#01-Formalization

Formalizations

August 11, 2025

➘ #01 Source Article

1) Language Domains

$G$ — Set of proposed God-concepts. Variables $g$ and $h$ refer to specific concepts within this set.

$s$ — Constant representing the honest seeker.

Predicates on $g$

$C(g)$ — Predicate meaning “ $g$ is coherent” (internally consistent without logical contradictions).

$K(g)$ — Predicate meaning “ $g$ is consistent with its attributed actions/record” (its descriptions and actions do not conflict).

$D(g)$ — Predicate meaning “ $g$ is desirable” (allegiance to g promotes human well-being or trustworthiness).

$Op(g)$ — Predicate meaning “ $g$ is open to independent evaluation” (no prohibition on questioning).

$Cir(g)$ — Predicate meaning “ $g$ is circular” (assessment of $g$ depends only on g’s own authority).

$W(g)$ — Predicate meaning “ $g$ is worthy of worship or allegiance.”

$H(g)$ — Predicate meaning “commitment to $g$ is high-stakes” (significant consequences for the seeker).

Policies attached to a concept

$Forbid(g)$ — Predicate meaning “ $g$ forbids independent questioning.”

$SelfOnly(g)$ — Predicate meaning “ $g$ requires only its own standards to assess itself.”

Doxastic / Normative

$B_s(g)$ — Predicate meaning “seeker $s$ believes in or commits allegiance to $g$ .”

$E_s(g)$ — Predicate meaning “seeker $s$ has critically evaluated $g$ on the relevant dimensions.”

$R_s(g)$ — Predicate meaning “it is rationally permissible for seeker $s$ to believe $g$ .”

Deontic operator (relative to $s$ )

$O_s(\varphi)$ — Means “seeker $s$ ought that φ.” This marks obligation claims.

2) Core Axioms / Premises

$\forall x , [ H(x) \rightarrow O_s(E_s(x)) ]$

— P1 (High-stakes duty): For any entity $x$ , if commitment to x is high-stakes, then seeker $s$ ought to evaluate $x$ .

$\forall g \in G , H(g)$

— P2 (Theistic stakes): All God-concepts in set $G$ are high-stakes commitments.

$\forall g , [ E_s(g) \rightarrow ( C(g) \wedge K(g) \wedge D(g) \wedge Op(g) \wedge \neg Cir(g) ) ]$

— P3 (What “evaluated” entails): If seeker $s$ has evaluated $g$ , then $g$ must be coherent, consistent with its record, desirable, open to independent evaluation, and not circular.

$\forall g , [ (\neg C(g) \vee \neg K(g) \vee \neg D(g) \vee \neg Op(g) \vee Cir(g) ) \rightarrow \neg R_s(g) ]$

— P4 (Failure bars rational permission): If $g$ fails on coherence, consistency, desirability, openness, or non-circularity, then it is not rationally permissible for $s$ to believe $g$ .

$\forall g , [ R_s(g) \rightarrow E_s(g) ]$

— P5 (Permission requires evaluation): If it is rationally permissible for $s$ to believe $g$ , then $s$ must have evaluated $g$ .

$\forall g , [ W(g) \rightarrow Op(g) ]$

— P6 (Worthiness welcomes scrutiny): If $g$ is worthy of worship, then it must be open to independent evaluation.

$\neg Op(g) \rightarrow \neg W(g)$

— Contraposition of P6: If $g$ is not open to independent evaluation, then $g$ is not worthy of worship.

$\forall g , [ B_s(g) \rightarrow W(g) ]$

— P7 (Belief aiming at allegiance presupposes worthiness): If $s$ believes in $g$ , then $g$ must be worthy of worship.

$\forall g , [ Forbid(g) \rightarrow \neg Op(g) ]$

— P8a (Forbidding ⇒ closed): If $g$ forbids independent questioning, then $g$ is not open to evaluation.

$\forall g , [ SelfOnly(g) \rightarrow Cir(g) ]$

— P8b (Self-standard ⇒ circular): If $g$ allows only its own standards for assessment, then $g$ is circular.

$P8 := P8a \wedge P8b$

— Optional: P8 can be defined as the conjunction of P8a and P8b.

$\exists h \in G , [ h \neq g \wedge Competes(h,g) ] \rightarrow O_s(E_s(g))$

— P9 (Plurality trigger): If there exists another God-concept $h$ in G that is different from $g$ and competes with it, then $s$ ought to evaluate $g$ .

$PB_s(g) \wedge O_s(E_s(g)) \rightarrow (\text{evaluate post-formation})$

— P10 (Stability of permissibility): Even if belief in $g$ is properly basic $(PB_s(g))$ and $s$ ought to evaluate $g$ , then evaluation should still occur after belief formation.

3) Immediate Derivations

$\forall g \in G , O_s(E_s(g))$

— D1 (Obligation to evaluate God-concepts): From P2 and P1, for all God-concepts $g$ in G, seeker $s$ ought to evaluate $g$ .

$R_s(g) \rightarrow [ C(g) \wedge K(g) \wedge D(g) \wedge Op(g) \wedge \neg Cir(g) ]$

— D2 (What permission entails): From P5 then P3, if it is rationally permissible for $s$ to believe $g$ , then $g$ must be coherent, consistent with its record, desirable, open to independent evaluation, and not circular.

$Forbid(g) \rightarrow \neg Op(g) \rightarrow \neg R_s(g)$

— D3a (Forbid defeats permission): From P8a and P4, if $g$ forbids questioning, then it is not open to evaluation, and therefore it is not rationally permissible to believe $g$ .

$SelfOnly(g) \rightarrow Cir(g) \rightarrow \neg R_s(g)$

— D3b (SelfOnly defeats permission): From P8b and P4, if $g$ requires only its own standards for assessment, then it is circular, and therefore it is not rationally permissible to believe $g$ .

$\neg Op(g) \rightarrow \neg W(g)$

— D4a (Closed concepts are unworthy): From the contrapositive of P6, if $g$ is not open to evaluation, then it is not worthy of worship.

$B_s(g) \rightarrow W(g)$

— D4b (Belief presupposes worthiness): From P7, if $s$ believes in $g$ , then $g$ must be worthy of worship.

Conclusion from D4: If $g$ is closed to evaluation ( $\neg Op(g)$ ), then it fails the worthiness requirement for belief, so belief is blocked.

$\forall g \in G , ( O_s(E_s(g)) \ \wedge \ ( [ \neg C(g) \vee \neg K(g) \vee \neg D(g) \vee \neg Op(g) \vee Cir(g) ] \rightarrow \neg R_s(g) ) )$

— D5 (Headline norm): For every $g$ in $G$ , seeker $s$ ought to evaluate $g$ , and if $g$ fails in coherence, consistency, desirability, openness, or non-circularity, then it is not rationally permissible to believe $g$ .

4) Fitch-style Proof of the Central Theorem

Goal 1: $\forall g , O_s(E_s(g))$

— Target claim A: For every God-concept g, seeker $s$ ought to evaluate $g$ .

$\forall g , ( [ \neg C(g) \vee \neg K(g) \vee \neg D(g) \vee \neg Op(g) \vee Cir(g) ] \rightarrow \neg R_s(g) )$

— Target claim B: For every God-concept g, if $g$ is incoherent, inconsistent with its record, undesirable, closed to scrutiny, or circular, then belief in $g$ is not rationally permissible for $s$ .

$\forall g , H(g)$

— From P2: All God-concepts are high-stakes for the seeker.

$\forall x , [ H(x) \rightarrow O_s(E_s(x)) ]$

— From P1: Anything high-stakes should be evaluated by $s$ .

$\text{Let } g \text{ be arbitrary.}$

— Begin universal generalization by reasoning with an arbitrary but fixed $g$ .

$H(g)$

— Instantiation of P2 for the arbitrary $g$ : commitment to $g$ is high-stakes.

$O_s(E_s(g))$

— Modus ponens on P1 with H(g): therefore $s$ ought to evaluate $g$ .

$\forall g , O_s(E_s(g))$

— Universal generalization: since $g$ was arbitrary, the obligation to evaluate holds for all $g$ .

$\forall g , ( [ \neg C \vee \neg K \vee \neg D \vee \neg Op \vee Cir ] \rightarrow \neg R_s )$

— From P4 directly: the failure-conditions bar rational permission for any $g$ ; this establishes Target claim B.

$\square$ — Goal 1 established: both the universal obligation to evaluate and the failure-bar on permission are derived.

Goal 2 (Corollaries for prohibitions/circularity):

$Forbid(g) \rightarrow \neg R_s(g)$

— Target claim (a): If $g$ forbids independent questioning, then it is not rationally permissible for $s$ to believe $g$ .

$SelfOnly(g) \rightarrow \neg R_s(g)[$

— Target claim (b): If $g$ demands only self-referential standards, then it is not rationally permissible for $s$ to believe $g$ .

$Forbid(g)$

— Assume, for conditional proof of (a), that $g$ forbids questioning.

$Forbid(g) \rightarrow \neg Op(g)$

— From P8a: forbidding implies lack of openness.

$\neg Op(g)$

— Modus ponens: thus $g$ is not open to independent evaluation.

$(\neg C(g) \vee \neg K(g) \vee \neg D(g) \vee \neg Op(g) \vee Cir(g)) \rightarrow \neg R_s(g)$

— From P4: any failure on the listed dimensions defeats rational permission.

$\neg R_s(g)$

— Modus ponens using the specific disjunct $\neg Op(g)$ : rational permission is defeated; hence $Forbid(g) \rightarrow \neg R_s(g)$ .

$SelfOnly(g)$

— Assume, for conditional proof of (b), that $g$ requires only its own standards.

$SelfOnly(g) \rightarrow Cir(g)$

— From P8b: self-standard implies circularity.

$Cir(g)$

— Modus ponens: thus $g$ is circular.

$(\neg C(g) \vee \neg K(g) \vee \neg D(g) \vee \neg Op(g) \vee Cir(g)) \rightarrow \neg R_s(g)$

— From P4: circularity is among the defeat conditions.

$\neg R_s(g)$

— Modus ponens using the specific disjunct Cir(g): rational permission is defeated; hence $SelfOnly(g) \rightarrow \neg R_s(g)$ .

$\square$ — Goal 2 established: prohibitions and self-referential standards each defeat rational permission.

Goal 3 (Unworthiness under prohibition):

$\neg Op(g) \rightarrow \neg W(g)$

— From the contraposition of P6: lack of openness entails lack of worthiness.

$Forbid(g) \rightarrow \neg Op(g)$

— From P8a: forbidding questioning implies not open.

$Forbid(g) \rightarrow \neg W(g)$

— Hypothetical syllogism on the two prior lines: a forbidding concept is unworthy.

$B_s(g) \rightarrow W(g)$

— From P7: belief presupposes worthiness.

$\text{Thus, Forbid}(g) \text{ blocks the worthiness presupposed by belief.}$

— If $g$ forbids questioning, then it cannot meet the worthiness condition required by belief.

$\square$ — Goal 3 established: prohibitions undermine worthiness and thereby block belief that presupposes worthiness.

5) Compact Sequent Summary

$\vdash \ \forall g , O_s(E_s(g))$

— From P1 and P2, it follows that $s$ ought to evaluate every God-concept.

$\vdash \ \forall g , ( [ \neg C \vee \neg K \vee \neg D \vee \neg Op \vee Cir ] \rightarrow \neg R_s )$

— From P4, any failure on coherence, consistency, desirability, openness, or non-circularity defeats rational permission.

$Forbid(g) \vdash \neg R_s(g)$

— From P8a with P4, forbidding evaluation defeats rational permission.

$SelfOnly(g) \vdash \neg R_s(g)$

— From P8b with P4, self-referential standards defeat rational permission.

$Forbid(g) \vdash \neg W(g) \quad \text{and} \quad B_s(g) \vdash W(g)$

— From P6 (contraposed) and P7, forbidding undermines worthiness while belief requires worthiness; thus forbidding defeats belief-worthiness.

6) Optional “Properly Basic” Add-on

$PB_s(g)$

— Premise: s’s belief in $g$ begins as properly basic (non-inferentially formed).

$\exists h \in G , [ h \neq g \wedge Competes(h,g) ] \rightarrow O_s(E_s(g))$

— From P9: the existence of competing peer concepts triggers an obligation to evaluate $g$ .

$O_s(E_s(g))$

— When the plurality condition is met, $s$ is obligated to evaluate $g$ even if belief was properly basic.

$PB_s(g) \wedge O_s(E_s(g)) \rightarrow (\text{evaluate post-formation})$

— From P10: proper basicality does not cancel the duty to evaluate; evaluation proceeds after belief onset.

$R_s(g) \rightarrow [ C(g) \wedge K(g) \wedge D(g) \wedge Op(g) \wedge \neg Cir(g) ]$

— By D2, continued rational permissibility requires coherence, consistency with record, desirability, openness, and non-circularity.

Conclusion: even with proper basicality, plural competition and high stakes sustain a duty to evaluate, and permission still requires $C \wedge K \wedge D \wedge Op \wedge \neg Cir.$

— Properly basic onset does not exempt $g$ from meeting the same rational-permission profile.

7) Plain-English Gloss

$\forall g \in G , \big( O_s(E_s(g)) \ \wedge \ ( [ \neg C(g) \vee \neg K(g) \vee \neg D(g) \vee \neg Op(g) \vee Cir(g) ] \rightarrow \neg R_s(g) ) \big)$

— For any proposed God-concept $g$ , the honest seeker $s$ ought to evaluate it; and if $g$ is incoherent, inconsistent with its record, undesirable, closed to scrutiny, or defended circularly, then believing $g$ is not rationally permissible.

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. Language Domains – What the Symbols Mean

The document starts by defining the “language” of the formalization — essentially the symbols, constants, and predicates that will be used in the logical statements.

✓ Variables and constants

G – the set of all proposed God-concepts. Each God-concept is represented by a variable like g or h.
s – a constant standing for the honest seeker (a person trying sincerely to figure out the truth).

✓ Predicates (properties that can be true or false of a God-concept g)
Each predicate is a function that outputs true or false depending on g’s properties:

C(g) – Coherence: g is internally logically consistent (no contradictions).
K(g) – Consistency with record: g’s described actions or history don’t conflict with one another.
D(g) – Desirability: allegiance to g promotes human well-being or trustworthiness.
O(g) – Openness: g can be evaluated without restriction.
Cir(g) – Circularity: g can only be evaluated by its own authority (self-referential).
W(g) – Worthy of worship: g merits allegiance.
H(g) – High-stakes: committing to g has major consequences for s.

✓ Policy predicates

F(g) – Forbids questioning.
S(g) – Requires only its own standards.

✓ Doxastic (belief-related) predicates

B(s,g) – s believes in g.
E(s,g) – s has evaluated g on all relevant dimensions.
P(s,g) – It is rationally permissible for s to believe in g.

✓ Deontic operator

O_s φ – “The seeker ought that φ” (obligation relative to s).

2. Core Axioms / Premises – The Rules the System Assumes

Each P# is a premise; the logic builds from these.

P1 – If something is high-stakes, s ought to evaluate it.
P2 – All God-concepts are high-stakes.
P3 – If s has evaluated g, g must meet all the key conditions: coherence, consistency, desirability, openness, and non-circularity.
P4 – If g fails any of those conditions, it is not rationally permissible to believe in g.
P5 – Rational permission to believe requires that evaluation has taken place.
P6 – Worthy-of-worship concepts are open to independent evaluation. (Contrapositive: If g is not open, it’s not worthy.)
P7 – If s believes in g, then g must be worthy of worship.
P8a – If g forbids questioning, it’s not open to evaluation.
P8b – If g requires only its own standards, it’s circular.
P9 – If another God-concept competes with g, s ought to evaluate g.
P10 – Even if belief starts as “properly basic,” evaluation is still required later.

3. Immediate Derivations – Quick Consequences of the Premises

These are logical “shortcuts” the author derives without a full proof:

D1 – From P1 + P2: s ought to evaluate every God-concept.
D2 – From P5 + P3: Rational permission to believe requires the five conditions to hold.
D3a – From P8a + P4: Forbidding questioning removes openness → fails permission.
D3b – From P8b + P4: Requiring self-standards → circular → fails permission.
D4a – From contrapositive P6: Closed concepts are unworthy.
D4b – From P7: Belief presupposes worthiness → closed concepts block belief.
D5 – Summary norm: Evaluate all; failure of any key condition blocks rational permission.

4. Fitch-Style Proof – Step-by-Step Demonstration

This is a formal method showing exactly how the conclusions follow from the premises.

Goal 1 – Show (A) s must evaluate all g, and (B) if g fails any key condition, belief is impermissible.
- Uses P2 + P1 to derive A.
- Uses P4 directly to derive B.
Goal 2 – Show forbidding questioning or requiring self-standards defeats rational permission.
- Forbidding → not open → fails P4 condition → no permission.
- Self-standard → circular → fails P4 condition → no permission.
Goal 3 – Show forbidding questioning makes g unworthy and thus belief impossible.
- Forbidding → not open → unworthy → fails belief-worthiness requirement.

5. Compact Sequent Summary – Condensed Logic Flow

Ought to evaluate all g (P1 + P2).
Any failure in the five conditions blocks permission (P4).
Forbidding or self-referentiality triggers such failures (P8a/b + P4).
Forbidding also destroys worthiness, which belief requires (P6 + P7).

6. “Properly Basic” Add-On – Handling Beliefs Held Without Evidence Initially

Some claim belief in God can start “properly basic” (not based on argument).
This framework says:

Even then, competing God-concepts (P9) and high-stakes status require evaluation.
P10 says evaluation must happen after belief onset.
Continued permission still depends on the five conditions (D2).

7. Plain-English Restatement

Any honest seeker ought to evaluate every proposed God-concept.
If the concept is incoherent, inconsistent with its record, undesirable, closed to scrutiny, or circular, then believing it is not rationally permissible.

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