What can be debated?

What can be debated?

October 20, 2023

Q1: Does debate assume an assertion that can be answered yes or no? #

What does it mean to ask the Q1 to another person? #

Decidable Statements #

Let’s consider a formalization of the idea that there exists a statement with a determinate truth value.

Definitions #

  • Let \( \mathcal{S} \) be the set of all statements.
  • Define the truth value function \( \tau: \mathcal{S} \rightarrow \{0, 1, \text{undefined}\} \) such that:
    • \( \tau(P) = 1 \) if \( P \) is true.
    • \( \tau(P) = 0 \) if \( P \) is false.
    • \( \tau(P) = \text{undefined} \) if \( P \) does not have a determinate truth value.

Main Statement #

The existence of a statement with a determinate truth value can be expressed as: \[ \exists P \in \mathcal{S} \, : \, \tau(P) \in \{0, 1\} \]

We say P is decidable statement.

is the definition truely capture the nature of debate? #

the weakness of the definition #

it assumes the truth value exists for the statement at hand #

What happens when there’s no observable truth value? #

In extreme case, let’s think about the statement “God exists”

Is there a truth value to the question “Does God exist” ?

I don’t know the answer.

  • I don’t know if it is true or false
  • I don’t know if there’s a truth value to this question
    • I don’t know if human has capability of answering the question.
    • I don’t know if the notion of God can be defined
    • I don’t know if the word “exist” can be defined
    • I don’t know if this statement is decidable. ie does P have property of \( \tau(P) \in \{0, 1\} \)

Then we are not allowed to have a debate on whether “God exists” if we use the definition of decidable statement.

So one can say, we have to be silent on this matter. But don’t we get some values out of the discussion on the question? Can you assert the question is not debatable? (Do you know for sure if the question can not be answered?) I say, the act of defining what is debatable defines what the person thinks what debatable is. It doesn’t mean there can be alternative definition.

it forces us to think of truth value #

Suppose a question: Is knowledge-a a common sense? It can be answered yes or no.

But one could argue the is rather inheritantly probablistic, even though people surely associates a boolean value (Yes/No) to the answer.

Ie, assuming truth value forces us to project our probablistic notion to discrete notion without us realizing it.

In another words, when differences arise (such as culture difference, say act of eating dogs), debate forces us to move to true/false dimension from continuous difference dimension.

hard to deal with situations when underlying assumption is different #

When party-a has a statement, party-b might think the statement is not a decidable statement.

It could be that party-b thinks some of assumptions that the statement is assuming is not decidable.

In this case, party-b isn’t able to decide yes / no to the statement. In order to proceed, they have to go over the assumptions of the statement at hand.

But it’s often difficult, because the statement itself assumes the decidability and its hard to debate against of the (often hidden or not-self-realized) assumptions

alternative definition of debate #

Debate is the communication device where two parties exchange their information on a statement.

Where the debate result can be interpreted as the gained insight on the difference of both parties.

Probabilistic statement #

Definition of Assessment Variables #

  • Let \( S \) be a statement or topic of debate.
  • \( I_A \) is a random variable representing party \( A \)’s probabilistic assessment of \( S \). Similarly, \( I_B \) represents party \( B \)’s probabilistic assessment of \( S \).

Associated Probability Distributions #

  • \( P_A(x) \) denotes the probability distribution of \( I_A \), i.e., how party \( A \) assigns probabilities to the various outcomes or interpretations of \( S \).
  • \( P_B(x) \) denotes the probability distribution of \( I_B \).

Entropy as a Measure of Uncertainty #

The entropy of a random variable quantifies the uncertainty or unpredictability associated with its outcomes. Let \( H(P) \) represent the entropy of a probability distribution \( P \). Therefore:

  • The entropy of \( I_A \) is given by \( H_A = H(P_A) \).
  • The entropy of \( I_B \) is given by \( H_B = H(P_B) \).

Here object is not to find \( \tau(P) \), it is to find the difference #

Difference in Expectation #

It is the aggregate measurement of observable in a distribution.

eg, what’s one’s expectation on return on an investment?

Often, the yes/no answer can be viewed as a projection from this aggregate assessment.

eg, Do you think president is doing well? yes/no

the entire probability distribution #

Here one learns about probability distribution of another party. eg, how one assigns the risk on each possible scenario

how one stands different from another #

where one can access another party’s standing on the shared distribution eg, based on risk distribution, one might consider a certain risk value to be too risky whereas another might consider it moderate risk.

Difference in Entropies #

The difference in entropies between the two parties’ assessments can be quantified as: \[ \Delta H = |H_A - H_B| \] This difference, \( \Delta H \), signifies the relative difference in uncertainty between the assessments of the two parties regarding statement \( S \).

It can be thought as a normalized aggregated difference on a statement. Normalized because we are not looking at the value of observables (it is normalized), and aggregated because we are expressing the difference as a single value.

eg, how one might measure riskness of an investment position differently from another (ie distribution is different)

weakness #

Although probablistic statement doesn’t preclude a decidable statement (one can use P ={0,1} distribution), it tends to lead to relativism where there’s no inherent good quality.

It could also lead to simple goal oriented thinking. Because the one thing one can hold onto in this frame is the goal which can’t be argued against. Remember, in decidable thinking, one could have many truth statements that he has to abide by, but here one can forego everything except the goal statement.

I suspect more competitive environment forces one to take the goal oriented thinking. So many goal oriented decision process involves probablistic thinking. (Poker Player, Business owner)

Different dimension of debate #

Note that decidable statement doesn’t preclude a probablistic thinking, ie one can build probablistic statement P and discuss the decidability of the statement. Still, it somehow tends to drive a thought process to a more deterministic thinking.

I find the following contrasts are reminiscent of platonic view vs relativism. It seems A, B are also reminiscent of classical physics vs quantum (?) physics.

decidable statement vs probablistic statement (A) #

axiom vs hypothesis #

Whether one takes the basis of argument as an axiom (a true statement) or as a hypothesis

induction vs hypothetical conditioning (B) #

The bridge between statement is considered decidable or it is a conditional statement based on assumptions.

find a true statement vs find difference #

objectives #

  • discover truth
  • persuasion
  • clarify difference
  • seek common ground
  • entertainment
  • make decision

changing my base assumption is good and hard #

When a person asks a question assuming yes / no, when he gets answer that it can’t be answered that way, or has to be framed differently, he’s actually getting much more information than he expected.

Where he expected to measure the strength of Yes / No in a domain, he’s getting the probability distribution itself needs change.

So when I ask a question what do you think? if someone answers something completely different, it indicates the possiblity of huge information for me.

Yet, I have a tendency to keep my present state. (inertia) Changing assumption takes an energy and we reluctant to do it unless we seek and envision a better return than cost.

Is one perspective better or truther than another? #

How you define ‘debate’ defines your perspective in some sense.

How do I define ‘better’ or ’truther’? (It depends on the context)

How about, is one perspective more flexible than another?

The answer to this is not about the inherent nature of the perspectives in mathematical sense, rather it’s about the manifested nature of them.

Decidable statements tend to be more fixed (truth). But it allows the coexistence of many such fixed statements.

Probabilistic statements tend to be more flexible. Yet it tends to fix a goal statement.

So it’s hard to say which one is more flexible, I can say one is flexible given the assumption (context). This reminds me of phyisical statement on small particles become conditional.

The word we describe something (here flexible, in physics position/speed) is not universal operator. It’s a conditional operator.

Which means …

  • I don’t know what I’m talking about
  • What I say is changable, because the condition we are using might turn out to be too terse.
  • My thoughts may be a just (linear) approximation on something, but this statement itself might be an approximation or not (where you take it as a truth statement). But I never know.
  • which leads to another thinking, as position / speed are entangled, our word / thinking may be entangled
  • and the ’entangled’ itself might be ’entangled’ or ‘xxxed’ (which we don’t have name because we have no concept as we had not known concept of ’entanglement’)
  • We can see ‘cultural relavatism’ but it can also be viewed as absolute relavatism in the sense that you regard each culture has absolute value on its own. Likewise, our word is an approximation to what we think we describe, but the word might look different from what we think it looks. Pictorially, where one thinks a strip which divides two sides, the actual shape might resemble mobius strip which doesn’t.
  • In physics, conservation of momentum needs a frame of reference. We need to make our stand to talk about something.