PEF model

The Polemical Effectiveness Formula (PEF) is an advanced evaluative model designed to quantify the effectiveness of apologetic arguments within structured theological discourse. Developed as a complementary tool to the Kaedah Istifham Mantiqi (Method of Logical Inquiry), the PEF model offers Muslim apologists a systematic approach to assess how effectively their questioning exposes logical inconsistencies, moral vulnerabilities, and foundational clashes in opposing theological positions, particularly during engagements with Christian missionaries. This model helps apologists measure not only the strength of their arguments but also their lasting impact and audience influence, providing a data-driven approach to debate refinement.

Formula

The PEF Score is calculated using the following weighted formula:

${\text{PEF Score}}=({\text{LP}}\times 0.30)+({\text{MV}}\times 0.30)+({\text{FC}}\times 0.25)+({\text{RR}}\times 0.15)$

where:

• ${\text{LP}}={\text{Logical Pressure}}$
• ${\text{MV}}={\text{Moral Vulnerability}}$
• ${\text{FC}}={\text{Foundational Clash}}$
• ${\text{RR}}={\text{Resistance Response}}$

Each variable is rated on a scale from 1 to 10 based on observed debate dynamics.

Scoring guidelines

Each component of the PEF formula is rated on a scale from 1 to 10. The following table guides how to assign these values based on debate performance:

Score	Logical Pressure (LP)	Moral Vulnerability (MV)	Foundational Clash (FC)	Resistance Response (RR)
10	Exposes undeniable core contradictions.	Reveals fatal moral flaws undermining the argument.	Demonstrates total irreconcilability between worldviews.	Opponent resorts to complete evasion or emotional reactions.
9	Forces major concessions due to clear inconsistencies.	Highlights severe ethical double standards.	Shows critical doctrinal incompatibility.	Opponent heavily dodges or shifts topics frequently.
8	Identifies strong contradictions with partial admissions.	Exposes clear moral incoherence recognized by the audience.	Establishes significant theological divergence.	Noticeable rhetorical deflection or metaphor use.
7	Reveals contradictions, though opponent attempts weak justification.	Raises impactful ethical concerns.	Highlights key doctrinal differences, downplayed by opponent.	Mild dodging or indirect answers.
6	Detects minor inconsistencies needing further probing.	Points out ethical weaknesses without destabilizing the argument.	Shows foundational differences with some ambiguity.	Occasional evasive responses.
5	Presents potential tensions but opponent deflects confidently.	Raises peripheral moral issues.	Moderate differences with room for harmonization.	Balanced responses with minor hesitation.
4	Weak logical pressure; opponent responds strongly.	Moral critique lacks significant impact.	Differences are acknowledged but trivialized.	Smooth and confident responses.
3	Minimal challenge; speculative argumentation.	Ethical points appear forced or irrelevant.	Foundational differences poorly articulated.	Fully cooperative opponent.
2	Barely any logical impact.	No meaningful ethical critique.	Fails to highlight core doctrinal differences.	Open and confident engagement.
1	No logical relevance; incoherent questioning.	No moral critique introduced.	No attempt to address foundational issues.	Opponent dominates the exchange confidently.

Derived metrics

The PEF model includes additional metrics to evaluate the broader impact and sustainability of polemical arguments. These metrics expand on the PEF Score by evaluating persuasion likelihood, argument sustainability, force impact, and rhetorical effectiveness.

Probability of persuasion

This metric estimates the likelihood of influencing the audience or opponent based on the PEF Score.

${\text{Probability}}~(\%)=\left({\frac {\text{PEF Score}}{10}}\right)\times 100$

This percentage reflects the likelihood of influencing either the opponent or the audience.

Probability (%)	Interpretation
90 – 100	Highly persuasive. Strong chance of shifting audience perception or forcing opponent concessions.
75 – 89	Persuasive. Noticeable impact on audience understanding; opponent under clear pressure.
50 – 74	Moderate influence. Some audience members may be swayed; opponent remains defensive but stable.
30 – 49	Low influence. Argument delivered but lacks strong impact.
Below 30	Minimal persuasive effect. Requires significant improvement in argument structure or delivery.

Effective Pressure

Accounts for the time decay of argument impact over prolonged discourse or multiple debate rounds.

${\text{Effective Pressure}}={\text{PEF Score}}\times e^{-kt}$

Where:

• $k$ = Decay constant
• $t$ = Time elapsed (in hours) or number of reply rounds

The decay constant $k$ typically reflects the pace and focus of the debate:

• Use $k=0.05$ for standard debates.
• Increase $k$ (e.g., to $0.07$) for fast-paced or shifting discussions.
• Decrease $k$ (e.g., to $0.03$) for formal or highly focused exchanges.

This value shows how much of the initial polemical force remains effective over time or across reply rounds.

• A value above 70% of the original PEF Score indicates sustained pressure.
• A drop below 50% suggests the argument’s impact is fading and may need reinforcement.
• Rapid decay may indicate over-reliance on a single point without follow-up.

Example

If initial PEF Score = 8.0, and after 3 rounds $t=3$, $k=0.05$:

${\text{Effective Pressure}}=8.0\times e^{-0.15}\approx 6.87$

This reflects strong, sustained pressure.

Force Magnitude

Represents the total polemical force exerted, focusing on logical, moral, and foundational dimensions (excluding opponent resistance).

$|{\vec {F}}|={\sqrt {{\text{LP}}^{2}+{\text{MV}}^{2}+{\text{FC}}^{2}}}$

Force Magnitude	Interpretation
>15	Exceptional polemical strength; argument combines high logical, moral, and foundational impact.
12 – 15	Strong force; effective across multiple dimensions.
9 – 11	Moderate force; effective but may lack depth in one area.
Below 9	Weak force; argument likely superficial or unbalanced.

Rhetorical Work Done

Measures the actual rhetorical impact in terms of shifting audience perception.

${\text{Work}}=|{\vec {F}}|\times \Delta B$

Where:

• $\Delta B$ = Belief displacement (rated from 0 to 1)

The value of $\Delta B$ represents the extent to which the audience or opponent's perception has shifted as a result of the argument. It is a subjective measure, but can be reasonably estimated based on:

• Observable audience reactions (e.g., agreement, applause, nodding, or engaged questioning).
• Feedback received after the debate, such as comments, reviews, or informal polls.
• Visible changes in the opponent's stance, including concessions, hesitations, or avoidance.

A $\Delta B$ value closer to 1 indicates a strong rhetorical impact, suggesting that the argument significantly influenced perceptions. A value near 0 reflects minimal impact.

Work Value	Interpretation
>12	Significant rhetorical impact; likely audience shift in perception.
8 – 12	Moderate influence; message was impactful but not transformative.
5 – 7	Limited effect; audience engagement present but shallow.
Below 5	Minimal rhetorical success; argument failed to resonate.

Application

The PEF model is primarily used in conjunction with structured questioning techniques, allowing apologists to evaluate both the immediate and sustained effectiveness of their arguments. By applying these metrics, practitioners can refine their approach in live debates, written exchanges, or post-discussion analyses. Practitioners are encouraged to use this model during post-debate reviews or live tracking with a scoring sheet for continuous improvement.

Practical example

The following example demonstrates how to apply the PEF model in a typical apologetic debate scenario.

Scenario

A Muslim apologist challenges a Christian missionary on the doctrine of Atonement. The following observations are made:

• The apologist exposes a major contradiction regarding justice through innocent sacrifice.
• Ethical concerns about punishing the innocent are highlighted and resonate with the audience.
• The missionary attempts to justify using metaphors but fails to reconcile core differences.
• The missionary frequently dodges direct answers.

Step 1: Assign scores

Based on the scoring guidelines:

• ${\text{LP}}=9$
• ${\text{MV}}=8$
• ${\text{FC}}=7$
• ${\text{RR}}=8$

Step 2: Calculate PEF Score

${\text{PEF Score}}=(9\times 0.30)+(8\times 0.30)+(7\times 0.25)+(8\times 0.15)$

${\text{PEF Score}}=2.7+2.4+1.75+1.2=8.05$

Step 3: Calculate Probability of Persuasion

${\text{Probability}}=\left({\frac {8.05}{10}}\right)\times 100=80.5\%$

Step 4: Calculate Effective Pressure

Assuming $k=0.05$ and after $t=2$ rounds:

${\text{Effective Pressure}}=8.05\times e^{-0.10}\approx 8.05\times 0.9048\approx 7.28$

Step 5: Calculate Force Magnitude

$|{\vec {F}}|={\sqrt {9^{2}+8^{2}+7^{2}}}={\sqrt {194}}\approx 13.93$

Step 6: Calculate Rhetorical Work Done

If audience engagement is high ($\Delta B=0.75$):

${\text{Work}}=13.93\times 0.75\approx 10.45$

Interpretation

• The apologist achieved a strong PEF Score ($8.05$), indicating high argument effectiveness.
• An 80.5% probability suggests strong persuasive impact.
• Effective Pressure remains high after two rounds ($7.28$), meaning sustained influence.
• Force Magnitude ($13.93$) reflects a well-balanced logical, moral, and foundational challenge.
• Rhetorical Work Done ($10.45$) indicates a significant shift in audience perception.

This analysis confirms that the argument was both strategically effective and rhetorically impactful.