#22-Formalization

August 28, 2025

➘ #22 Source Article

Symbolic Logic Formalization

$\forall x (Rational(x) \to Belief(x) \propto Evidence(x))$
Annotation: For any entity $x$ , if $x$ is rational, then the degree of $Belief(x)$ must be proportional to the degree of $Evidence(x)]$ . Rationality requires beliefs to scale with evidential support.

$\exists y (Faith(y) \to Belief(y) > Evidence(y))$
Annotation: There exists some $y$ such that if $y$ holds faith, then $Belief(y)]$ exceeds $Evidence(y)]$ . Faith is defined as overbelief—belief stronger than what evidence justifies.

$Rational(God) \wedge Commands(God, Faith) \to \bot$
Annotation: If $God$ is rational yet commands faith, a contradiction arises. A rational being cannot coherently demand belief that surpasses evidence.

$\forall x (Faith(x) \to Obedience(x) \wedge \neg Inquiry(x))$
Annotation: For any $x$ , if $x$ adopts faith, then $x$ is disposed toward $Obedience(x)]$ and away from $Inquiry(x)]$ . Faith functions as a mechanism of submission rather than rational investigation.

$Benevolent(God) \to \forall p (Person(p) \to Accessible(Salvation, p))$
Annotation: If $God$ is benevolent, then for every $Person(p)]$ , salvation must be accessible to $p]$ . A universally benevolent deity would not arbitrarily restrict access.

$\exists q (Person(q) \wedge \neg Accessible(Salvation, q))$
Annotation: There exists some $q$ such that $q$ is a person and salvation is not accessible to $q]$ . This reflects the exclusivist doctrine that denies salvation to many.

$Benevolent(God) \wedge Exclusivity \to \bot$
Annotation: If $God$ is benevolent, but salvation is exclusive, a contradiction follows. Benevolence is incompatible with exclusion of many from salvation.

$\forall x (Test(Faith(x)) \to Loyalty(x) \wedge \neg Rational(x))$
Annotation: For any $x]$ , if $Faith(x)]$ is required as a test, then it demonstrates $Loyalty(x)]$ but also entails $\neg Rational(x)]$ . Loyalty through faith comes at the cost of rationality.

$Rational(God) \to \neg Commands(God, \neg Rational(x))$
Annotation: If $God]$ is rational, then $God]$ cannot command irrational belief. To do so would undermine divine rationality.

$\forall c, p (c \neq p \to E[BS(c)] - E[BS(p)] = (c - p)^2 > 0)$
Annotation: For any credence $c]$ and evidential probability $p]$ , if $c]$ does not equal $p]$ , then the expected Brier score $E[BS(c)]$ is worse than $E[BS(p)]$ by $(c - p)^2]$ . Beliefs that depart from evidence are strictly less accurate.

$EU(Belief) = \sum_i P(H_i) \times Payoff(H_i) - Cost(Error)$
Annotation: The expected utility $EU]$ of holding a belief equals the weighted sum of possible payoffs under hypotheses $H_i]$ , discounted by the costs of error. Properly calibrated belief maximizes expected outcomes.

$Belief(x) > P(x) \Rightarrow EU(Belief) < EU(P)]$
Annotation: If $Belief(x)]$ is greater than the evidential probability $P(x)]$ , then the expected utility of believing is lower than the expected utility of proportioning belief to $P]$ . Overbelief yields predictable practical disadvantage.

$H_H : \text{The Bible is primarily a human-authored text}$
Annotation: Hypothesis $H_H]$ asserts that the Bible originates chiefly from human authors.

$H_D : \text{The Bible is divinely authored or directly guided by God}$
Annotation: Hypothesis $H_D]$ asserts that the Bible is authored or directly guided by a rational God.

$E : \text{Observed valorization of belief beyond evidence}$
Annotation: Evidence $E]$ is the recurring biblical promotion of belief that exceeds evidential support.

$LR = \frac{P(E \mid H_H)}{P(E \mid H_D)}$
Annotation: The likelihood ratio $LR]$ measures how much more expected the evidence $E]$ is under human authorship $H_H]$ versus divine authorship $H_D]$ .

Illustration: If $P(E \mid H_H) = 0.8]$ and $P(E \mid H_D) = 0.1]$ , then $LR = 8]$ .
Annotation: The evidence is eight times more likely under human authorship than divine authorship, favoring $H_H]$ over $H_D]$ .

Final Conclusion:

$(Rational(God) \wedge Benevolent(God)) \to (\neg Commands(God, Faith))$
Annotation: If $God]$ is both rational and benevolent, then $God]$ would not endorse faith understood as belief beyond evidence, since such endorsement leads to contradiction, diminishes accuracy, and predictably worsens human decision-making outcomes.

A Fitch-Style Proof.

∀x (Rational(x) → Belief(x) ∝ Evidence(x)) Premise
Annotation: Rationality requires beliefs to be proportional to evidence.
∃y (Faith(y) → Belief(y) > Evidence(y)) Premise
Annotation: Faith entails overbelief, i.e., belief exceeding evidential warrant.
Rational(God) Assumption
Annotation: Assume God is rational.
Commands(God, Faith) Assumption (for conditional proof)
Annotation: Assume God commands faith.
Faith(y) → Belief(y) > Evidence(y) From (2), existential instantiation
Annotation: For some entity y, faith requires belief beyond evidence.
Commands(God, Faith) → Faith(y) From (4), specification
Annotation: If God commands faith, then y is enjoined to hold faith.
Faith(y) From (4) and (6)
Annotation: By assumption, faith is commanded.
Belief(y) > Evidence(y) From (5) and (7)
Annotation: y’s belief exceeds evidence.
Rational(God) → (Belief(y) ∝ Evidence(y)) From (1), universal instantiation
Annotation: If God is rational, commanded belief must track evidence.
Belief(y) ∝ Evidence(y) From (3) and (9)
Annotation: Rationality demands proportional belief.
Belief(y) > Evidence(y) ∧ Belief(y) ∝ Evidence(y) From (8) and (10)
Annotation: Contradiction arises: belief both exceeds and matches evidence.
⊥ From (11)
Annotation: Logical contradiction.
¬(Rational(God) ∧ Commands(God, Faith)) From (3)–(12), conditional proof
Annotation: A rational God cannot coherently command faith.

◉ Final Conclusion:
If God is rational, He cannot command faith understood as belief beyond evidence.

◉ A plain English walkthrough of the Master Proof above.

The first premise states that any rational being forms beliefs in proportion to the evidence. This means if something has strong evidence, belief should be strong, and if evidence is weak, belief should be weak.
The second premise defines faith as belief that goes beyond what the evidence supports. So faith, by definition, is overbelief.
Next, we assume that God is rational. This is a common claim in theology—that God is perfectly rational.
For the sake of testing the argument, we also assume that God commands faith. This is the key assumption we want to explore.
From the definition of faith, we know that if God commands faith, then some person would have to believe more strongly than the evidence allows.
If God commands faith, then such a person is obligated to practice faith, meaning they will hold belief that exceeds evidence.
So, under the assumption, this person indeed believes more than the evidence supports.
But from the first premise, if God is rational, then any belief God commands must be in proportion to the evidence.
Therefore, the belief commanded by God should exactly match the evidence.
Now we face a contradiction: the commanded belief both exceeds the evidence and exactly matches the evidence at the same time.
A contradiction signals that our assumptions cannot all be true together.
Therefore, we conclude that the original assumption—that God is rational and yet commands faith—is impossible. It leads directly to contradiction.

Final Conclusion: If God is rational, He cannot command faith that exceeds evidence. The very definition of faith, combined with rationality, makes such a command incoherent.

◉ Narrative Summary

The core argument begins with the premise that rationality requires beliefs to be proportionate to the evidence. If someone is rational, their confidence in a claim rises or falls in line with how strong or weak the evidence is. Faith, however, is defined in this framework as going beyond evidence—believing more than the evidence justifies. This immediately creates tension: faith departs from the proportionality that rationality demands.

When we assume that God is rational, and at the same time assume that God commands faith, a problem emerges. If God commands faith, then a person who follows this command must hold a belief that exceeds the evidence. Yet if God is rational, then whatever belief God commands must exactly match the evidence. The believer is therefore placed in an impossible position—both required to go beyond evidence and to align strictly with it. This logical clash produces a contradiction.

From this contradiction, the conclusion follows: it cannot be the case that a rational God commands faith. To insist otherwise would mean accepting that rationality both demands proportional belief and permits overbelief simultaneously, which is incoherent. Thus, the narrative outcome is clear—if God is rational, He cannot endorse the kind of faith that exceeds evidence. The very concept of faith, as traditionally promoted in the Bible, is fundamentally incompatible with divine rationality.

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