Tuesday, March 26, 2013

What is ambiguity?

This essay is based on this theory.

What is ambiguity?

Or rather, how can one be conscious of ambiguity so as to prevent it from slowing down the educative process? Some dictionaries do not define ambiguity appropriately. They define it as ‘uncertainty in interpretation.’ The problem with this definition is that it does not express the quantitative nature of the term. A more appropriate definition which some dictionaries give is the ‘ability to express more than one interpretation.’ This definition is more accurate because it clearly expresses the quantitative aspect of the term with the phrase, ‘more than one.’ So lets further define the term ambiguity qualitatively so as to further reduce the ambiguity of the term ambiguity. Lets first employ philosophical logic then we’ll convert the philosophical statements into symbolic form.

Consider a situation in which a teacher teaches a subject matter to her student; this is a communication between two people, a transaction. A transaction has multiple events that occur against it. In this situation, the first event occurs when the teacher chooses an area in her knowledge network to teach. Note that some of the points and vectors in this area of her knowledge network are likely to be somewhat misaligned as compared to the Universe’s knowledge network; this difference is what Mathematicians call error. The next event occurs when the teacher translates her idea into words; and since no human is perfect, there is error here too and this is what we call ambiguity. Then the teacher speaks her idea to the student; and since no human is perfect, there is error here on the part of the teacher and on the student. This error is of many types; the teacher’s slurring of her words, the student’s lack of concentration, etc. Then the student attempts to make sense of the teacher’s statements and converts them to an idea; his version of her idea. There is error here too. At this point, the teacher must work towards decreasing this error. She asks the student a line of Socratic questions whose answers will provide the teacher with evidence as to what degree the student understood her idea. The questions serve to decrease the error in understanding by cyclically chipping away at the error through measurement. The students answers are the measurements while the teachers questions are the measuring devices. Lets now employ symbolic logic to define the objects in this transaction:

X = Teacher (female).
Y = Her student (male).
A = X’s idea. This is represented by a very small localized area of X’s knowledge network.
B = The Universe’s version of A.
C = X’s translation of A to English.
D = Y’s understanding of C.
∆E = The error between D and A, i.e. this is his misunderstanding of A. (∆ is pronounced delta and it means difference which in our case means error.)
Note that error can not be completely eliminated; it can only be minimized through the use of Numerical Methods. This is an axiom I learned from the field of Chaos Theory, a branch of Numerical Methods. This means that ∆E > 0.

Q = X’s questions to Y in order to determine D.
P = Y’s answers to Q

What is the object of the end goal of this transaction? It is D; the student’s understanding of A, the teacher’s idea. What is the end goal? We expect D to be as close to A as possible. So what is D’s relationship to A?

D = A +- ∆E. This means that the student’s understanding of the teacher’s idea is equal to the teachers idea plus or minus the error of the transaction.

D and A are trivial. That leaves only ∆E which is the error of the whole transaction. Since the transaction is composed of many events, the transaction error is the sum of the error from the events. Lets define the events and some attributes of the objects:

1. X thinks of A to teach to Y.
∆A = The error in X’s understanding of B. This is one of the terms that make up ∆E.

2. X converts A into its English language equivalent dubbed C.
∆C = The error in X’s translation of A to C. This is the ambiguity. This is a 2nd term that makes up ∆E.

3. X speaks C to Y.
∆S = The error in X’s speech to Y, as in the slurring of her words. This is a 3rd term that makes up ∆E.
∆H = The error in Y’s hearing of X, due to the lack of concentration. This is a 4th term that makes up ∆E.

4. Until ∆E ≃ 0, (The symbol ≃ means almost equal to.)

i. X asks Q to Y in order to determine D.
∆Q = The error in Y’s understanding of Q. Note that this is a sub-transaction in that it could contain more than one event and so each event comes with it another error term.

ii. Y answers X with A.
∆P = The error in X’s understanding of P.

So now lets use all the error terms we’ve just defined to determine ∆E.
∆E = ∆A + ∆C + ∆S + ∆H

Lets consider these error terms. Which of the 4 types of error do we as teachers have direct control over? Only ∆C and ∆S. But ∆S is trivial; the simple rule is to enunciate your words. This leaves us with only ∆C. Note that when the teacher translates her idea into words, she must realize the fact that the student’s knowledge network is quite different than hers. This means that any word in his vocabulary, which is part of his knowledge network, could have a slightly different meaning than the same word in her vocabulary, which is part of her knowledge network. Thus any one of her statements can be misunderstood by him. Lets dig deeper. Lets define some objects and their attributes of the event that results in ∆C:
Ci = A statement from C.
n = The number of statements in C.
As an example, if n = 3, then C = (C1, C2, C3).

Cij = one possibility that Ci can mean.
m = The number of possibilities that Ci could mean.
If m > 1, then Ci is dubbed ambiguous, meaning the statement could be interpreted in more than one way.
As an example, if n = 1 and m = 4, then C1 = (C11, C12, C13, C14).

Ui = Y’s version of Ci.

Lets assume, for simplicity, that the student only considered one of the many interpretations, i.e. he made an unconscious assumption, i.e. an assumption in which he was not aware that he was assuming because he could not imagine the other possibilities.

∆Ui = The error in Ui as compared to Ci. His misunderstanding of Ci due to the ambiguity of the teachers statement. To reiterate, this error is irrespective of the other types of error, ∆A, ∆S, and ∆H.

Therefore Ui = Ci - ∆Ui

So the total ambiguity error of the transaction, ∆C, is the sum of the ambiguity error of each event in the transaction, i.e. each statement in the communication:
∆C = ∆U1 + ∆U2 + ∆U3 … + ∆Un

So how do we decrease ∆C, the ambiguity error of her entire argument? It seems that we should decrease m, the number of possible interpretations of a Ci, the teacher’s statement. Or rather, we should decrease the average m across all the statements of a communication; lets dub this mAve. What happens if we decrease mAve to almost 1? Then we will have practically removed all error in ambiguity of the teachers’ argument, ∆C. But what if mAve is large enough to cause a large ∆C? Lets consider a statement in which m = 2. The teacher asks a Socratic question with the intention of revealing the difference between the 2 possibilities, i.e. the error, thereby removing all error in ambiguity of the teacher’s statement, ∆Ci. How far can this be taken? Or rather, how high can mAve reach while the teacher still retains the ability to use a Socratic line of questions in order to reduce the ambiguity to practically zero? Well that depends on how powerful the Socratic line of questions is.


What is assumption?

Continuing with the same situation, can the student be trained such that he does not assume thereby causing him to ask questions himself rather than relying on the teacher to expose the misunderstanding? Remember that Ui is Y’s version of Ci and that we assumed for simplicity, that the student only considered one of the many interpretations, i.e. he made an unconscious assumption, one in which he was not aware that he was assuming because he did not imagine the other possibilities. Why doesn't a student imagine the other possibilities? It is because he has not yet learned the logic of assumption. Lets create a similar situation mimicking the previous one but with one change; the student does not assume and instead realizes the other possibilities. At this point, the student is to ask the questions while the teacher answers in such a way to reveal ∆E, the difference between D and A. Note that the teacher does not necessary need to answer the questions with answers. Instead she can use questions as the answers, thereby allowing the student to derive the answers himself, and since the student knows better than the teacher about what he understands, he is more likely to produce more appropriate questions to more accurately reveal the difference between D and A. Therefore the student’s line of Socratic questions to decrease ∆E would be a shorter list of questions as compared to the teacher’s line of Socratic questions. In other words, once the student learns the logic of assumption, then the entropy of the educative process is further decreased and thus learning occurs faster. So how does the teacher teach the child the logic of assumption? (flag till later)


(later) This was my confusion as I defined it in my theory of knowledge. This confusion is the problem.

The problem: How do we learn to minimize assumptions?

The solution: Learn philosophy by reading and discussion.

I wrote this in another thread a few days ago:
Before coming to this site, I was a pretty good thinker, but not really. I used to make thinking mistakes like employing empiricism, reductionism, anthropomorphism, justificationism, etc. And while I was making these thinking mistakes I had no idea that I was doing this of course, so many [maybe most] of the conclusions that I would draw were wrong [and many of these were incorrect assumptions]. And since I've been on this [BoI] email list I've learned what these things are and so now I don't make these thinking mistakes as much. And I can also notice them in other people's arguments, although this is still very limited too. The other people on this site are still finding my thinking mistakes. And each time that they reveal one of my thinking errors, I learn that thing even more, and so my thinking skill improves.

So now I know what philosophy is. Its the practice of thinking without thinking mistakes in a way that tries to reduce mistakes.

Join the discussion group or email comments to rombomb@gmail.com

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