What is ambiguity? A definition involving philosophical and symbolic logic

(This is an exert from my theory of knowledge. If you are not comfortable with the math, please still do read the very end because I switch back to words.)

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 minus the error of the transaction.
(mtngrl123 pointed out that I forgot to include the positive side of the error +∆E, so I've corrected it below. Thanks mtngrl123 )
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 could 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? (finish later)

Welcome RomBomb
Interesting analogy. A question for you....why is error only considered as subtracting something from a whole. When in relaity error can also mean erroneous information added on. Therefore we must look at not only error in itself but rather a margin of error.

Mental gymnastics...why?
I personally don't believe human interaction can be reduced to math and logic, sorry...

I agree with frmthhrt. Human interaction is fiendishly complex with a nearly infinite number of variables, and it can never be analyzed with complete precision. I don't think it could be figured out through formulas.

One more observation....it is in the differences in interpretations that fresh insights can be found too.

frmthhrt wrote: Mental gymnastics...why?
I personally don't believe human interaction can be reduced to math and logic, sorry...

I like why not questions more than why questions. Don't you?

My method of theorizing helps me learn, which makes me more successful. So I assume it could helps others so that they may be more successful. Do you disagree? If so, why?

Nucky wrote:I agree with frmthhrt. Human interaction is fiendishly complex with a nearly infinite number of variables, and it can never be analyzed with complete precision. I don't think it could be figured out through formulas.

I agree with the first part that "Human interaction is fiendishly complex with a nearly infinite number of variables."

But did I suggest that my theory is precise? If so, please quote me. Oops. I just realized that my Theory of Knowledge (How the mind learns), which defines my method of theorizing, wasn't included in this thread. Here is it: https://docs.google.com/document/d/1zttc-vzxF_Ckx0i40R7v2FkXUZvjjQOQcCwF80Y_Vkw/edit?hl=en_US.

I ask you the same question I asked frmthhrt:

My method of theorizing helps me learn, which makes me more successful. So I assume it could helps others so that they may be more successful. Do you disagree? If so, why?

melodiccolor wrote:One more observation....it is in the differences in interpretations that fresh insights can be found too.

Thank you

mtngrl123 wrote:Welcome RomBomb
Interesting analogy. A question for you....why is error only considered as subtracting something from a whole. When in relaity error can also mean erroneous information added on. Therefore we must look at not only error in itself but rather a margin of error.

Thanks for this great insight. I was thinking the same thing as I was reading the article one last time before submitting it here. But I chose not to fix it because it would take too long. So thanks for keeping me honest.

So this is what I wrote:
D = A - ∆E. This means that the student’s understanding of the teacher’s idea is equal to the teachers idea minus the error of the transaction.

And mtngrl123 is saying that I forgot to include the + ∆E. So the correct statement is:

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.

And I've edited the original post for this correction.

Thanks again mtngrl123. And you said you think you might not be capable (or something like that). Why did you think that?

rombomb wrote:
My method of theorizing helps me learn, which makes me more successful. So I assume it could helps others so that they may be more successful. Do you disagree? If so, why?

I think that it could help some people, as long as they're not taught that it's the only way of learning. I don't think that you're saying that it is the only way of learning though.

Once this gets going, perhaps research could be done on the types of people this method helps and also how and why it helps these types of people.

Nucky wrote:

rombomb wrote:
My method of theorizing helps me learn, which makes me more successful. So I assume it could helps others so that they may be more successful. Do you disagree? If so, why?

I think that it could help some people, as long as they're not taught that it's the only way of learning. I don't think that you're saying that it is the only way of learning though.

Once this gets going, perhaps research could be done on the types of people this method helps and also how and why it helps these types of people.

The research is done. My types will love my method. We are INTPs. Like Albert Einstein, Charles Darwin, and Thomas Jefferson.

And I'm sure that at least 3 or 7 more of the 16 personality types will do well with my method too. Why do I say 3 or 7? Lets consider that a riddle. Who can crack it?

Dominant: Introverted thinking (Ti)

Ti seeks precision, such as the exact word to express an idea. It notices the minute distinctions that define the essence of things, then analyzes and classifies them. Ti examines all sides of an issue, looking to solve problems while minimizing effort and risk. It uses models to root out logical inconsistency.[12] Ti is calm, articulate, and aware of the forces that bind reality together. As introverted Thinkers, INTPs spend the majority of their time and energy ordering the interior, logical world of principles and generalizations in an effort to understand.

Auxiliary: Extraverted intuition (Ne)

Ne finds and interprets hidden meanings, using “what if” questions to explore alternatives, allowing multiple possibilities to coexist. This imaginative play weaves together insights and experiences from various sources to form a new whole, which can then become a catalyst to action.[13] Ne gives INTPs a grasp of the patterns of the world around them. They use their intuition to amalgamate empirical data into coherent pictures, from which they can derive universal principles. INTPs frequently puzzle over a problem for hours on end, until the answer suddenly crystallizes in a flash of insight.

Tertiary: Introverted sensing (Si)

Si collects data in the present moment and compares it with past experiences, a process that sometimes evokes the feelings associated with memory, as if the subject were reliving it. Seeking to protect what is familiar, Si draws upon history to form goals and expectations about what will happen in the future.[14] Si gives INTPs the potential for keen observation. They use this function to gather empirical data, use physical tools, perceive physical relationships, and support their internal logic with a rich sense of space.

Inferior: Extraverted feeling (Fe)

Fe seeks social connections and creates harmonious interactions through polite, considerate, and appropriate behavior. Fe responds to the explicit (and implicit) wants of others, and may even create an internal conflict between the subject’s own needs and the desire to meet the needs of others.[15] Fe drives the INTP to desire harmony in community. At their most relaxed, INTPs can be charming and outgoing among friends, or when they have a clearly defined role in the group. When under stress, however, INTPs can feel disconnected from the people around them, unable to use their extraverted Feeling to reach out to others. As their inferior function, Feeling can be a weak point; when threatened they will hide behind a wall of stoic logic. This can lead them to bottle up their emotions to preserve reason and harmony; but a failure to deal with these concealed emotions can lead to inappropriate outbursts.

Not so simple; did you read the other types, the other complementary functions? Te, Ni, Se, Fi....it is not as simple or clear cut as one site's description of one type. You need to do alot more research into it.

Another riddle (or what ever this is called). What is the primary difference between Einstein, Darwin, and Jefferson? And I'm referring to the way their minds work (think education).

Do research to determine what? Which types would do well learning with my method? I'm sure thats true. I'm just conjecturing.

Or are you saying that I can't be sure that I'm an INTP? I have another source: https://www.mbticomplete.com/ (paid $60).

I've also included my own notes on each bullet point in italics.

INTPs are independent problem solvers who excel at providing a detached, concise analysis of an idea or situation. They have a high regard for competence in themselves and others. They ask the hard questions, challenging others and themselves to find new logical approaches. INTPs are mainly interested in ideas and have little liking for small talk. They value autonomy, home, family, and health. (My brother agreed.)


Characteristics of INTPs
• INTPs are theoretical and abstract, interested more in ideas than in
social interaction. (Yes I am.)
• They seek to develop logical explanations for everything that interests them. (And guess what. Everything interests me. Well not really.)
• They have an unusual ability to focus in depth to solve problems in their
area of interest. (Yes I'm unusual.)
• Their natural skepticism leads them to doubt or question many statements
or ideas until they can be convinced of their logical consistency. (Yes. Although it took me a long time to figure out how to do this in all parts of my life instead of just in the fields that I learned from my education background which was Math, Physics, Chemistry, and Biology.)
• They love to spend time creating models of or forming theories about
complex systems. (Yes I do.)

INTPs with others
• INTPs tend to have a small circle of close friends and enjoy being with
others who like to discuss ideas. (Yes I do.)
• They value people of intelligence and competence. (Yes I do.)
• They love to theorize and discuss abstractions with others. (What do you think?)
• INTPs prize precision in communication and dislike redundancy or stating
the obvious. (Ya but I'm getting betting at realizing when people need redundancy. I measure their facial expressions looking for the confused look. This requires redundancy but with a new perspective. I use Socratic questions to reveal the misunderstanding. Wait thats what this article is about. I'm entered a different sort of redundancy. This is weird.)
• They are usually quiet and reserved, although they can be quite talkative
about areas in which they have a lot of knowledge. (What do you think?)
• They can become so absorbed in an idea that they ignore or lose track
of others. (This is not good for my girls. )

INTPs at work
• INTPs bring great energy, intensity, and focus to researching or analyzing a
problem that arouses their curiosity. (I wrote the first draft of the 'What is ambiguity' article in its entirety in an hour maybe two. I can't remember and I don't pay attention to time that much anymore. Too engrossed to allow myself to pay attention to time.)
• They are more interested in the challenge of solving problems than in seeing
the solutions put to practical use. (Yes I create a successful model and then when its time for implementation, I hire someone to do it because I get bored and I'm ready to go to the next thing.)
• They quickly see inconsistencies and illogic and enjoy taking apart
and reworking ideas. They naturally build complex theoretical systems
and models. (What do you think?)
• Depending on their interests, INTPs may be good at pure science, research,
mathematics, or engineering. (Yes, yes, yes, and no. Engineering bores me. There is no theory involved. Its only practice. Theory is 2nd order knowledge. Practice is 1st order knowledge. 1st order is boring to me.)
• They may become scholars, teachers, or abstract thinkers in fields such as
economics, philosophy, and psychology. (Yep all of these and more.)
• They find it difficult to work on routine tasks and are often not satisfied
managing people. (Yep. I gave up managing people about 10 years ago when I hired a GM. He does it now.)

Potential blind spots for INTPs
• If INTPs have not developed their Thinking preference, they may jump
from insight to insight, never analyzing them with a critical eye or
integrating them into a whole. (I integrating everything into my Theory of Knowledge on 7/21/2011.)
• If they rely too much on their Thinking, they may overlook what they and
others care about. They may decide that something is not important just
because it isn’t logical to care about it. (Like religion or God.)
• If they have not developed their Intuition preference, INTPs may have no
reliable way of taking in information and may become immersed in their
internal logical systems. Then they find it difficult to communicate or act on
their ideas. (I'm not sure what this means. Somebody help me.)
• Also, they are in danger of gaining too little knowledge and experience of
the world. If so, their thinking will be done in a vacuum and nothing may
come of their ideas. (Not me. I was lucky, but definitely bloomed 10 years later than I should have bloomed. Read my theory of knowledge.)
• INTPs want to state the truth precisely, but sometimes they make it so
complicated that not everyone can follow them. (I'm getting better because my writing ability has dramatically increased recently.)
• They often find it hard to express appreciation, which can cause difficulties
in their work or personal relationships. (I'm getting over this. You all have helped.)

I was reflecting on my chat with melodiccolor. I wanted to explain what I meant by 'human language.'

Everyone in my life has always made me feel alien. Every time that I had a human interaction that caused this feeling, I would withdraw. I felt so alone. No one to talk to. No one to relate to. I had no friends. People liked me. But they didn't know what was inside of me. And I of course had no idea what was inside of them.

I've always felt that my mind works more like a computer than a human mind; although that is only because I never met anyone like me.

During my years running businesses, I slowly learned how to relate to people. I started to use 'human language'; language that others understood. The language that I was comfortable with didn't make sense to others. So I felt very alien. It was a chore for me to write like them. I had to do it very very consciously. But the more that I do it, the stronger the habit becomes, i.e. I will have created a habit of writing, and thus thinking, in 'human language.' Habits are saved in the unconscious.

Now I'm trying to rewire my brain so that I may better relate to people. I'm rewiring it for empathy. I'm not suggesting that I could ever acquire empathic powers like some of you, but I do believe that I can get a bit closer to you all, however so slightly. My theory of knowledge explains that rewiring is possible and that it can be fast. Most psychologists would disagree that it can be done quickly, but that is only because they are assuming that it has to happen accidentally and unconsciously. I am doing it very systematically (consciously) so this allows me to do it much faster. Again this is all part of my theory of knowledge.

I'd like to include my description of me that I posted on my facebook page (I wrote this a few weeks ago before coming to this forum):
I've always considered myself a simple guy and I realize that others probably disagree. This introspection is new for me. It is so fascinating. But to me, everything is fascinating, even if only from one perspective; and it is experience and systematic reflection that reveal those perspectives.

Thanks for listening.

I wish you the best of results in this endeavor. It is hard to be so isolated.

melodiccolor wrote:I wish you the best of results in this endeavor. It is hard to be so isolated.

I should have clarified. I don't feel isolated anymore. My empathic teacher/nanny and I have been opening up to each other for 1.5 years now and I've been learning a lot about myself and others and of course how to understand and interact with people. It has even helped me with my relationship with my mom. Which brings me to this story:

When I was 5 or so, I have a memory of my mom not hugging me anymore. I felt so bad. Decades went by. I finally talked to my mom about it. She was shocked. She always thought that I didn't want hugs. She said that I told her that. Of course I can't remember that part. But she was very wrong. That conversation ended with a lot of hugging and crying from both sides. Our relationship is so strong now. This was about 3 months ago now. Now we don't have any trouble talking about our emotions. So guess how I was able to open up to her?

It happened when I showed her my Self-Psychotherapy article (https://docs.google.com/document/d/1oWccP4_X31L_pmeou46aXAwTLW9gU0pfFDnjk9jM53Y/edit?hl=en_US). She cried right in front of me while reading it. Her painful memory was about her mom who died of diabetes and my mom blamed herself for something that happened. A bit of background. My grandma lived with us. Then she decided to go back to Syria because she was depressed here. And my mom blames herself for letting her go. My grandma suffered horribly before her death. And my mom wanted to be there for her and she wasn't.

After the psychotherapy session, she was no longer sad about it. And then immediately after that is when I brought up my painful memory of my mom not hugging me.

We are all better now.