#13-Formalization

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➘ #13 Source Article

Symbolic Logic Formalization

1.\quad R(x,y) = \text{x has a personal relationship with y}
Annotation: Definition of relational predicate.

2.\quad \forall x \forall y \ (R(x,y) \rightarrow NPK(x,y))
Annotation: Genuine relationships require nonpublic knowledge.

3.\quad R(b,j) \rightarrow NPK(b,j)
Annotation: If a believer b has a relationship with Jesus j, then that believer must have nonpublic knowledge of Jesus.

4.\quad \neg \exists b \ (NPK(b,j))
Annotation: No believer actually has nonpublic knowledge of Jesus.

5.\quad \forall b \ (R(b,j) \rightarrow NPK(b,j))
Annotation: Restating that the relationship claim requires NPK.

6.\quad \neg \exists b \ (NPK(b,j))
Annotation: Restating that no such knowledge exists.

7.\quad \therefore \ \neg \exists b \ (R(b,j))
Annotation: Therefore, no believer has a genuine relationship with Jesus.

8.\quad (\neg \exists b \ (R(b,j))) \Rightarrow H_{Alt}
Annotation: Best explanation: believers’ experiences are psychological, not evidence of divine intimacy.

The Fitch-style natural deduction of the core result.

0.\ \ \text{Vocabulary: }R(x,y),\ NPK(x,y);\ \text{constant } j \text{ denotes Jesus}
Annotation: Predicate R(x,y) means “x has a personal relationship with y”; predicate NPK(x,y) means “x has nonpublic knowledge about y”; constant j names Jesus.

1.\ \ \forall x\forall y\ \big(R(x,y)\ \rightarrow\ NPK(x,y)\big)\quad(Premise)
Annotation: In ordinary relationships, having a genuine personal relationship entails privileged, nonpublic knowledge. This is the paper’s relational baseline.

2.\ \ \neg\exists b\ NPK(b,j)\quad(Premise)
Annotation: Empirical claim from the analysis: no believer b verifiably has nonpublic knowledge about j (reports reduce to public scripture or subjective states).

3.\ \ \exists b\ R(b,j)\quad(Assumption\ \text{for}\ \neg\text{-Intro})
Annotation: Begin a subproof by assuming there exists some believer with a personal relationship to j, with the goal of deriving a contradiction.

│ 4.\ \ R(c,j)\quad(\exists\text{-Elim on }3)
Annotation: Introduce a new arbitrary witness c such that R(c,j), discharging no prior dependencies (standard witness condition for \exists\text{-Elim}).

│ 5.\ \ R(c,j)\ \rightarrow\ NPK(c,j)\quad(\forall\text{-Elim on }1)
Annotation: Instantiate line 1 at x:=c, y:=j to get the conditional for this specific c and j.

│ 6.\ \ NPK(c,j)\quad(\rightarrow\text{-Elim on }4,5)
Annotation: From R(c,j) and R(c,j)\rightarrow NPK(c,j), infer NPK(c,j) (modus ponens).

7.\ \ \exists b\ NPK(b,j)\quad(\exists\text{-Intro on }6)[
Annotation: From NPK(c,j), generalize to “there exists some believer with nonpublic knowledge of j”.

8.\ \ \bot\quad(\text{Contradiction from }2,7)
Annotation: Lines 2 and 7 yield \bot since they assert both \neg\exists b\ NPK(b,j) and \exists b\ NPK(b,j).

9.\ \ \neg\exists b\ R(b,j)\quad(\neg\text{-Intro discharging }3-8)

Annotation: By negation-introduction: because assuming \exists b\ R(b,j) led to \bot, conclude that no believer has a personal relationship with j.

Likelihoodist corollary that pairs with the Fitch result.

H_1:\ \exists b,R(b,j)\ \ \wedge\ \ \forall x\forall y,(R(x,y)\rightarrow NPK(x,y))
Annotation: H_1 says at least one believer has a genuine personal relationship with Jesus and such relationships produce nonpublic knowledge. This mirrors the paper’s baseline for genuine relationships.

H_2:\ \neg\exists b,R(b,j)
Annotation: H_2 says no believer has such a relationship; experiences are understood without appeal to an external interacting agent.

E := E_1 \land E_2 \land E_3 \land E_4
Annotation: E is the observed evidential bundle: E_1 no verified nonpublic knowledge; E_2 no guided artifact/discovery; E_3 no convergent private revelations; E_4 no above-chance predictive success.

\mathcal{L}(E;H_2,H_1)\ :=\ \frac{P(E\mid H_2)}{P(E\mid H_1)}
Annotation: Likelihood ratio comparing how expected E is under H_2 versus H_1.

\forall i\in{1,2,3,4}:\ \ P(E_i\mid H_2)\ >\ P(E_i\mid H_1)
Annotation: Each component of the observed absence (no NPK, no artifacts, no convergence, no predictive wins) is more expected if there is no genuine relationship than if there is one.

\text{Modeling assumption: }P(E\mid H_k)=\prod_{i=1}^{4}P(E_i\mid H_k)\ (conservative independence for illustration)
Annotation: To display the mechanics transparently, treat the components as approximately independent; this only strengthens the qualitative comparison and is not required for the direction of favoring.

\Rightarrow\ \ \mathcal{L}(E;H_2,H_1)\ =\ \prod_{i=1}^{4}\frac{P(E_i\mid H_2)}{P(E_i\mid H_1)}\ >\ 1
Annotation: Because every factor in the product exceeds 1, the overall likelihood ratio exceeds 1.

E\ \text{ favors } H_2 \ \text{over}\ H_1\ \ \text{iff}\ \ \mathcal{L}(E;H_2,H_1)>1
Annotation: Likelihoodist verdict rule: when \mathcal{L}>1, the observed evidence favors H_2 over H_1.

Given the paper’s evidential survey, \ \mathcal{L}(E;H_2,H_1)\ \gg\ 1\ \Rightarrow\ \text{"}H_2\text{ is strongly favored over }H_1\text{"}

Annotation: With all four absences documented in the paper, the comparative support is large, so E strongly favors H_2.

◉ A plain English walkthrough of the symbolic logic above.

  1. Vocabulary setup:
    We define R(x,y) as “person x has a personal relationship with y,” and NPK(x,y) as “person x has nonpublic knowledge of y.” Jesus is denoted by j.
  2. Premise 1 – The relational standard:
    In any genuine personal relationship, one party has nonpublic knowledge of the other. Formally: \forall x \forall y (R(x,y) \rightarrow NPK(x,y)).
    Translation: If you really know someone personally, you should know things outsiders don’t.
  3. Premise 2 – The empirical observation:
    No believer actually has nonpublic knowledge of Jesus. Formally: \neg \exists b (NPK(b,j)).
    Translation: Despite centuries of claims, no one produces unique insider information about Jesus.
  4. Assumption – Try granting the believer’s claim:
    Suppose some believer does have a personal relationship with Jesus: \exists b R(b,j)].
  5. From the assumption, pick a witness:
    Let’s call that believer c. So we assume R(c,j)].
  6. Apply the relational rule:
    From Premise 1, R(c,j) \rightarrow NPK(c,j)].
    Translation: If c has the relationship, c must have insider knowledge.
  7. Derive the consequence:
    From R(c,j)] and R(c,j) \rightarrow NPK(c,j)], we infer NPK(c,j)].
    Translation: If c really has the relationship, then c must have nonpublic knowledge of Jesus.
  8. Generalize:
    That means there exists at least one believer with such knowledge: \exists b NPK(b,j)].
  9. Contradiction arises:
    But this clashes with Premise 2 (\neg \exists b NPK(b,j)]). So we’ve hit a contradiction.
  10. Conclusion – Negation introduction:
    Since assuming \exists b R(b,j)] led to contradiction, we conclude \neg \exists b R(b,j)].
    Translation: No believer has a genuine personal relationship with Jesus.
Likelihoodist Corollary Walkthrough
  1. Hypothesis H_1: At least one believer has a personal relationship with Jesus, and such relationships entail insider knowledge.
  2. Hypothesis H_2: No believer has such a relationship; experiences are self-generated.
  3. Evidence E: The observed facts — no nonpublic knowledge, no artifact discoveries, no convergent private revelations, no successful predictions.
  4. Likelihood ratio: Compare P(E \mid H_2) to P(E \mid H_1).
  5. If people really had a relationship with Jesus (H_1), the evidence E would be extremely unlikely.
  6. If no one has that relationship (H_2), E is exactly what we’d expect.
  7. Result: The ratio strongly favors H_2.
  8. Translation: The absence of insider evidence is much more likely if there is no personal relationship with Jesus, so the data point decisively away from H_1.

So in plain English:
✓ If real relationships exist, we’d expect special evidence.
✓ No such evidence appears.
✓ Assuming it does exist leads straight into contradiction.
✓ Therefore, no believer has a genuine relationship with Jesus.
✓ And the observed evidence is vastly more likely on the “no relationship” hypothesis.

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