#02-Formalization

August 11, 2025

➘ #02 Source Article

1) Language Domains — Symbols & Meanings

Domains & variables
$G$ : set of all proposed deity-concepts. Each deity is represented by $g$ or $h$ .
$s$ : constant for the honest seeker (a person sincerely pursuing truth).
$m$ : the medium through which the deity is said to reveal itself.

Predicates on $g$
$H(g)$ : “High stakes” — commitment to $g$ carries significant consequences.
$C(g)$ : $g$ is internally coherent (no logical contradictions).
$K(g)$ : $g$ is consistent with its attributed history and actions.
$O(g)$ : $g$ is open to independent investigation (no restrictions).
$NC(g)$ : $g$ is assessable without circular reasoning (uses external standards).
$Cl(g)$ : $g$ is sufficiently clear for a reasonable agent to reliably form/maintain accurate belief.
$AimR(g)$ : $g$ claims to aim at a personal relationship with agents.
$CanCl(g)$ : $g$ has the ability to provide clarity adequate to stakes and aims.
$W(g)$ : $g$ is worthy of allegiance.

Policy predicates
$F(g)$ : $g$ forbids independent questioning.
$SOnly(g)$ : $g$ allows only its own standards for judgment.

Medium predicates
$Medium(g,m)$ : $g$ ’s primary self-revelation uses medium $m$ .
$Ambig(m)$ : $m$ is ambiguity-prone (reasonable interpreters disagree widely).
$Access(m)$ : $m$ is widely accessible across cultures, times, and abilities.
$Robust(m)$ : $m$ resists corruption/distortion.
$Better(m_1,m_2)$ : $m_1$ is superior to $m_2$ in clarity, access, and robustness.
$Avail(g,m)$ : $m$ is feasible for $g$ to use.
$Book(m)$ : $m$ has the defining traits of book-based revelation.

Doxastic / deontic predicates
$B(s,g)$ : $s$ believes in $g$ .
$E(s,g)$ : $s$ has evaluated $g$ against relevant tests.
$Perm(s,g)$ : it is rationally permissible for $s$ to believe in $g$ .
$O_s(\varphi)$ : “The seeker ought that $\varphi$ ” (obligation relative to $s$ ).

Competition
$Comp(g,h)$ : $g$ and $h$ make incompatible claims about reality.

2) Core Axioms / Premises

$P1$ — $\forall x ; (H(x) \rightarrow O_s(E(s,x)))$
If a commitment has high stakes, the honest seeker ought to evaluate it.
This is a general principle of responsible inquiry: the more is at risk, the more pressing the duty to investigate.

$P2$ — $\forall g \in G ; H(g)$
Every deity-concept is high-stakes.
All proposed gods, by their nature, involve significant consequences (eternal, moral, existential).

$P3$ — $\forall g ; (E(s,g) \rightarrow (C(g) \land K(g) \land O(g) \land NC(g) \land Cl(g)))$
Evaluation means testing for coherence, consistency with record, openness, non-circularity, and clarity.
Passing these ensures the concept can survive rational scrutiny.

$P4$ — $\forall g ; (\neg(C(g) \land K(g) \land O(g) \land NC(g) \land Cl(g)) \rightarrow \neg Perm(s,g))$
Failure on any test blocks rational permission.
Even one failed property removes the rational basis for belief.

$P5$ — $\forall g ; (Perm(s,g) \rightarrow E(s,g))$
Permission to believe requires prior evaluation.
You can’t be rationally permitted to believe without having checked.

$P6$ — $\forall g ; (W(g) \rightarrow O(g))$
Worthy-of-allegiance concepts must be open to evaluation.
Contraposition: closed concepts are unworthy.

$P7$ — $\forall g ; (B(s,g) \rightarrow W(g))$
Belief presupposes worthiness.
If a belief demands allegiance, the object must be worthy.

$P8a$ — $\forall g ; (F(g) \rightarrow \neg O(g))$
Forbidding questioning closes evaluation.

$P8b$ — $\forall g ; (SOnly(g) \rightarrow \neg NC(g))$
Requiring only self-standards makes evaluation circular.

$P9$ — $\forall g \forall h ; ((g \neq h \land Comp(g,h)) \rightarrow O_s(E(s,g)) \land O_s(E(s,h)))$
Competing claims trigger duty to evaluate both.

$P10$ — $\forall g ; ((AimR(g) \land CanCl(g) \land H(g)) \rightarrow O_s(Cl(g)))$
If a god aims at relationship, has power to clarify, and stakes are high, clarity is obligatory.

$P11$ — $\forall g \forall m ; ((Medium(g,m) \land Access(m) \land Robust(m) \land \neg Ambig(m)) \rightarrow Cl(g))$
A medium with access, robustness, and no ambiguity delivers clarity.

$P12$ — $\forall g \forall m ; ((Medium(g,m) \land (Ambig(m) \lor \neg Access(m) \lor \neg Robust(m))) \rightarrow \neg Cl(g))$
A medium with any of these flaws lacks clarity.

$P13$ — $\forall g \forall m_1 \forall m_2 ; ((Medium(g,m_2) \land Avail(g,m_1) \land Better(m_1,m_2) \land H(g)) \rightarrow \neg Cl(g))$
Choosing an inferior medium when a better one is available under high stakes means clarity fails.

$P14$ — $\forall m ; (Book(m) \rightarrow (Ambig(m) \lor \neg Access(m) \lor \neg Robust(m)))$
Book-like media tend toward ambiguity, limited access, and fragility.

$P15$ — $\forall g ; ((AimR(g) \land CanCl(g) \land \neg Cl(g)) \rightarrow Tension(g))$
If a relational god can clarify but does not, there is an explanatory tension.

$P16$ — $\forall g ; (Perm(s,g) \rightarrow (C(g) \land K(g) \land O(g) \land NC(g) \land Cl(g)))$
Permission to believe requires all five properties to persist.

3) Immediate Derivations

$D1$ — From $P1$ + $P2$ : $O_s(E(s,g))$ for all $g$ .
The seeker has a duty to evaluate every deity-concept.

$D2$ — From $P5$ + $P3$ : $Perm(s,g) \rightarrow (C(g) \land K(g) \land O(g) \land NC(g) \land Cl(g))$ .
Permission requires passing all tests.

$D3a$ — From $P8a$ + $P4$ : $F(g) \rightarrow \neg Perm(s,g)$ .
Forbidding questioning removes openness → fails test → no permission.

$D3b$ — From $P8b$ + $P4$ : $SOnly(g) \rightarrow \neg Perm(s,g)$ .
Self-standard → circularity → fails test → no permission.

$D4$ — From contraposition of $P6$ and $P7$ : $\neg O(g) \rightarrow \neg W(g)$ ; $B(s,g) \rightarrow W(g)$ ; hence $\neg O(g) \rightarrow \neg B(s,g)\text{-worthiness}$ .
Closed concepts cannot be worthy, so belief is blocked.

$D5$ — From $P12$ + $P4$ : Deficient medium ⇒ $\neg Perm(s,g)$ .
A flawed medium destroys clarity, hence permission.

$D6$ — From $P13$ + $P4$ : Avoidable deficiency ⇒ $\neg Perm(s,g)$ .
Choosing a worse medium than available defeats clarity and permission.

$D7$ — From $P14$ + $P12$ + $P4$ : $Book(m) \land Medium(g,m) \rightarrow \neg Perm(s,g)$ .
Book-like media are presumptively deficient and block permission unless exceptional.

$D8$ — From $P10$ + $P15$ : $AimR(g) \land CanCl(g) \land H(g) \land \neg Cl(g) \rightarrow Tension(g)$ .
A personal, able god who leaves things unclear creates hiddenness tension.

4) Fitch-Style Proof — Central Theorem

Theorem T:
For any $g \in G$ :

$O_s(E(s,g))$ .
$\neg(C(g) \land K(g) \land O(g) \land NC(g) \land Cl(g)) \rightarrow \neg Perm(s,g)$ .
Deficient or dominated medium ⇒ $\neg Perm(s,g)$ .

Proof:

$\forall g ; H(g)$ ( $P2$ ).
$\forall x ; (H(x) \rightarrow O_s(E(s,x)))$ ( $P1$ ).
Let $g$ be arbitrary. From 1 and 2 → $O_s(E(s,g))$ . ⟨1⟩
From $P4$ : $\neg(C(g) \land K(g) \land O(g) \land NC(g) \land Cl(g)) \rightarrow \neg Perm(s,g)$ . ⟨2⟩
Assume $Medium(g,m) \land (Ambig(m) \lor \neg Access(m) \lor \neg Robust(m))$ . From $P12$ → $\neg Cl(g)$ . From (2) → $\neg Perm(s,g)$ .
Assume $Medium(g,m_2) \land Avail(g,m_1) \land Better(m_1,m_2) \land H(g)$ . From $P13$ → $\neg Cl(g)$ . From (2) → $\neg Perm(s,g)$ .
Therefore 1–3 hold for all $g$ . ∎

5) Corollaries

$Cor.1$ — From $D3a$ and $D3b$ : $F(g) \lor SOnly(g) \rightarrow \neg Perm(s,g)$ .
Blocking questions or using circular standards removes permission.

$Cor.2$ — From $D8$ : $AimR(g) \land CanCl(g) \land H(g) \land \neg Cl(g) \rightarrow Tension(g)$ .
A relational aim with unfulfilled clarity obligation signals inconsistency.

$Cor.3$ — From $D7$ : $Book(m) \land Medium(g,m) \rightarrow \neg Perm(s,g)$ unless $Access(m) \land Robust(m) \land \neg Ambig(m)$ .
Book-based revelation must overcome its default flaws to permit belief.

$Cor.4$ — From $P9$ and $D1$ : $Comp(g,h) \rightarrow O_s(E(s,g)) \land O_s(E(s,h))$ .
Competing gods must each be evaluated; failure of one blocks its permission.

6) Compact Sequent Summary

$H(g) \Rightarrow O_s(E(s,g))$ .
$E(s,g) \Rightarrow (C(g) \land K(g) \land O(g) \land NC(g) \land Cl(g))$ ; failure ⇒ $\neg Perm(s,g)$ .
$F(g) \lor SOnly(g) \Rightarrow \neg Perm(s,g)$ .
Deficient/dominated medium ⇒ $\neg Perm(s,g)$ .
$Book(m) \land Medium(g,m) \Rightarrow \neg Perm(s,g)$ (prima facie).
$AimR(g) \land CanCl(g) \land \neg Cl(g) \Rightarrow Tension(g)$ .

7) Plain-English Gloss

If a god concept has high stakes, the honest seeker must evaluate it. Passing all five rational tests — coherence, consistency with record, openness, non-circularity, and clarity — is required for rational permission to believe. A god that forbids questions, insists on its own authority, or uses a medium that is ambiguous, inaccessible, or fragile fails the clarity test. If a better medium is available but unused, permission is lost. Book-like revelation is presumptively disqualifying unless its weaknesses are somehow neutralized. A god aiming for relationship, with the ability to provide clarity, yet failing to do so, creates a serious tension between stated aim and actual method. The presence of competing god-concepts requires evaluation of each, and any that fail cannot be rationally believed.

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.

Language Domains

Before we can build the logic, we need to define the “language” — the set of symbols and terms the proof will use. This ensures precision and removes any guesswork about what each letter or symbol means.

$G$ : The set of all proposed deity-concepts.
Think of this as the master list of “gods” people claim exist, each with different traits.
$g, h$ : Variables that stand for particular members of $G$ .
If $g$ = “the Christian God” and $h$ = “the Islamic God,” they are two different items in the set.
$s$ : The “honest seeker.”
This is a placeholder for any person who genuinely wants to find the truth about these claims.
$m$ : A “medium” of revelation.
This is the way a deity is said to reveal itself — for example, a book, a voice, a vision, or some universal inner sense.

Predicates are properties or relationships we can assert about these variables.
For example, $H(g)$ means “g is high-stakes,” and $Medium(g,m)$ means “g uses medium m as its primary self-disclosure.”

Core Axioms / Premises

The $P1$ – $P16$ list contains the foundational rules of the system. They are written in formal logic so their implications are crystal clear.

$P1$ : $\forall x (H(x) \rightarrow O_s(E(s,x)))$
If something has high stakes, the seeker is obligated to evaluate it.
In human terms: if the consequences are big (e.g., eternal salvation or damnation), you should investigate thoroughly.
$P2$ : $\forall g \in G ; H(g)$
All deity-concepts have high stakes.
This removes any claim that “my god isn’t high-stakes, so you don’t need to check.”
$P3$ : $E(s,g) \rightarrow (C(g) \land K(g) \land O(g) \land NC(g) \land Cl(g))$
Evaluation means checking five criteria: coherence, record-consistency, openness, non-circularity, and clarity.
$P4$ : $\neg(C \land K \land O \land NC \land Cl) \rightarrow \neg Perm(s,g)$
Failing even one of the five tests removes rational permission to believe.
$P5$ : $Perm(s,g) \rightarrow E(s,g)$
You can’t be rationally permitted to believe without evaluation.
$P6$ : $W(g) \rightarrow O(g)$
If a deity is worthy of allegiance, it must be open to investigation.
$P7$ : $B(s,g) \rightarrow W(g)$
Believing in a deity assumes it is worthy of allegiance.
$P8a$ : $F(g) \rightarrow \neg O(g)$
If a deity forbids questioning, it is not open to evaluation.
$P8b$ : $SOnly(g) \rightarrow \neg NC(g)$
If a deity only allows its own standards to judge itself, it fails non-circularity.
$P9$ : $Comp(g,h) \rightarrow O_s(E(s,g)) \land O_s(E(s,h))$
If two gods make incompatible claims, you must evaluate both.
$P10$ : $AimR(g) \land CanCl(g) \land H(g) \rightarrow O_s(Cl(g))$
If a god aims at relationship, can provide clarity, and stakes are high, clarity is obligatory.
$P11$ & $P12$ :
These define when a medium counts as clear and when it fails — based on accessibility, robustness, and freedom from ambiguity.
$P13$ : Choosing a worse medium when a better one is available is a clarity failure.
$P14$ : Book-like media are presumed to have clarity problems unless proven otherwise.
$P15$ : If a relational god can clarify but doesn’t, that’s a hiddenness tension.
$P16$ : Permission to believe requires that all five criteria remain satisfied over time.

Immediate Derivations

These are quick logical consequences of the premises — things we can state with confidence without a long proof.

Example:

$D1$ says the seeker must evaluate every deity.
$D3a$ says that if a god forbids questioning, you cannot rationally believe in it.
$D7$ says that if a god’s primary revelation is book-like, you are presumptively not permitted to believe unless the book is unusually clear, accessible, and robust.

Fitch-Style Proof

This section is a formal walk-through showing how the main theorem follows step-by-step from the premises.
It’s the “show your work” part of the argument, leaving no gap where someone could say “but how did you get from there to here?”

Corollaries

These are “spin-off” results that come almost for free once the main theorem is established.
Example:

$Cor.3$ directly targets book-based revelation as presumptively disqualifying unless its weaknesses are overcome.

Compact Sequent Summary

This condenses the entire argument into a chain of conditionals — a high-level blueprint of the reasoning.

Plain-English Gloss

This section removes the symbols entirely but keeps the logical content.
It’s meant for quick understanding without needing to parse $\forall$ or $\rightarrow$ syntax, yet it is faithful to the rigorous version above.

Recent posts

✓ Plantinga’s Abandonment of Credence

February 10, 2026

Alvin Plantinga’s “Warrant” isn’t an epistemic upgrade; it’s a design for inaccuracy. My formal proof demonstrates that maximizing the binary status of “knowledge” forces a cognitive system to be less accurate than one simply tracking evidence. We must eliminate “knowledge” as a rigorous concept, replacing it with credencing—the honest pursuit…
✓ The Great Gaslighting Grift

February 1, 2026

This article critiques the stark gap between the New Testament’s unequivocal promises of answered prayer and their empirical failure. It examines the theological “bait-and-switch” where bold pulpit guarantees of supernatural intervention are neutralized by “creative hermeneutics” in small groups, transforming literal promises into unfalsifiable, psychological coping mechanisms through evasive logic…
✓ Theology’s Floating Fortress

January 31, 2026

This article characterizes theology as a “floating fortress”—internally coherent but isolated from empirical reality. It details how specific theological claims regarding prayer, miracles, and scientific facts fail verification tests. The argument posits that theology survives only through evasion tactics like redefinition and metaphor, functioning as a self-contained simulation rather than…
✓ Parsimony and Christianity

January 27, 2026

This post applies parsimony (Occam’s Razor) to evaluate Christian Theism. It contrasts naturalism’s high “inductive density” with the precarious “stack of unverified assumptions” required for Christian belief, such as a disembodied mind and omni-attributes. It argues that ad hoc explanations for divine hiddenness further erode the probability of theistic claims,…
✓ Grounding Ways of Knowing

January 19, 2026

Modern apologists argue that religious belief is a rational map of evidence, likening it to scientific frameworks. However, a deeper analysis reveals a stark contrast. While science adapts to reality through empirical testing and falsifiability, theology insulates belief from contradictory evidence. The theological system absorbs anomalies instead of yielding to…
✓ Childlike Faith

January 19, 2026

This post critiques the concept of “childlike faith” in religion, arguing that it promotes an uncritical acceptance of beliefs without evidence. It highlights that while children naturally trust authority figures, this lack of skepticism can lead to false beliefs. The author emphasizes the importance of cognitive maturity and predictive power…
✓ The Amalekite Infants

January 19, 2026

This analysis examines the agonizing moral conflict presented by the explicit biblical command to slaughter Amalekite infants in 1 Samuel 15:3. Written from a skeptical, moral non-realist perspective, it rigorously deconstructs the various apologetic strategies employed to defend this divine directive as “good.” The post critiques common evasions, such as…
✓ The Evidence-Mapping Illusion

January 17, 2026

Modern Christian apologetics claims faith is based on evidence, but this is contradicted by practices within the faith. Children are encouraged to accept beliefs uncritically, while adults seeking evidence face discouragement. The community rewards conformity over inquiry, using moral obligations to stifle skepticism. Thus, the belief system prioritizes preservation over…
✓ The Divine Judgment Evasion

January 17, 2026

In the realm of Christian apologetics, few topics generate as much palpable discomfort as the Old Testament narratives depicting divinely ordered genocide. While many believers prefer to gloss over these passages, serious apologists feel compelled to defend them. They must reconcile a God described as “perfect love” with a deity…
✓ Answered Prayers?

January 10, 2026

This post examines various conditions Christians often attach to prayer promises, transforming them into unfalsifiable claims. It highlights how these ‘failsafe’ mechanisms protect the belief system from scrutiny, allowing believers to reinterpret prayer outcomes either as successes or failures based on internal states or hidden conditions. This results in a…
✓ Categorical Labels Rarely Reflect Rationality

January 8, 2026

In public discourse, labels such as “atheist,” “agnostic,” and “Christian” often oversimplify complex beliefs, leading to misunderstandings. These tags are low-resolution summaries that hinder rational discussions. Genuine inquiry requires moving beyond labels to assess individual credences and evidence. Understanding belief as a gradient reflects the nuances of thought, promoting clarity…
✓ Why an Intelligible Universe?

January 6, 2026

The featured argument, often employed in Christian apologetics, asserts that the universe’s intelligibility implies a divine mind. However, a meticulous examination reveals logical flaws, such as equivocation on “intelligible,” unsubstantiated jumps from observations to conclusions about authorship, and the failure to consider alternative explanations. Ultimately, while the universe exhibits structure…
✓ How Myths Emerge

January 1, 2026

The piece discusses how historical figures like Jesus and Alexander the Great undergo “legendary inflation,” where narratives evolve into more than mere history, shaped by cultural needs and societal functions. As communities invest meaning in these figures, their stories absorb mythical elements and motifs over time. This phenomenon illustrates how…
✓ Mythical Husk over the Jesus Seed

January 1, 2026

This post argues against extreme views in debates about the historical Jesus, emphasizing the distinction between the theological narrative shaped by scriptural interpretation and the existence of a human core. It maintains that while the Gospels serve theological purposes, they do not negate the likelihood of a historical figure, supported…
✓ The Semantic Creep of Faith

December 19, 2025

Hebrews 11:1 is often misquoted as a clear definition of faith, but its Greek origins reveal ambiguity. Different interpretations exist, leading to confusion in Christian discourse. Faith is described both as assurance and as evidence, contributing to semantic sloppiness. Consequently, discussions about faith lack clarity and rigor, oscillating between certitude…
✓ AI for Christian Apologists

December 18, 2025

This post emphasizes the importance of using AI as a tool for Christian apologetics rather than a replacement for personal discernment. It addresses common concerns among Christians about AI, advocating for its responsible application in improving reasoning, clarity, and theological accuracy. The article outlines various use cases for AI, such…

J on ✓ Theology’s Floating Fortress
Thanks for another interesting piece (as usual). I might add that even the “flying” fortress theologians have tried to build…
Phil Stilwell on ✓ Parsimony and Christianity
Good insights! “William Philosophers” definitely refers to William of Ockham.
J on ✓ Parsimony and Christianity
Given that I read a collection of Russell’s works on religion a few years ago, the reference to the teapot…
J on ✓ Grounding Ways of Knowing
Alright, thanks for the insights. While I’ve read works on ethics and ethical theories (including some of Immanuel Kant’s more…
Phil Stilwell on ✓ Grounding Ways of Knowing
Hi J, I’m a moral non-realist. I hold that there are no legitimate moral obligations since a moral realm has…
J on ✓ Grounding Ways of Knowing
Hi Phil: Apologies if this isn’t really related to skepticism, but I was wondering: What are your thoughts on the…
J on The Mirage of the Cosmic Rulebook:
Has Juan considered the problems with divine command theory and could he address the following?: a.) What makes Christian versions…
Phil Stilwell on The Mirage of the Cosmic Rulebook:
Juan has been blocked. I ran out of both daylight and patience.A debriefing has been added above in the light…
Phil Stilwell on The Mirage of the Cosmic Rulebook:
Check the new formalization section in grey above. Fourty minutes.
J. GonzalezRamos on The Mirage of the Cosmic Rulebook:
Phil, an hour to “get honest” about what exactly? I pointed out the contradiction between your self-identification as a moral…

Phil Stilwell

A Deep Dive into Common Faith-Based Concepts & Claims