#28-Formalization

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➘ #28 Source Article

Symbolic Logic Formalization

1.; \text{Gradient: } \Pr(H \mid E) = \frac{\Pr(E \mid H)\Pr(H)}{\Pr(E \mid H)\Pr(H) + \Pr(E \mid \lnot H)\Pr(\lnot H)}
This expresses Bayes’ theorem: the rational credence in hypothesis H given evidence E is the weighted balance of how well H explains the evidence compared to its rival. Belief should increase or decrease proportionally to this ratio.

2.; \text{Complementarity: } d(H \mid E) = 1 - b(H \mid E)
Here, doubt is defined as the mathematical complement of belief. If belief in H is 0.7, then doubt is 0.3. Suppressing doubt without increasing evidence falsifies the proportional balance.

3.; \text{Bayes Factor: } BF = \frac{\Pr(E \mid H)}{\Pr(E \mid \lnot H)}
The Bayes factor represents how strongly the evidence favors H over \lnot H. Only when this ratio is sufficiently large does belief rationally rise.

4.; \frac{\Pr(H \mid E)}{\Pr(\lnot H \mid E)} = BF \times \frac{\Pr(H)}{\Pr(\lnot H)}
This odds form shows that the posterior odds equal the prior odds multiplied by the Bayes factor. Without strong evidence or priors, doubt remains rational.

5.; \text{Minimal Rationality Norm: } (BF \text{ modest}) \Rightarrow 0 < d(H \mid E) < 1
If the Bayes factor is modest, then rationality requires some degree of doubt. Commands to eliminate doubt contradict this evidential norm.

6.; \forall H, E ; [ ; \text{If } \Pr(H \mid E) < 1 ;; \Rightarrow ;; d(H \mid E) > 0 ; ]
For any hypothesis and evidence, if belief is less than certainty, then doubt must be greater than zero. This establishes doubt as the necessary rational complement.

7.; (\text{Anti-doubt exhortations}) ;\Rightarrow; (\exists H, E); [, d(H \mid E) = 0 ;\wedge; \Pr(H \mid E) < 1 ,]
Scriptural or theological exhortations against doubt effectively demand zero doubt even where the evidence does not warrant certainty.

8.; (\exists H, E); [, d(H \mid E) = 0 ;\wedge; \Pr(H \mid E) < 1 ,] ;\Rightarrow; \text{Over-belief}
This creates the condition of over-belief: assigning more confidence than the evidence justifies.

9.; \text{Over-belief } ;\Rightarrow; \lnot \text{Truth-tracking}
Over-belief by definition undermines truth-tracking mechanisms, because it misrepresents the evidential balance.

10.; \therefore (\text{Anti-doubt norms}) ;\Rightarrow; \lnot \text{Truth-tracking}
Therefore, norms that prohibit doubt undercut rational inquiry and prevent agents from aligning belief with reality.

A Fitch-Style Proof.

B(H \mid E) := \mathrm{P}(H \mid E)
Annotation: This denotes the degree of belief in hypothesis H given evidence E.

D(H \mid E) := 1 - B(H \mid E)
Annotation: Doubt is defined as the complement of belief; if belief is 0.7, doubt is 0.3.

O(H) := \tfrac{\mathrm{P}(H)}{\mathrm{P}(\neg H)} \quad ; \quad O(H \mid E) := \tfrac{\mathrm{P}(H \mid E)}{\mathrm{P}(\neg H \mid E)}
Annotation: Prior and posterior odds compare the likelihood of H versus its negation \neg H.

\mathrm{BF}_{H,\neg H}(E) := \tfrac{\mathrm{P}(E \mid H)}{\mathrm{P}(E \mid \neg H)}
Annotation: The Bayes factor measures how strongly the evidence E favors H over \neg H.

\text{Mod}(H,E) := \text{``priors not extreme and } 1 < \mathrm{BF}_{H,\neg H}(E) < k \text{ for finite } k''
Annotation: A condition of modest evidential support: neither overwhelming nor negligible.

\text{TT} := \text{Truth-tracking}
Annotation: The target epistemic norm: beliefs should proportionally track evidence.

\text{OB}(H,E) := \text{Over-belief at } (H,E) \equiv (D(H \mid E) = 0 ;\wedge; B(H \mid E) < 1)
Annotation: Over-belief occurs when doubt is eliminated despite incomplete evidence.

N := \text{General anti-doubt norm}
Annotation: The norm under critique: commands or policies that prohibit doubt.

Premises
  1. O(H \mid E) = \tfrac{\mathrm{P}(E \mid H)}{\mathrm{P}(E \mid \neg H)} \cdot O(H)
    Annotation: Posterior odds equal prior odds times the likelihood ratio.
  2. D(H \mid E) = 1 - B(H \mid E)
    Annotation: Doubt is mathematically defined as the complement of belief.
  3. \forall H,E ;; (\text{Mod}(H,E) \Rightarrow 0 < B(H \mid E) < 1)
    Annotation: With modest evidence, rational belief is strictly between 0 and 1.
  4. \forall H,E ;; (B(H \mid E) < 1 \Rightarrow D(H \mid E) > 0)
    Annotation: Whenever belief is less than certainty, rational doubt must remain positive.
  5. N \Rightarrow \forall H,E \in \mathcal{S} ;; (D(H \mid E) = 0)
    Annotation: Anti-doubt norms prescribe eliminating doubt in their scope of application.
  6. \exists H^{\ast},E^{\ast} \in \mathcal{S} ;; \text{Mod}(H^{\ast},E^{\ast})
    Annotation: There exist actual modest-evidence cases inside the norm’s domain.
Goal

\therefore ; N \Rightarrow \lnot \text{TT}
Annotation: The objective is to show that anti-doubt norms undermine truth-tracking.

Fitch-Style Derivation
  1. O(H \mid E) = \tfrac{\mathrm{P}(E \mid H)}{\mathrm{P}(E \mid \neg H)} \cdot O(H)
    Annotation: Rational credence is anchored to Bayesian proportionality.
  2. D(H \mid E) = 1 - B(H \mid E)
    Annotation: Doubt is always the complement of belief.
  3. \forall H,E ;; (\text{Mod}(H,E) \Rightarrow 0 < B(H \mid E) < 1)
    Annotation: With modest support, belief cannot rationally reach 0 or 1.
  4. \forall H,E ;; (B(H \mid E) < 1 \Rightarrow D(H \mid E) > 0)
    Annotation: If belief is less than certain, doubt is guaranteed to be positive.
  5. N \Rightarrow \forall H,E \in \mathcal{S} ;; (D(H \mid E) = 0)
    Annotation: Under the anti-doubt norm, doubt is mandated to be zero.
  6. \exists H^{\ast},E^{\ast} \in \mathcal{S} ;; \text{Mod}(H^{\ast},E^{\ast})
    Annotation: There is at least one modest-evidence case inside the norm’s scope.
  7. Assume N
    Annotation: Suppose the anti-doubt policy is in effect.
  8. \forall H,E \in \mathcal{S} ;; (D(H \mid E) = 0)
    Annotation: If the norm is adopted, doubt must vanish in all such cases.
  9. Choose H^{\ast},E^{\ast} \in \mathcal{S} with \text{Mod}(H^{\ast},E^{\ast})
    Annotation: Consider a concrete modest-evidence scenario within the norm’s reach.
  10. 0 < B(H^{\ast} \mid E^{\ast}) < 1
    Annotation: By definition of modest support, belief here is strictly partial.
  11. D(H^{\ast} \mid E^{\ast}) > 0
    Annotation: Rationality requires non-zero doubt in this case.
  12. D(H^{\ast} \mid E^{\ast}) = 0
    Annotation: Yet the anti-doubt norm demands eliminating all doubt.
  13. \bot
    Annotation: This yields a contradiction between rational requirement and policy.
  14. \text{OB}(H^{\ast},E^{\ast})
    Annotation: This is a case of over-belief: certainty-like posture without evidential warrant.
  15. \text{OB}(H^{\ast},E^{\ast}) \Rightarrow \lnot \text{TT}
    Annotation: Over-belief undermines truth-tracking because it distorts proportionality.
  16. \lnot \text{TT}
    Annotation: Therefore truth-tracking fails in this case.
  17. N \Rightarrow \lnot \text{TT}
    Annotation: Concluding: anti-doubt norms necessarily undermine truth-tracking.

◉ A plain English walkthrough of the Master Proof above.

Step 1: Setting the Stage

We begin with the Bayesian framework.

  • Belief in a claim should rise or fall depending on how strongly the evidence supports it.
  • Doubt is simply the flip side of belief. If you believe something 70% strongly, you still doubt it 30%.
  • Therefore, unless the evidence is completely decisive, some doubt must remain.
Step 2: The Norm in Question

Certain anti-doubt norms—like biblical exhortations that say “do not doubt”—demand that people erase doubt entirely.

  • In logical terms: under such a norm, doubt is required to be zero, no matter how incomplete the evidence is.
Step 3: A Realistic Case

But the world gives us many modest-evidence cases—situations where the evidence is mixed, ambiguous, or only moderately supportive.

  • In those situations, the Bayesian rule says: “Belief should be partial, not absolute.”
  • That means: if belief is partial, some non-zero doubt is required.
Step 4: The Collision

Now imagine applying the anti-doubt rule in one of these modest-evidence cases.

  • On the rational side: doubt should be greater than zero (because the evidence doesn’t settle the matter).
  • On the norm side: doubt must be zero (because the exhortation forbids it).
  • This creates a direct contradiction: rationality requires doubt, but the norm forbids it.
Step 5: Over-Belief

When you eliminate doubt in a case where the evidence is incomplete, you fall into over-belief.

  • Over-belief is a kind of epistemic distortion: you act as though the evidence is stronger than it really is.
  • That undermines what we call truth-tracking—the ability to let your beliefs reflect the world as it actually is.
Step 6: The Conclusion

Therefore:

  • Any general rule that prohibits doubt will, in realistic cases, lead to over-belief.
  • Over-belief breaks the proportional link between evidence and belief.
  • And that means anti-doubt norms undermine truth-tracking.
Walkthrough Summary
  1. Belief should match evidence, and doubt is the necessary complement.
  2. Anti-doubt rules forbid doubt entirely.
  3. But many real cases involve incomplete evidence, where doubt is rationally required.
  4. Forcing doubt to zero in those cases produces a contradiction.
  5. This contradiction generates over-belief—pretending the evidence is stronger than it is.
  6. Over-belief destroys truth-tracking.
  7. Thus: anti-doubt norms are incompatible with rational, evidence-sensitive belief.

◉ Narrative Summary

The argument begins with a simple observation: belief and doubt are not enemies, but complements. Whenever evidence for a claim is less than decisive, some degree of doubt is rationally required. This is because in the Bayesian framework, the strength of belief in a hypothesis rises or falls proportionally to the support of the evidence, and doubt is simply the remaining gap. If you believe something 70% strongly, then 30% of rational space is reserved for doubt. Only when evidence is overwhelming can doubt be eliminated.

Against this proportional model stand anti-doubt norms—such as those found in certain scriptural exhortations—that command people to erase doubt entirely. These norms operate as if evidence must always produce unqualified belief. Taken as a general policy, they demand that doubt be set to zero even in cases where the evidence is weak, ambiguous, or modest.

The tension emerges clearly when we consider a realistic case. Suppose someone confronts a claim supported only modestly by the evidence—enough to make it somewhat plausible but not certain. Rationality says: “Hold partial belief and retain some doubt.” The anti-doubt rule, by contrast, says: “Eliminate doubt.” These two directives cannot both be satisfied. Rationality requires some doubt, but the norm forbids it.

What follows is over-belief—the adoption of a level of confidence that the evidence does not justify. Over-belief is not just an error of degree; it is a distortion that breaks the link between belief and reality. By demanding the suppression of doubt where the evidence is incomplete, anti-doubt norms effectively dismantle truth-tracking. They cause people to treat weak or mixed evidence as if it were decisive, severing the proportional connection between evidence and credence.

The conclusion is therefore unavoidable: any system or worldview that prohibits doubt under conditions of incomplete evidence undermines the very conditions of responsible inquiry. Far from stabilizing belief, anti-doubt norms force it off balance, producing confidence that outstrips reality. By contrast, rehabilitating doubt as the rational partner of belief preserves proportionality, safeguards truth-tracking, and sustains the discipline of evidence-sensitive reasoning.

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