Scoring System for Arguments: A Hierarchical Approach

Objective:

To establish a scoring system for evaluating arguments based on their hierarchical relationship to a main conclusion. Arguments closer to the conclusion have more weight, with weight halving for each level of removal.

Definitions:

• n: The level of an argument, indicating its distance from the conclusion.

• Level 1: Direct arguments.
• Level 2: Arguments supporting or opposing level-1 arguments, and so on.

N_A,n: Number of arguments for the conclusion at level n.

N_D,n: Number of arguments against the conclusion at level n.

Correct Equation:

The score for the conclusion is calculated using this summation:

$$\text{<span style="background-color: white;">Score</span>}<span\; style="background-color:\; white;">=</span>\underset{\mathrm{<span\; style="background-color:\; white;">n</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}{\overset{\mathrm{<span\; style="background-color:\; white;">\infty </span>}}{<span\; style="background-color:\; white;">\sum </span>}}<span\; style="background-color:\; white;">(</span><span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">N</span>}}_{\mathrm{<span\; style="background-color:\; white;">A</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">-</span>{\mathrm{<span\; style="background-color:\; white;">N</span>}}_{\mathrm{<span\; style="background-color:\; white;">D</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">2</span>}}^{\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">)</span>$$

Explanation:

• The summation (

  ${\sum }_{n=1}^{\mathrm{\infty }}$

  ) iterates over all levels of arguments from the closest (n=1) to the furthest (n approaches infinity).
• At each level n, we find the net number of arguments (for minus against) and multiply it by the weight

  $2^{1-n}$

  (or

  $\frac{1}{2^{n-1}}$

  ), which decreases by half for each successive level.

Rationale for Correction:

• The initial formulation mistakenly treated A_n and D_n as the sum of argument scores rather than counts.
• By correcting this to represent the number of arguments and multiplying by the appropriate weight, we ensure each argument's influence is accurately reflected in the score.

Example:

• Suppose there are 3 arguments for the conclusion at level 1 (N_A,1 = 3) and 1 against (N_D,1 = 1).
• At level 2, there are 2 arguments for (N_A,2 = 2) and none against (N_D,2 = 0).

The score calculation would be:

$$\text{<span style="background-color: white;">Score</span>}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">2</span>}}^{\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">+</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">0</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">2</span>}}^{\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">0.5</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">3</span>}$$

This results in a score of 3, correctly accounting for the weighted contributions from each argument level.

Navigating the Hierarchy of Beliefs: A Score-Based Argument Evaluation System

Here’s a complete, polished version that integrates clarity, accurate mathematical representation, and actionable insights for your audience:

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Belief Score System: Evaluating Arguments

This framework introduces a relational database system to evaluate beliefs and conclusions by scoring them based on their supporting and opposing arguments. Users can submit beliefs as reasons to support or oppose other beliefs, creating a hierarchical structure where conclusions depend on the strength of their underlying assumptions.

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Core Algorithm

Equation #1: Conclusion Score ($CS$)

$$CS(C) = \frac{\sum_{i} \left( LS(A(C, i)) \cdot BS(A(C, i)) \right) - \sum_{j} \left( LS(D(C, j)) \cdot BS(D(C, j)) \right)}{\sum_{i} BS(A(C, i)) + \sum_{j} BS(D(C, j))}$$

Where:

• $C$: The conclusion being evaluated.
• $A(C, i)$: The $i$-th argument supporting $C$.
• $D(C, j)$: The $j$-th argument opposing $C$.
• $LS$: Linkage Score, measuring how strongly an argument supports/opposes $C$ (range: 0 to 1).
• $BS$: Belief Score, calculated recursively for arguments based on their own supporting and opposing arguments.

Base Case:

For root arguments with no supporting assumptions, $BS = 1$ if valid and $BS = 0$ if invalid.

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Explanation

Numerator:

• Weighted difference between supporting and opposing arguments, scaled by their relevance ($LS$).
• If opposing arguments outweigh supporting ones, the numerator will be negative.

Denominator:

• Sum of all argument scores (supporting + opposing), ensuring $CS(C)$ remains between -1 and 1.

Recursive Nature:

• $BS(A)$ is calculated using the same formula, allowing the score to cascade through hierarchies of arguments.

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Example

Conclusion: "It was good for us to join WWII."

1. Supporting Argument ($A_1$): "Nazis committed genocide."
   • $LS = 0.9, BS = 0.95$
   • Contribution: $0.9 \cdot 0.95 = 0.855$
2. Opposing Argument ($D_1$): "War causes many deaths."
   • $LS = 0.7, BS = 0.8$
   • Contribution: $0.7 \cdot 0.8 = 0.56$

Calculation:

$$CS(C) = \frac{0.855 - 0.56}{0.95 + 0.8} = \frac{0.295}{1.75} \approx 0.169$$

Result: $CS(C) \approx 0.169$, indicating moderate support for the conclusion.

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Additional Scoring Features

Uniqueness Score:

To manage redundancy, arguments deemed semantically identical are grouped and weighted to reduce overrepresentation.

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Other Factors Affecting Conclusion Scores

1. Monetary Investment:

Beliefs can receive scores based on collective investment:

$$MoneyScore(B) = \frac{M(B)}{\text{AverageInvestment}}$$

Where $M(B)$ is the money invested in belief $B$, and $\text{AverageInvestment}$ is the total money divided by the number of beliefs.

2. Legal Influence:

$$LegalScore = \frac{\text{Laws Supporting} - \text{Laws Opposing}}{\text{Total Laws}}$$

This normalizes the influence of laws supporting or opposing a conclusion.

3. Logical Verification:

• Verified logical assessments from certified logic professors ($.edu$ affiliations) add credibility to arguments.

4. Media and Cultural Support:

• Media like books, films, and expert opinions are integrated using a linkage score for relevance and quality.

5. Up/Down Votes:

• Users vote on attributes such as logic, clarity, originality, and relevance. These scores feed into $BS$ calculations.

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Practical Examples

1. Cultural Beliefs (e.g., Burqas):

To assess societal norms, calculate the difference between countries enforcing and banning burqas, normalized by the total number of countries.

2. Moral Dilemmas (e.g., Shooting Intruders):

Aggregate state laws supporting/opposing actions like shooting intruders to evaluate societal consensus.

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Potential Challenges

1. Technical Development:
   • SQL and PHP expertise are needed to implement the database and algorithms effectively.
2. Scalability:
   • Managing large, hierarchical datasets and ensuring computational efficiency.

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Call to Action

This system aims to create a transparent, scalable platform for evaluating beliefs and conclusions. With your support, we can build this tool to promote evidence-based reasoning and foster informed decision-making.

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This refined version provides a cohesive explanation, aligning mathematical rigor with practical applications. Let me know if you'd like to focus on specific implementation aspects or provide visual aids for this system!