Symbolic Logic Reconstruction

The hypothesis that a prayer-answering God exists.
Annotation: This states the core theistic claim: God exists and answers prayer in accordance with the New Testament promises.

The hypothesis that no prayer-answering God exists (naturalism).
Annotation: This is the naturalistic alternative where prayer outcomes are indistinguishable from chance.

New Testament promises (e.g., Matthew 7:7–8; John 14:13–14; Mark 11:24) assert unqualified causal links between prayer and outcome.
Annotation: The scriptures explicitly say “everyone who asks receives,” “whatever you ask, I will do,” etc.

Empirical evidence shows prayer outcomes are indistinguishable from chance (STEP trials, meta-analyses, sociological comparisons).
Annotation: In reality, large-scale studies and lived experience reveal no measurable effect of prayer beyond natural probabilities.

---

Step 1: Predictions under each hypothesis

Annotation: If a prayer-answering God exists, the probability of observing outcomes different from chance is far higher than under naturalism.

Annotation: If naturalism is true, we expect prayer results to align with randomness, exactly what is observed.

---

Step 2: Actual evidence

Annotation: This is the central observed fact—prayer does not produce results exceeding chance expectations.

Annotation: The observed evidence is much less likely if a prayer-answering God exists than if naturalism is true.

---

Step 3: Likelihoodist comparison

Annotation: The likelihood ratio compares how well each hypothesis predicts the evidence.

Annotation: The ratio strongly favors naturalism, since the evidence is far more expected under than under .

---

Final Conclusion

Annotation: Given the indistinguishability of prayer outcomes from chance, the likelihoodist comparison decisively supports naturalism () over theism ().

---

Fitch-Style Natural Deduction (Likelihoodist Core)

◉ Formal vocabulary

Annotation: denotes “a prayer-answering God exists and the New Testament promises are literally reliable about outcomes.”

Annotation: denotes “no prayer-answering God exists” (naturalism).

Annotation: denotes “observed world in which prayer outcomes are indistinguishable from chance.”

Annotation: is the probability measure; conditionals like are likelihoods.

Annotation: is the likelihood ratio .

Annotation: and are informal comparative magnitudes (“much greater than,” “much less than”) sufficient for likelihoodist verdicts.

Principles and background commitments

If , then favors over .
Annotation: is the likelihood principle used explicitly in the paper’s reasoning.

If , then prayer produces outcome patterns that depart detectably from chance, i.e., and thus .
Annotation: codifies the unqualified New Testament promises as predictive commitments.

If , then prayers have no causal efficacy beyond natural baselines, i.e., .
Annotation: Under naturalism, chance-level prayer outcomes are exactly what we should expect.

.
Annotation: Large trials, meta-analyses, and everyday observations support as the empirical datum emphasized in the paper.

◉ Fitch-style derivation

1. 
   Annotation: From observation, holds.
2. 
   Annotation: From scripture-based predictions, if then .
3. 
   Annotation: From naturalistic background, if then .
4. (from 2, 3, by conjunction of predictive commitments)
   Annotation: Collecting the predictive contrasts supplied by and .
5. (from 4, by arithmetic comparison)
   Annotation: The evidence is far less likely under than under .
6. (definition)
   Annotation: By definition of the likelihood ratio .
7. (from 5, 6, by substitution)
   Annotation: The likelihood ratio massively disfavors relative to .
8. (from 5 via )
   Annotation: If (the evidence that prayer outcomes are indistinguishable from chance) holds, then the comparative relation states that is evidentially favored over .

Corollary (optional Bayesian posterior statement)

9. (Bayes’ theorem on odds)
   Annotation: Posterior odds equal likelihood ratio times prior odds.
10. (from 7, 9)
    Annotation: Unless priors are overwhelmingly against , pushes posterior credence toward .
11. (from 7–10, given non-extreme priors)
    Annotation: The posterior favors over once is taken into account.

◉ Auxiliary derivation on unfalsifiability retreat (capturing a second pillar)

A1.
Annotation: To reconcile with , apologists append .

A2.
Annotation: The stipulation schema makes every observational outcome consistent with .

A3.
Annotation: If no possible observation discriminates, then loses predictive specificity.

A4.
Annotation: Under universal compatibility, there is no prospective evidence that would favor .

A5.
Annotation: This captures the paper’s claim that retreat to stipulations erases testability and collapses the empirical content of prayer promises.

◉ Summary conclusion

is strongly favored over , and stipulative rescues of forfeit testability.
Annotation: The observed parity of prayer outcomes with chance supports over , and attempts to immunize from disconfirmation undermine its evidential standing, as argued in the paper.