#20-Formalization

August 25, 2025

➘ #20 Source Article

Symbolic Logic Formalization

$P_1$ Rational belief requires confidence to be proportionate to the degree of evidential support.
Annotation: Belief strength must scale with evidence (e.g., 60% evidence → 0.6 confidence).

$P_2$ Biblical faith is typically depicted as binary (1 or 0), demanding unwavering certainty regardless of evidential support.
Annotation: Faith functions like an on/off switch, immune to evidence.

$P_3$ If a belief system disregards evidence-proportionate calibration, it fails to track truth reliably.
Annotation: Rationality requires beliefs that adapt with shifting evidence; rigidity misrepresents reality.

$P_4$ Biblical faith stigmatizes doubt, treating it as weakness rather than rational recalibration to evidence.
Annotation: Doubt is rational in evidentialist frameworks, but condemned in binary faith.

$P_5$ Any epistemic model that valorizes certainty independent of evidence is epistemically rigid and inferior to evidence-responsive rationality.
Annotation: Certainty divorced from evidence fosters distortion, bias, and poor decisions.

$C$ Therefore, biblical faith cannot be reconciled with core rationality and is epistemically inferior to evidence-proportionate belief.
Annotation: Rational belief requires proportionality and adaptability; binary faith promotes static conviction, making them incompatible.

An expanded syllogistic chain version.

$D_1$ Let $E(p)$ be the evidential support in $[0,1]$ for proposition $p$ ; let $C(p)$ be an agent’s credence in $p$ in $[0,1]$ .
Annotation: $E(p)$ measures how strongly the available evidence supports $p$ ; $C(p)$ is the confidence level the agent assigns to $p$ .

$D_2$ Define truth-tracking for a system $S$ as: $\mathsf{TT}(S)\iff(\forall p,\ C(p)\approx E(p))\ \wedge\ (\forall p,e,\ C_{t+}(p)=C_t(p\mid e))$ .
Annotation: A system is truth-tracking when credences roughly match evidential degrees and update by Bayesian conditionalization on new evidence $e$ .

$D_3$ Define the binary-faith model $\mathsf{BF}$ by: (i) $\forall p\in\mathcal{F},\ C(p)\in{0,1}$ ; (ii) for disconfirming $e$ , typically $C_{t+}(p)=C_t(p)$ .
Annotation: In the biblical binary-faith pattern, credence in faith-targets $p$ is “on/off” and tends not to diminish when contrary evidence appears.

$D_4$ Let the relation $\prec_{\mathsf{epi}}$ mean “is epistemically inferior to.”
Annotation: For systems $S_1,S_2$ , $S_1\prec_{\mathsf{epi}}S_2$ abbreviates: $S_1$ systematically yields less accurate, less evidence-responsive credences than $S_2$ .

$P_1$ Rational-belief systems $\mathsf{RB}$ satisfy truth-tracking: $\mathsf{RB}\Rightarrow\mathsf{TT}$ .
Annotation: Core rationality requires proportionality to evidence and Bayesian updating; hence any genuinely rational system is truth-tracking.

$P_2$ If a system is not truth-tracking while an alternative is, then it is epistemically inferior: $\neg\mathsf{TT}(S_1)\ \wedge\ \mathsf{TT}(S_2)\ \Rightarrow\ S_1\prec_{\mathsf{epi}}S_2$ .
Annotation: Failing to align with and update to evidence makes a system worse, epistemically, than one that does.

$P_3$ $\mathsf{BF}$ violates the proportionality and updating conditions in $\mathsf{TT}$ by (3-i) and (3-ii).
Annotation: Because $C(p)$ is fixed at $0$ or $1$ irrespective of $E(p)$ , and because $C(p)$ resists reduction under disconfirming $e$ , $\mathsf{BF}$ does not track truth as defined.

$C_1$ From $P_3$ and $D_2$ : $\neg\mathsf{TT}(\mathsf{BF})$ .
Annotation: Given its binary, evidence-resistant structure, $\mathsf{BF}$ fails the truth-tracking criterion.

$C_2$ From $P_1$ , there exists a rational system $\mathsf{RB}$ with $\mathsf{TT}(\mathsf{RB})$ . From $C_1$ and $P_2$ : $\mathsf{BF}\prec_{\mathsf{epi}}\mathsf{RB}$ .
Annotation: Because rational systems truth-track while $\mathsf{BF}$ does not, $\mathsf{BF}$ is epistemically worse than evidence-proportionate rational belief.

$C_F$ Therefore: $\mathsf{BF}\prec_{\mathsf{epi}}\mathsf{RB}$ and cannot be reconciled with core rationality.
Annotation: Binary biblical faith is epistemically inferior to evidence-proportionate belief and stands in structural conflict with core rationality’s requirements.

◉ A plain English walkthrough of the symbolic logic above.

1. Setting up the terms.
We begin by distinguishing two key quantities:
✓ E(p) = how much the evidence supports a proposition p, on a scale from 0 to 1.
✓ C(p) = how confident a person is in p, again on a scale from 0 to 1.
This gives us a way to measure whether confidence is aligned with evidence.

2. What it means to track truth.
A belief system is truth-tracking if two conditions hold:
✓ Confidence matches the evidence (if the evidence is 70%, confidence should be about 0.7).
✓ Confidence updates properly when new evidence appears, using a rule like Bayesian conditionalization.

3. What binary faith looks like.
Biblical faith can be formalized as a model (BF) where:
✓ Confidence is either 0 or 1 (on/off, nothing in between).
✓ Once belief is “on,” it doesn’t really change even if new evidence goes against it.
This means it doesn’t scale with evidence or update in response to it.

4. The comparison standard.
We also define a comparison: one system is epistemically inferior to another if it tracks truth less reliably—meaning it produces beliefs that are less responsive to evidence.

5. Rational belief systems.
By contrast, rational belief systems (RB) are defined as truth-tracking. They obey proportionality (confidence reflects the evidence) and updating (beliefs change when new evidence arises).

6. Evaluating binary faith.
Since binary faith (BF) locks confidence at either 0 or 1 and resists updating, it fails both conditions required for truth-tracking.

7. Drawing the conclusion.
From this, it follows that:
✓ Rational systems (RB) are truth-tracking.
✓ Binary faith (BF) is not truth-tracking.
✓ Therefore, BF is epistemically inferior to RB.

8. Final outcome.
Thus, binary faith is structurally incompatible with core rationality. It prioritizes certainty over evidence and cannot be reconciled with the principle that belief strength should proportionately mirror evidential support. vastly more probable given cultural conditioning than given free will or divine fairness.

◉ Narrative Summary

The contrast between rational belief and biblical faith can be formalized in terms of how each system aligns with evidence. Rational belief operates on the principle that confidence should mirror the degree of evidential support. If the evidence for a proposition is partial—say sixty percent—then one’s confidence should be calibrated to roughly that same degree. Moreover, rational belief requires responsiveness: when new evidence emerges, beliefs must be adjusted in proportion, typically through Bayesian updating. In this sense, rational systems are truth-tracking, because they continuously recalibrate to the evidential landscape.

Binary faith, by contrast, departs sharply from this model. It treats confidence as an on–off switch, allowing only two settings: absolute belief or absolute disbelief. Once belief is affirmed, the system resists recalibration even when contrary evidence arises. Doubt is not regarded as a signal to re-balance conviction but rather as a weakness to be overcome. Because of this rigidity, binary faith fails the basic test of truth-tracking—it neither scales confidence proportionally to evidence nor permits the adjustments that rational belief demands.

When the two systems are compared, the epistemic hierarchy becomes clear. Rational belief, by virtue of its proportionality and updating rules, tracks truth more faithfully. Binary faith, by resisting proportionality and rejecting updating, systematically misrepresents reality. It is therefore epistemically inferior to rational belief. This incompatibility reveals why biblical faith cannot be reconciled with core rationality: where rationality prioritizes humility, adaptability, and evidence-responsiveness, binary faith valorizes static certainty, even at the cost of truth.

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