➘ #33 Source Article

Bayesian Comparisons

A3. Bayesian Comparison: Likelihood Discipline

Annotation: The likelihood ratio for hypothesis relative to evidence is the probability of observing if is true, divided by the probability of observing if is false.

Annotation: If the ratio is greater than one, the evidence is more expected under than under its alternatives, so it counts in favor of .

Annotation: If the ratio is less than one, the evidence is less expected under , so it counts against .

Annotation: Posterior odds for hypothesis given evidence equal the prior odds multiplied by the product of likelihood ratios across all independent evidence families. This reflects Bayesian updating.

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A4. Application to Atonement Hypotheses

Annotation: This model claims harm (blood) is strictly necessary for forgiveness.

Annotation: This model claims non-harmful governance mechanisms suffice for forgiveness credibility.

Annotation: This model prioritizes union, healing, and restored relationship, not harm.

Annotation: This model emphasizes liberation from oppressive powers, not harm necessity.

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A5. Expected Likelihood Patterns

Annotation: The existence of effective non-harm signals is less expected under the penal-substitution model, so it weakens it.

Annotation: The fact that justice pluralism functions without harming innocents also weakens penal-substitution necessity.

Annotation: These same evidences strengthen the governance sufficiency model.

Annotation: They also strengthen the relational repair model.

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A6. Toy Illustration of Comparative Fit

Annotation: Modal independence evidence favors governance and relational repair over penal substitution.

Annotation: Non-harm signaling strongly disfavors penal substitution and supports governance/relational repair.

Annotation: Justice pluralism further tilts support away from penal substitution.

Annotation: Social-psychological outcomes (favoring restorative approaches) again weaken penal substitution and strengthen alternatives.

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A Fitch-Style Proof.

A1. Scope Fallacy Derivation (Fitch-Style)

Annotation: Inside the cultic or institutional system , the rule applies to all relevant members. This is a bounded, context-specific claim.

Annotation: The system and its rule are divinely instituted. This gives authority to the system but does not automatically make necessary across all worlds.

Annotation: The necessity thesis asserts that holds in every possible world, not just within

Annotation: To validly infer , one would need an additional bridge principle linking divine institution to modal necessity.

Annotation: No accepted principle allows the inference from (“God instituted this rule”) to (“the rule holds in all possible worlds”). Many divine rules are contextual or pedagogical.

Annotation: Since the bridge principle is absent, premises [/latex] do not entail .

Annotation: The conclusion is that necessity has not been established; remains a local rule within .

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A2. Omnipotence–Compassion Coherence (Reductio)

Annotation: Assume omnipotence (no external constraints on divine action).

Annotation: Assume perfect compassion (no gratuitous harm required before reconciliation).

Annotation: Suppose, for contradiction, that forgiving without harm is impossible—that harm is strictly necessary.

Annotation: If harm is necessary, the constraint must be either external (an outside law) or internal (God’s own nature requires it).

Annotation: If the necessity is external, omnipotence is denied.

Annotation: If the necessity is internal, compassion is redefined as harm-first, which contradicts perfect compassion.

Annotation: Either way, the assumption leads to a denial of omnipotence or compassion.

Annotation: But we began with both omnipotence and compassion as true.

Annotation: By contradiction, it follows that forgiving without harm is possible. Therefore, harm is not strictly necessary for pardon.

◉ A plain English walkthrough of the Master Proof above.

1. The Scope Fallacy

We begin with the claim that within a particular religious or institutional system , a rule is enforced. That much is true—it is part of that system’s practice. Even if we add the premise that God himself instituted this system, it still does not follow that is true in every possible world.

Why? Because rules that govern specific communities—dietary restrictions, ritual practices, or even speed limits—do not automatically extend to all contexts. To make that leap, one would need a bridge principle: “If God institutes a rule, then it is necessarily true across all possible worlds.” But that bridge is not accepted. Many divinely instituted rules (such as temporary sacrificial rituals or symbolic acts) are contextual, not universal. Therefore, the inference from “this is a rule within God’s chosen system” to “this is a metaphysical necessity for all worlds” fails. This is the scope fallacy.

2. Omnipotence–Compassion Coherence

Next, consider God’s attributes of omnipotence (, no external constraints on his ability) and perfect compassion (, no gratuitous harm required for reconciliation).

Suppose, for the sake of argument, that harm is absolutely necessary before God can forgive. There are only two ways this could be true:

✓ Either there is some external law binding God, which would contradict (since an omnipotent being cannot be limited by something outside of himself).

✓ Or God’s own nature requires that forgiveness must always be conditioned on prior harm. But that would redefine in a way that contradicts the very idea of perfect compassion.

In both cases, we reach contradictions. Thus, given and , it must be possible for God to forgive without requiring harm. This does not mean God must always do so, only that forgiveness without harm is logically possible. The claim that harm is strictly necessary collapses under its own weight.

3. Bayesian Comparison

Finally, we compare competing hypotheses about atonement using Bayesian tools.

Under Penal Substitution (), harm or blood is strictly necessary. If that were true, we would not expect to see effective non-harmful reconciliation methods in human practice.

But in fact, we do observe such practices: truth commissions, public confessions, restitution, and symbolic covenantal acts. These work without spilling blood.

Evidence families—modal independence (), credible non-harm signals (), justice pluralism (), and restorative outcomes ()—all turn out to be less likely under and more likely under alternatives such as Governance sufficiency () and Relational repair ().

The Bayesian updating shows that posterior odds shift away from harm-necessity models and toward sufficiency models. The evidence cumulatively stacks up against penal substitution being logically or practically necessary.

Putting It All Together

The proof as a whole demonstrates:

✓ Logical error (scope fallacy): Moving from “rule in a system” to “universal necessity” is invalid.

✓ Attribute coherence ( and ): The necessity of harm contradicts divine omnipotence and compassion.

✓ Comparative evidence (Bayesian model): Real-world reconciliation practices make harm-necessity unlikely.

Therefore, the thesis stands: forgiveness without harm is possible, coherent, and better supported by logic and evidence than blood-dependent necessity.

◉ Narrative Summary

The claim that forgiveness must be blood-dependent rests on a fragile chain of reasoning. At its root lies a scope fallacy. Rules that apply within a given religious or institutional system do not automatically extend into every possible world. One may point to a divinely instituted ritual within a covenant, but such rituals have always been contextual and symbolic. To infer from their existence that forgiveness across all realities must require harm is to mistake the local for the universal. Without a defensible bridge principle—one that shows any divinely instituted rule to be a necessary truth across worlds—the claim collapses into an invalid inference.

Even if this gap could be closed, the attributes traditionally ascribed to God further undermine the necessity thesis. Omnipotence rules out external constraints that would bind God to a harm-first framework. Perfect compassion rules out an internal nature that conditions reconciliation on prior injury. To suppose otherwise is to dilute either omnipotence or compassion. A reductio demonstrates the point: assume harm is strictly necessary, and one is forced either to deny God’s omnipotence or to redefine compassion so that it requires gratuitous harm. Both moves contradict the very attributes defenders of harm-necessity wish to preserve. The coherence of omnipotence and compassion together entails that forgiveness without harm is possible.

Beyond these logical considerations, empirical and comparative evidence tilts the scales against harm-necessity. Societies across history have implemented reconciliation without innocent suffering. Truth commissions, public confessions, restitution systems, and covenantal rites all serve as credible signals that reform has occurred. These practices embody plural forms of justice—restorative, distributive, deterrent—without blood. When tested under Bayesian comparison, such evidence families consistently yield likelihood ratios that count against penal substitution while supporting governance and relational repair models. The cumulative effect is decisive: the world we observe is one in which harm is not required for credible forgiveness.

Taken together, the argument unfolds in three stages: first, the alleged necessity is shown to rest on a scope error; second, the claim is made incoherent by God’s own attributes; and third, the comparative evidence of human reconciliation points away from violence and toward sufficiency. Forgiveness without harm is not only possible but also more consistent with divine attributes and human experience. The insistence on blood as a prerequisite is not a universal truth but a contingent tradition, one that can be revised without loss of its pedagogical or symbolic functions.