this to be because of the author's supplementary sentences about ‘holes’ and ‘hobbits’ to not, in fact, be about “Iahitgtlah”. Rather, they are about H*. Imagine reading a story as a sort of conversation between the reader and the author. According to Stalnaker, “To engage in conversation is, essentially, to distinguish among alternative possible ways that things may be.”(322) We can understand that each assertion S in H*, where “Iahitgtlah” is S and succeeding sentences are S’,S’’,... contain information about H*. They are means of expressing propositions, the purpose of which for Stalnaker is to “make such distinctions”. I put forth that to express a proposition, to make an assertion, is to update the presuppositions of the world. (the presuppositions of the world are not the world itself, but rather what is possible in the world. If the story says that hobbits currently live in holes, and then says that no hobbits exist, then a presupposition, an assumption about the world, appears to be taken away. We flounder in confusion for a while until we figure out how to re-configure the world so that we can more clearly understand it.)

Stalnaker says “The presuppositions define the limits of the set of alternative possibilities among which speakers intend their expressions of propositions to distinguish.”(322) That is, a presupposition is a limit on the world. To presuppose is to believe that the world that the conversation is about, out of all the worlds that the conversation could be about, is not a world where a presupposition is false. Importantly, the ‘worlds the conversation could be about’ are infinite in nature. This is because these limits are lower boundaries on H*, where the expansion of H* means the continuation of reading the story, and can be updated without deletion of possible storylines. For example, even if a character dies, they can come back to life, but they would still have been dead. If they time travel, they still would have time traveled. Another way to phrase this is to say that S establishes a limit on H* where H* must contain ““Ithitgtlah”. S’ expands that limit so that H* must contain ““Iahitgtlah” and S’. The limit of S’ contains certain worlds that were possible for the limit of S, but the limit of S allows worlds where S’ was a different sentence and so H* would have been different. These ‘certain worlds’ that are allowed in S but not S’ does not mean that S’ is smaller than S however. The limit of S’ can be updated by S’’, and S’’’, and so on and so forth for the entire extent of the story of H*. This reflects the intuitive understanding that in each sentence in the story, we learn more about the story of The Hobbit, but  what happens next is unknown. There is an infinite limit of possibility. Thus, S is a lower boundary because it is a starting point for clarification about H*, not an ending point where the story must eventually end at (an upper boundary), or results in a limit on the type, though perhaps not the number, of possible events in the story where only certain things may happen in the story and no more than those things (an encircling limit). S is updatable because S’ is a limit on the worlds in H* that only contains worlds that are allowed under S as well. We can say that if we understand the story so far for H* as we have read in the book, we know what limits the S assertions have for H*. We clearly understand what has happened; where the story has been, where it’s at, but not where it’s going. If we understand reading H* as a conversation, then certain statements Stalnaker makes such as “A conversation is a process taking place in an ever-changing context”, and “To make an assertion is to reduce the context set in a particular way” can be more formally defined as the actions of updating the limits of H*.(323) (The difference between presuppositions and assumption is, now I think, not quite so clear. I think that perhaps a presupposition is before a statement, but an assumption is a limit in a world. I may have been using the wrong word in this paper when I wrote it. 'lower boundary' is italicized because it is an important part of how we understand what is going on in the story. We read a story in a linear fashion, but stories can also wrap around themselves. If we understood a revealed lie or a historical change or time travel in a tale as reversing limits, then we would not have an accurate representation of reading through a story. This is because when we read a new sentence, when we find a new lim in H*, we also have what was written before in our heads. This is all just a fancy way of saying: a story read from front to back is not the same story as when it is read from back to front.)

...Human participants in a conversation may understand limits as being vague or fuzzy. We can accept laws or rules without knowing exactly where the boundaries are - though we often have a tendency to test those rules. We can have an idea of a definition of something but not be able to explain it, as for when we say: ‘I’ll know it when I see it’. ...(This just says that there are ways to speak which are not rigid. I used the word 'limits' because I think that these limits tend to act like limits in calculus, where you can get very, very close to them but not quite touch. There are almost always ways to re-interpret a sentence to mean something different, or to understand that a sentence, like a poem, may say more then you thought it did the first, second, or hundredth time that you read it.)

...to say something that all participants in the conversation already know. You can say a presupposition, or have an unstated assertion be a presupposition. To explicitly state and bring out something that was already presupposed in the conversation could be done for many reasons ... This doesn’t mean that any particular sentence is an updating S, but possibly that a particular idea is an updating S. The words of the sentence are less important than the meaning or content of the sentence. A sentence “Iahitgtlah” may have a very different result on the limits of H* at the beginning of the story compared to the middle of the story, as the limits on H* that result from S =/= LimS’, even if the words of S and S’ are the same. ...(This just says that if you repeat something, if you say “Stop! Stop! Stop!”, then each word is a distinct Lim. Each word reinforces the message or suggests that there should be something more to understand about the message. That is, we sometimes have reasons for repeating ourselves. I included this bit in here because at this point the paper started focusing more on Stalnaker, and the worry about re-using words was a noticeable part of his paper. )

...If the limits are vague but work well enough to result in a H*, then H* is also vague, but works well enough. So what is working well enough? Working well enough that any truth value gaps can be ignored. I explained earlier that I interpreted this statement as possibly being the claim that limits resulting from an S are vague or fuzzy. This could simply be because humans are limited minds, but this could also be for the practical purpose of getting along with the work. In conversation it is often worthwhile to accept a statement even if you are unsure what it means, and figure it out later. The unknown piece may be explained in a limit update, or be unimportant for understanding the meaning of any future limit update. Two things may be said about ‘unknown pieces’ here. First, that they are almost always not completely unknown. The sentence: “FRI% )DSL, ^$K!” is certainly a much stranger sentence than something like “See the S-olmo run”. Secondly, those sentences can hide unknowns. “Ithitgtlah” for example contains the unknown piece ‘hobbit’, but other known pieces like ‘hole’ and ‘lived’. It is graspable, and works as an S that places a limit.  As long as we accept the unknown and don’t demand an immediate investigation into it, we can proceed in the conversation. We can fill in background, but we rarely feel the need to tie up every loose end. If we did not do this, we would exclude all sorts of possible statements. How would we say something new, or talk about what was over the next hill? ...(The way we use limits is pragmatical, not rigidly logical. Thus comes from the idea of limits as being like or similar to in some way as I understand them to be, limits in calculus as well, or numbers in a number line. If we imagine two limits or two numbers, then there is always a limit or number which can lie between those two points. If everything is a unique line, then there is always a unique line in between those lines. The new line is possible, and so if we didn't practically ignore that, then a new sentence in a book could mean almost anything, if we stretched our minds far enough. Stalnaker takes about certain conditions for rational communication, and my talk here about our tendencies in speech with each other is an attempt to work out what some of those ideas might be.)

[Stalnaker] “One must be able to tell what a statement says independently of any facts that might be relevant to determining its truth.”(327) This could be understood as the idea that the H* that S is about need not be the actual H*. Imagine if you will, another book ‘The Hobbit’ that was a different story than H*. (Perhaps the readers copy is a misprint or a fake) Call this new book the Verified H*(VH*). That the reader thinks S is about VH* but S is actually about H*, does not have any bearing on the meaning of S. S says what it says regardless of the perceived status of what S talks about. One could say that H* is the totality of all S in H*, so that if another reader reads S and thinks of a limit on VH*, then S expresses the same limit in both cases, but the fact of a particular H being in that limit may be different in different worlds. There is an interesting aside to this, when Stalnaker talks about Sherlock Holmes. He says, of the statement “Sherlock Holmes does not exist”, that “Then, the proposition will be necessarily false...since the domain of no possible world contains the actual person we call Sherlock Holmes”.(330) I generally agree with him when we talk about Sherlock Holmes the actual person, but Sherlock Holmes the character is a different beast. (This was one of the more confusing bit of the paper I think. This may be partly because Stalnaker talks about the meaning of truth, and I''m not entirely certain what it is to be true. A truth-value does not seem to be the same concept, though its commonly understood that they are somehow linked together. In the context of how I am talking in this paper, truth si probably something about a certain type of link between the statement and the matters of fact, or what is the case. This is obviously complicated by by assumption of belief in everything. For example, if you believe everything, then something like justified belief is not bound to lead to truth. As well, I stated that you can justify everything. Hence my talk about the existence of Sherlock Holmes, and my general idea that “That the reader thinks S is about VH* but S is actually about H*, does not have any bearing on the meaning of S.” We don't know what reality is, and statements about H* are statements about the world, not about reality. So, VH actually has no bearing on the meaning of H*, unless we change world, so that when we talk about H* we really intend to be talking about VH* - but by the way that worlds work, if VH is not in the world, then we cannot know we are talking about VH. If VH is in the world, then everything we say still makes sense and has the same meaning, but is a lie.)

All worlds in which H* exists, we can imagine that hobbits exist. By the third principle, the meaning of ‘hobbit’ does not depend on the status of any particular H. Any proposition S that talks about a hobbit creates a limit on a H where ‘hobbits exist’. So any assertion about hobbits asserts that hobbits exist. What about an assertion that ‘hobbits do not exist’? It is true that that creates an H without hobbits, but if the reader understands that assertion, he then understands something about hobbits - that they do not exist in H. Without knowing what hobbits are or what the word refers to. They have a negative existence. They can be said to have the same sort of existence that a “a nasty, dirty, wet” hobbit hole would have. As we recall the second sentence S’, that the hole the hobbit from S lived in was “Not a nasty, dirty, wet hole, filled with the ends of worms…”, we can see that the  “a nasty, dirty, wet'' hobbit hole is in a world between S and S’, perhaps one where Bilbo was a worm or a rabbit. After the reader of The Hobbit reads S, it could be the case in H* that there is such a hole. After the reader reads S’, it could not be the case for H*. The existence of hobbits, in a H where hobbits do not exist, is in a world before any S limitation on H. In fact anything at all might exist in the worlds-under-all-limits. Principle three suggests there may be a way of understanding this space as necessary for S to have meaning, that S may not result in a limit sprung full-grown, but a limit update of some sort. Perhaps we carry presuppositions into all H’s, though this of course leaves us asking how we form the first limit or presupposition necessary for understanding our first S. What can we learn from silence? (This may also be where we get into talk about the difference between presuppositions and assumptions. Presuppositions may be in silence, for example. When I talk about 'without knowing..hobbits..”, then what I think I am doing is taking grief with te idea that unicorns do not exist. I think unicorns do exist, like Santa

Leaving hobbits behind, we can use our new tools to review two conundrums. First, how does ‘Damn John got the promotion’(DJ) differ from ‘John got the promotion’(J)? We can deduce J from DJ, but not DJ from J. Consider DJ and J as telling us about some set of possible worlds W, so that DJ and J both create a limit on W. DJ contains more information about W than J, if ‘Damn’ contains info, so DJ is in the position of S’ when J is in the position of S. DJ must be in the position of S’ because it contains more information about W than J, so the limit of J contains worlds that are not in the limit of DJ. If DJ is like S’, then it is an update on and S like J. Remember that limits are only updated, not destroyed, so if we remove some information from DJ, we have the result of some new limit that was contained in DJ, which is J. We could also remove different information from D and have another statement like K (Damn John). It is possible for an S’ and an S to not be related in this way if they talk about unrelated things, but because they talk about an H, S’, if you understand the limit on H fully and clearly, no matter how much information you remove, will never result in some S* that is outside the limits of S. Misunderstandings of that sort result from misunderstandings of the H and the limits on H that S’ puts forth, not from any misunderstanding of the words of S’, but from misunderstanding the meaning of S’. In the same way to understand the relation of DJ and J, we need to understand the W they are about. 

Our second conundrum is understanding the difference between “Fire is hot”(F) and “It is true that Fire
is hot”.(TF) Both of them result in the same limit on a set of possible worlds. Where they differ is that TF expresses a presupposition of F, in that TF says F, essentially saying F again. We can look at this event through the view of the first principle of rational communication, the presumption of nontrivial. We might then say that the first principle says that if we follow the rule, there is some unstated S that expresses a limit that is a part of any limit in rational conversation we engage in, where we attempt to understand the worlds that the conversation is concerned with as being inside that limit. We could further say that this is some STF limit that is already in F, in a conversation. This extension S however only exists in a conversation. Outside of a conversation it may not exist, as if I shouted it to the empty sky. In like fashion, if TF is presupposition of F, then we should, by removing information from F, when F is a S’, have the result of a S like TF. So what happens if we remove the information in F? Well, we remove the information from what - from the world that F is about. By removing the words we are left with the context, and the rules of the world F is about. If F is about an empty world of empty sky and empty words, then there is nothing left. If F is about a world of conversation, then we are left with the rules of conversation. We could say that personal analysis of conversational rules is normally done by reviewing ideas about what we should and shouldn’t say, or by looking at examples of proper and effective engagement. Perhaps we can try being silent or saying nonsense next time we are in a rational communication and then analyze the awkwardness and empty space to figure out the rules. As far as F and TF are concerned, both sentences will mean the same thing, but they will have different meaning in an utterance in a conversation, as for the listener, understanding TF when they already know F will result in a search for a different limit on the world of conversational concern for TF than F, because of principle one, the presumption of non-triviality in a rational communication. (I;ll fix the spacing later. For now – I included these two paragraphs in the paper to give examples of actually usign my ideas to understand something. I called these two issues 'conundrums', and as I rememebr my tacher, marking up the page, noticed that I didn't actually tall us why these conundrums were conundrums. This is a reminder that it's usually good to have someone critique your papers really hard before you submit them, because as the riter you'll just glance over things that you think are obvious. In retrospect, I feel that the reason I called these conundrum was because I had read or heard somewhere about thse being problems, and so I just assumed for some reason that what I was talking about was obvious. It wasn't. Anyways the two conundrums I was trying to get ate were first the issue of the use of adjectives and truth values. See, 'J is X' and 'That damn J is X' seem to be both true, if J is X is true, but say different things. I tried to get around this by taking issues of truth out of the equation, as I'm not sure what truth is, and having the statements be about limits on worlds instead. The second conundrum is the idea that if I say 'X' and I say 'X is true', then I say different things, bt I assert the same thing to be true. Again, this is because truth is, on some views, supposedly about a link between the statement and a state of reality. It is an issue of how a truth can take different forms. I tried to get around this by making the issue about worlds instead of reference cases in reality. We can see that both these 'conundrums' are pretty similar to each other, in that they are 'conundrums' because we can say things in different ways. If there is only one reality and one perfect way to say something is true, then it would appear that, if we can say the truth in different ways, that fictions, or a world, is more complex than reality. Either that, or we don't understand what it means for a statement to be true. They are conundrums because they are confusing under certain definitions of truth. If we understand them to be limits, then this is obviously the case, and they become understandable again. Three final statements: One, when I wrote this piece my teacher said that I hadn't properly understood Stalnaker's rules of rational conversation. He may have been right, but what we can also get out of that concern is an understanding that the basic ideas about words as limits on worlds can be separated from the rules of rational conversation. This is probably work for another day, but my formal works are probably a combination of two types of rules. The basic ones, and the ones for rational conversation. I should disentangle them at some point. Secondly, the general idea for his talk about limits in worlds is, I suspect, actually just another version of two-dimensional semantics. Thirdly, I made some attempts at formalizing this:

-Partial Formalization-

H = Set of possible worlds

W = a world

H* = The H of the book The Hobbit. How the story might turn out. 

S = a sentence about H

LimS = result of applying S to H

H* =:= {S,S’,S’’,...}

S = S iff S:H->H in S

W = {...x...H*...y...}

W’= {...x’...H*...y’...}

H* in W = H* in W’

{S,S’,S’’,...} in W = {S,S’,S’’,...} in W’

S: H* -> H* in LimS

S’: H* -> H* in LimS’

For H, S =/= S’, and LimS’=(LimS + LimS’)

LimS in H* = LimS in VH* iff S=S 

DJ =S’ and J=S if W=H

LimF = (LimF + LimT) = (LimF + LimT + LimS) for W when W = a rational communication

F - F = {∅}

LimF - LimF = (LimT + LimS) in W

Works Cited

Tolkein R., J R. The Hobbit, or, There and Back Again. Ballantine Books, 1982. 

Stalnaker, R. C. (1978). Assertions. In P. Cole (Ed.), Syntax and Semantics 9: Pragmatics (pp. 315-332). New York: Academic Press.