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.