#04-Formalization

August 13, 2025

➘ #04 Source Article

Language — Domains, constants, predicates

Domains: hypotheses $\mathcal{H}={H_1,H_2}$ ; texts/media $\mathcal{M}$ .
Constants: $b\in\mathcal{M}$ (the Bible); evidence proposition $E$ .
Hypotheses:
$H_1$ : “Divine authorship with competence and compassion-orientation” (omnipotence/omniscience, intent to promote human well-being via generally usable, testable guidance).
$H_2$ : “Human authorship constrained by ordinary ancient horizons.”
Core predicates/functions:
$\mathrm{CG}(m)$ : “ $m$ contains concise, generalizable, empirically testable, high-value preventive guidance (e.g., sanitation, water purification, inoculation-like practices, basic disaster risk reduction).”
$\mathrm{Obs}(E)$ : “the observed content state equals $E$ .”
$P(,\cdot\mid\cdot,)$ : standard conditional probability.
$\mathrm{LR}(E;H_1,H_2)=\dfrac{P(E\mid H_1)}{P(E\mid H_2)}$ : likelihood ratio (Bayes factor).

Operationalization of the evidence

Define the explanandum as the specific absence claim:
$E;:=;\neg \mathrm{CG}(b)$ .
Informally: the Bible lacks concise, generalizable, empirically testable preventive guidance whose historical implementation would have substantially reduced needless suffering (while containing only partial, largely ritual or non-generalizable hygiene content).

Bridge principles (predictive commitments)

Guidance-expectation under $H_1$ : If an agent is omniscient and compassion-oriented with intent to assist across eras and cultures, then the probability that its decisive text includes $\mathrm{CG}$ is high:
$\mathbf{BP1:}\quad P(\mathrm{CG}(b)\mid H_1);\text{is high, hence};P(\neg \mathrm{CG}(b)\mid H_1);\text{is low.}$
Constraint under $H_2$ : If a text is authored by ordinary ancient communities without modern scientific theory or population-level intervention models, then omission of $\mathrm{CG}$ is probable:

$\mathbf{BP2:}\quad P(\neg \mathrm{CG}(b)\mid H_2);\text{is high, hence};P(\mathrm{CG}(b)\mid H_2);\text{is low.}$

Formal premises from the paper’s comparative claims

Empirical observation (expectation gap):
$\mathbf{P1:}\quad \mathrm{Obs}(E);\wedge;E=\neg \mathrm{CG}(b).$
Predictive asymmetry (from BP1–BP2):
$\mathbf{P2:}\quad P(\neg \mathrm{CG}(b)\mid H_1)\ll 1\quad\text{and}\quad P(\neg \mathrm{CG}(b)\mid H_2)\approx 1.$
Auxiliary-defense non-improvement (inscrutability, free-will, cultural accommodation, progressive revelation, selective preservation, soul-making do not raise $P(E\mid H_1)$ to parity with $P(E\mid H_2)$ without ad-hoc cost):
$\mathbf{P3:}\quad \forall A\in\mathcal{A},;P(E\mid H_1\wedge A)\le P(E\mid H_1)+\varepsilon,;\text{with};\varepsilon;$ small and outweighed by added complexity.
Here $\mathcal{A}$ is the set of offered theological auxiliaries; their introduction either leaves $P(E\mid H_1)$ essentially unchanged or reduces overall predictive adequacy via penalized complexity.

Likelihood result

By definition,
$\mathrm{LR}(E;H_1,H_2)=\dfrac{P(E\mid H_1)}{P(E\mid H_2)}=\dfrac{P(\neg \mathrm{CG}(b)\mid H_1)}{P(\neg \mathrm{CG}(b)\mid H_2)}.$
From $\mathbf{P2}$ , with $P(\neg \mathrm{CG}(b)\mid H_1)$ small and $P(\neg \mathrm{CG}(b)\mid H_2)$ large,
$\mathbf{C1:}\quad \mathrm{LR}(E;H_1,H_2)\ll 1.$
From $\mathbf{P3}$ , for any auxiliary $A$ ,
$\mathrm{LR}_A(E;H_1,H_2)=\dfrac{P(E\mid H_1\wedge A)}{P(E\mid H_2)}\le \dfrac{P(E\mid H_1)+\varepsilon}{P(E\mid H_2)}\ll 1,$
so auxiliaries do not close the gap without incurring ad-hoc penalties that further reduce overall plausibility.

Evidential conclusion (likelihoodist)

Given observation $E$ and the predictive asymmetry,
$\mathbf{C2:}\quad E;\text{strongly favors};H_2;\text{over};H_1;\text{(i.e.,};\mathrm{LR}(E;H_1,H_2)\ll 1\text{).}$
If one updates priors by Bayes’ theorem, then for any priors with $0<P(H_1),P(H_2)<1$ ,
$\mathbf{C3:}\quad \dfrac{P(H_1\mid E)}{P(H_2\mid E)}=\mathrm{LR}(E;H_1,H_2)\cdot\dfrac{P(H_1)}{P(H_2)}\ll \dfrac{P(H_1)}{P(H_2)},$
so the posterior odds shift toward $H_2$ .

Minimal expansion of $\mathrm{CG}$ for clarity (schematic)

Let $\mathcal{G}$ be a set of cross-culturally feasible, empirically testable, high-impact preventive principles (e.g., sanitation, potable-water protocols, isolation/handwashing, proto-inoculation, basic agricultural rotation, floodplain management). Write:
$\mathrm{CG}(m);:=;\exists S\subseteq\mathcal{G};\big(\mathrm{Concise}(S)\wedge \mathrm{Generalizable}(S)\wedge \mathrm{Testable}(S)\wedge \mathrm{Included}(S,m)\big).$
Observation instantiated:
$E\iff \neg \exists S\subseteq\mathcal{G};\big(\mathrm{Concise}\wedge \mathrm{Generalizable}\wedge \mathrm{Testable}\wedge \mathrm{Included}(S,b)\big).$
Predictive commitments:

$H_1\Rightarrow P(\exists S\subseteq\mathcal{G};\mathrm{Included}(S,b));\text{high};\quad H_2\Rightarrow P(\exists S\subseteq\mathcal{G};\mathrm{Included}(S,b));\text{low}.$

Summary syllogism (schematic, likelihood form)

$\mathbf{S1:};E=\neg \mathrm{CG}(b);\text{is observed.}$
$\mathbf{S2:};P(E\mid H_1)\ll P(E\mid H_2).$
$\mathbf{S3:};\Rightarrow \mathrm{LR}(E;H_1,H_2)\ll 1.$

$\mathbf{S4:};\Rightarrow E;\text{is strong comparative evidence for};H_2;\text{over};H_1.$

Plain English Gloss:

Given that the Bible’s lack of clear, broadly useful preventive guidance is far more expected if it was written by ordinary humans than if it were authored by an all-knowing, compassion-oriented deity, this absence strongly counts against $H_1$ and for $H_2$ , unless one alters $H_1$ with extra assumptions that themselves need separate justification.

Here’s a set of technical explanations and reader-friendly notes for the symbolic logic in the content above, allowing a non-specialist to follow the reasoning without losing the rigor. We’ll move section by section, unpacking the notation and argument flow.

1. Domains, constants, and hypotheses

Technical:
We define a set of possible explanations (hypotheses) $\mathcal{H}={H_1,H_2}$ .
$H_1$ : The Bible was authored (or decisively inspired) by an omniscient, omnipotent, compassion-oriented being intending to promote human well-being with broadly applicable, testable guidance.
$H_2$ : The Bible was produced by human authors constrained by the cultural and scientific limits of their time.
We also define $\mathcal{M}$ as the set of possible texts, and $b$ as the actual Bible in that set. $E$ stands for the observed state of the Bible’s content.

Reader-friendly note:
We start by setting out two competing explanations for where the Bible came from:
✓ Under $H_1$ , it’s the work of a perfect, all-knowing author who wants to help humanity across all times and places.
✓ Under $H_2$ , it’s the work of ordinary people with no special foresight.
We also label “the Bible” as $b$ and “the evidence we see in it” as $E$ .

2. The key predicate: $\mathrm{CG}(m)$

Technical:
$\mathrm{CG}(m)$ means “text $m$ contains concise, generalizable, empirically testable, high-value preventive guidance.”
This includes principles like sanitation, clean water, vaccination-like measures, or other knowledge that—if followed—would prevent large-scale suffering across cultures and centuries.

Reader-friendly note:
We have a yes-or-no test for whether a text contains the kind of practical, universally useful advice that could save lives—things like “boil water before drinking” or “wash hands to prevent disease.” If a text has such advice, it satisfies $\mathrm{CG}$ .

3. Defining the evidence $E$

Technical:
$E := \neg \mathrm{CG}(b)$ means that the Bible $b$ does not contain such concise, universally applicable, testable guidance.

Reader-friendly note:
Our actual observation is that the Bible lacks clear, practical instructions that could have prevented huge amounts of human suffering—no unambiguous, easily testable advice that works across times and cultures.

4. Bridge principles: what each hypothesis predicts

Technical:
Under $H_1$ , the probability $P(\mathrm{CG}(b)\mid H_1)$ should be high because an omniscient, compassion-oriented author would know and want to include this guidance. Therefore $P(\neg \mathrm{CG}(b)\mid H_1)$ should be low.
Under $H_2$ , $P(\neg \mathrm{CG}(b)\mid H_2)$ should be high because ordinary ancient authors lacked the relevant scientific knowledge.

Reader-friendly note:
If God wrote the Bible, we’d expect it to have this kind of advice—so finding it missing would be surprising.
If humans wrote it long ago, we’d expect it not to have this advice—so finding it missing would be exactly what we’d expect.

5. The observed “expectation gap”

Technical:
We observe $E = \neg \mathrm{CG}(b)$ .
From the bridge principles, $P(E\mid H_1) \ll P(E\mid H_2)$ .
This means the likelihood ratio $\mathrm{LR}(E;H_1,H_2) = \frac{P(E\mid H_1)}{P(E\mid H_2)} \ll 1$ .

Reader-friendly note:
We’ve got a mismatch between what $H_1$ would predict and what we actually see: the absence of practical guidance is far more in line with the “human authorship” explanation than with the “God authorship” explanation. This gap is what makes the evidence count strongly in favor of $H_2$ .

6. Considering “auxiliary hypotheses”

Technical:
Some defenders of $H_1$ add auxiliary explanations (e.g., “God chose not to give such guidance to preserve free will” or “God worked within the cultural norms of the time”).
Formally, for any auxiliary $A \in \mathcal{A}$ , $P(E \mid H_1 \wedge A) \le P(E \mid H_1) + \varepsilon$ , where $\varepsilon$ is small and outweighed by increased complexity or reduced independent plausibility.

Reader-friendly note:
Some people try to adjust $H_1$ so the lack of guidance seems less surprising—but these adjustments either don’t change the prediction much or make the explanation more complicated and less believable.

7. Minimal expansion of $\mathrm{CG}$

Technical:
$\mathrm{CG}(m) := \exists S\subseteq\mathcal{G};(\mathrm{Concise}(S) \wedge \mathrm{Generalizable}(S) \wedge \mathrm{Testable}(S) \wedge \mathrm{Included}(S,m))$
where $\mathcal{G}$ = set of universally useful, testable preventive principles.

Reader-friendly note:
Formally, a text has $\mathrm{CG}$ if there’s at least one set of principles that are short, work everywhere, can be tested, and are actually written in it.

8. The conclusion

Technical:
From the likelihood ratio being much less than 1, the evidence $E$ strongly favors $H_2$ over $H_1$ .
Bayesian updating shows that whatever your prior odds were, the posterior odds shift toward $H_2$ .

Reader-friendly note:
Given what we see in the Bible, it’s much more reasonable to think it was written by ordinary people than by an all-knowing, compassion-oriented deity. The evidence pushes our confidence toward the human authorship explanation.

Comparative Table of Symbolic Logic, Plain-English Interpretation, and Real-World Analogies for the ‘God and Needless Suffering’ Argument

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