➘ #28 Source Article
Symbolic Logic Formalization
This expresses Bayes’ theorem: the rational credence in hypothesis 
Here, doubt is defined as the mathematical complement of belief. If belief in
The Bayes factor represents how strongly the evidence favors 
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.
If the Bayes factor is modest, then rationality requires some degree of doubt. Commands to eliminate doubt contradict this evidential norm.
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.
![6.; \forall H, E ; [ ; \text{If } \Pr(H \mid E) < 1 ;; \Rightarrow ;; d(H \mid E) > 0 ; ]](https://s0.wp.com/latex.php?latex=6.%3B+%5Cforall+H%2C+E+%3B+%5B+%3B+%5Ctext%7BIf+%7D+%5CPr%28H+%5Cmid+E%29+%3C+1+%3B%3B+%5CRightarrow+%3B%3B+d%28H+%5Cmid+E%29+%3E+0+%3B+%5D&bg=ffffff&fg=000&s=0&c=20201002)
Scriptural or theological exhortations against doubt effectively demand zero doubt even where the evidence does not warrant certainty.
![7.; (\text{Anti-doubt exhortations}) ;\Rightarrow; (\exists H, E); [, d(H \mid E) = 0 ;\wedge; \Pr(H \mid E) < 1 ,]](https://s0.wp.com/latex.php?latex=7.%3B+%28%5Ctext%7BAnti-doubt+exhortations%7D%29+%3B%5CRightarrow%3B+%28%5Cexists+H%2C+E%29%3B+%5B%2C+d%28H+%5Cmid+E%29+%3D+0+%3B%5Cwedge%3B+%5CPr%28H+%5Cmid+E%29+%3C+1+%2C%5D&bg=ffffff&fg=000&s=0&c=20201002)
This creates the condition of over-belief: assigning more confidence than the evidence justifies.
![8.; (\exists H, E); [, d(H \mid E) = 0 ;\wedge; \Pr(H \mid E) < 1 ,] ;\Rightarrow; \text{Over-belief}](https://s0.wp.com/latex.php?latex=8.%3B+%28%5Cexists+H%2C+E%29%3B+%5B%2C+d%28H+%5Cmid+E%29+%3D+0+%3B%5Cwedge%3B+%5CPr%28H+%5Cmid+E%29+%3C+1+%2C%5D+%3B%5CRightarrow%3B+%5Ctext%7BOver-belief%7D&bg=ffffff&fg=000&s=0&c=20201002)
Over-belief by definition undermines truth-tracking mechanisms, because it misrepresents the evidential balance.
Therefore, norms that prohibit doubt undercut rational inquiry and prevent agents from aligning belief with reality.
A Fitch-Style Proof.
Annotation: This denotes the degree of belief in hypothesis
Annotation: Doubt is defined as the complement of belief; if belief is 0.7, doubt is 0.3.
Annotation: Prior and posterior odds compare the likelihood of 
Annotation: The Bayes factor measures how strongly the evidence
Annotation: A condition of modest evidential support: neither overwhelming nor negligible.
Annotation: The target epistemic norm: beliefs should proportionally track evidence.
Annotation: Over-belief occurs when doubt is eliminated despite incomplete evidence.
Annotation: The norm under critique: commands or policies that prohibit doubt.
Premises
Annotation: Posterior odds equal prior odds times the likelihood ratio.
Annotation: Doubt is mathematically defined as the complement of belief.
Annotation: With modest evidence, rational belief is strictly between 0 and 1.
Annotation: Whenever belief is less than certainty, rational doubt must remain positive.
Annotation: Anti-doubt norms prescribe eliminating doubt in their scope of application.
Annotation: There exist actual modest-evidence cases inside the norm’s domain.
Goal
Annotation: The objective is to show that anti-doubt norms undermine truth-tracking.
Fitch-Style Derivation
Annotation: Rational credence is anchored to Bayesian proportionality.
Annotation: Doubt is always the complement of belief.
Annotation: With modest support, belief cannot rationally reach 0 or 1.
Annotation: If belief is less than certain, doubt is guaranteed to be positive.
Annotation: Under the anti-doubt norm, doubt is mandated to be zero.
Annotation: There is at least one modest-evidence case inside the norm’s scope.- Assume
Annotation: Suppose the anti-doubt policy is in effect.
Annotation: If the norm is adopted, doubt must vanish in all such cases.- Choose
with
Annotation: Consider a concrete modest-evidence scenario within the norm’s reach.
Annotation: By definition of modest support, belief here is strictly partial.
Annotation: Rationality requires non-zero doubt in this case.
Annotation: Yet the anti-doubt norm demands eliminating all doubt.
Annotation: This yields a contradiction between rational requirement and policy.
Annotation: This is a case of over-belief: certainty-like posture without evidential warrant.
Annotation: Over-belief undermines truth-tracking because it distorts proportionality.
Annotation: Therefore truth-tracking fails in this case.
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
- Belief should match evidence, and doubt is the necessary complement.
- Anti-doubt rules forbid doubt entirely.
- But many real cases involve incomplete evidence, where doubt is rationally required.
- Forcing doubt to zero in those cases produces a contradiction.
- This contradiction generates over-belief—pretending the evidence is stronger than it is.
- Over-belief destroys truth-tracking.
- 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.
Free of Faith 
Leave a comment