Formalization of Core Hypotheses
An omnipotent, clarity-seeking God intervenes in the public world without natural constraint.
Annotation: Hypothesis 
states that God performs public miracles, unconstrained by physical laws, aiming for recognition.
No such deity intervenes; miracle reports arise from ordinary processes (chance, bias, error, psychosomatic effects).
Annotation: Hypothesis 
is naturalism/non-intervention: all miracle reports can be explained through natural or psychological mechanisms.
Predictions from Each Hypothesis
![H_e \Rightarrow (\exists M),[M \in { \text{improbable lawful events} ,\cup, \text{genuine law-violations}}]](https://s0.wp.com/latex.php?latex=H_e+%5CRightarrow+%28%5Cexists+M%29%2C%5BM+%5Cin+%7B+%5Ctext%7Bimprobable+lawful+events%7D+%2C%5Ccup%2C+%5Ctext%7Bgenuine+law-violations%7D%7D%5D+&bg=ffffff&fg=000&s=0&c=20201002)
Annotation: If is true, we should observe both highly improbable but physically lawful events and unmistakable violations of physical law.
![H_a \Rightarrow (\forall M),[M \in { \text{fringe-of-possibility lawful events only}}]](https://s0.wp.com/latex.php?latex=H_a+%5CRightarrow+%28%5Cforall+M%29%2C%5BM+%5Cin+%7B+%5Ctext%7Bfringe-of-possibility+lawful+events+only%7D%7D%5D+&bg=ffffff&fg=000&s=0&c=20201002)
Annotation: If is true, miracle claims will cluster at the edge of ordinary processes (rare recoveries, coincidences, psychological effects), but never cross into genuine impossibility.
Empirical Observation
Annotation: The actual observed miracle data consist overwhelmingly of events that are improbable yet possible under natural law, with a conspicuous absence of public, instrumented impossibilities.
Likelihood Comparison
Annotation: The probability of seeing only fringe lawful events is much higher under naturalism than under divine-interventionism.
Bayesian Update Formulation
Annotation: The posterior odds favoring over are determined by the likelihood ratio and the prior odds. Since , the evidence strongly tilts the posterior toward .
Auxiliary Hypotheses Issue
Annotation: Theists often modify with auxiliary assumptions. These adjustments shrink predictive boldness and make the hypothesis unfalsifiable, thereby reducing its explanatory power.
Conclusion
Annotation: Given the distribution of miracle claims (clustering at the fringe of possibility with no documented impossibilities), the comparative likelihood strongly supports naturalism/non-intervention over divine public intervention.
Fitch-style derivation of the paper’s core argument
Vocabulary and target claim
An omnipotent, clarity-seeking God publicly intervenes without natural constraint.
Annotation: says God acts in public, aiming to be recognized, and is not bound by physical laws.
No such deity intervenes; miracle reports arise from ordinary processes.
Annotation: is naturalism/non-intervention; reports come from coincidence, bias, error, placebo, psychosocial mechanisms.
Observed miracle claims overwhelmingly cluster among improbable-yet-lawful events and lack public, instrumented law-violations.
Annotation: 
summarizes the data pattern emphasized in the paper: fringe-of-possibility clustering, no verified impossibilities.
Annotation: Likelihood notation; 
is the probability of seeing if 
were true.
Target comparative claim:
Annotation: If is much more probable on than on , then supports over (Law of Likelihood).
Predictions
Annotation: Under , a public, recognition-aiming deity yields a mixed portfolio: both rare lawful events and at least some public, unmistakable impossibilities.
Annotation: Under , we should see fringe lawful events (coincidence, remission, expectancy effects), but not genuine impossibilities.
Observational premise
Observed 
s are overwhelmingly lawful-improbable 
no public, instrumented law-violations
Annotation: The data exhibit fringe clustering and lack verified impossibilities in modern, instrumented contexts.
Likelihood assignments justified by 1–3
Annotation: On , is expected: high-trial environments produce rare lawful outliers; placebo/expectancy and selection effects abound. So 
is large.
Annotation: On , the absence of any well-documented impossibilities alongside persistent fringe clustering is surprising; thus 
is small.
Annotation: The observed distribution strongly favors at the level of likelihoods.
Law of Likelihood and posterior odds
Annotation: Law of Likelihood: data favor the hypothesis that renders them more probable. (Strictly, “favors” is comparative, not absolute.)
Annotation: Posterior odds equal the Bayes factor times the prior odds. With a large likelihood ratio, posteriors tilt toward unless priors are extremely lopsided.
Annotation: Ceteris paribus, the evidence moves credence toward .
Auxiliary-loading (dilution of predictive content)
Annotation: A common move is to add auxiliaries to to align it with .- Adding auxiliaries that suppress clear impossibilities
reduced predictive boldness and discriminability
Annotation: As is insulated to fit ambiguous data, it becomes less testable and loses comparative power. This does not raise in a principled way; it flattens distinct predictions.
Fitch-style subproof structure (textual layout)
Annotation: Begin subproof under .
Annotation: Under , at least some public impossibilities are expected.
Annotation: The observed record reports none with adequate documentation.
Annotation: Within this subproof, conflicts with the live expectation under (not a strict contradiction, but a strong likelihood penalty).
Annotation: Exit the subproof; we have justified line 5.
Annotation: Begin subproof under .
Annotation: Under , no impossibilities are expected.
Annotation: Matches the prediction.
Annotation: Within this subproof, aligns with ; we justify line 4.
Annotation: Exit the subproof; we have justified the high likelihood under .
Annotation: We have the comparative evidential verdict that the paper defends.
Note on scope (consistency with the paper)
Scope: The inference is comparative over the miracle-testimony stream alone, not over “all evidence for/against theism.
Annotation: As in the paper, the claim is explicitly limited to the evidential weight of contemporary miracle testimony considered by itself. Other arguments for theism are not evaluated here.
◉ A plain English walkthrough of the symbolic logic above.
The competing hypotheses
- : If there is an omnipotent, clarity-seeking God who acts in public without natural constraint, then we should see both unusual but lawful events (like rare recoveries) and at least some unmistakable impossibilities (like breaking the laws of physics).
- : If no such God intervenes, then all miracle reports should fall into the category of improbable-yet-lawful events, with no genuine law-breaking.
What we actually observe ()
- The data show a consistent pattern: miracle claims cluster on the fringe of possibility. They involve rare but naturally possible events (recoveries, coincidences, subjective experiences), but there are no verified public cases of impossible events (such as violations of conservation laws or moving mountains).
Predictions compared to the data
- On , the absence of impossibilities is surprising. If God’s purpose is clarity, we should see at least some unmistakable miracles that cannot be mimicked by natural chance.
- On , the absence of impossibilities is exactly what we would expect, since human psychology, coincidence, and placebo already generate improbable-but-lawful outcomes.
Likelihood assignment
- Therefore, the probability of the observed data given is high.
- The probability of the observed data given is low.
- So, the likelihood ratio
is very large.
Implication under the Law of Likelihood
- Since is much more expected on than on , the data count as evidence in favor of over .
Bayesian update
- Posterior odds depend on both the prior odds and the likelihood ratio. Even if one starts with moderate priors favoring , the strong likelihood advantage for shifts the balance toward .
Auxiliary hypotheses issue
- Some defenders of add auxiliary assumptions: God prefers ambiguity, avoids coercion, or only rarely intervenes.
- But these moves weaken predictive power. Instead of explaining why impossibilities are absent, they redefine so that almost any data “fit.” This makes less testable and less discriminating than .
Overall conclusion
- Given the actual distribution of miracle claims, the data are strongly aligned with and poorly aligned with .
- Thus, contemporary miracle testimony — considered on its own — provides comparative support for naturalism/non-intervention over the hypothesis of a clarity-seeking, miracle-working deity.
◉ Prose Version
When we compare the two hypotheses about miracles, the difference comes down to what each would lead us to expect. If an omnipotent and clarity-seeking God were active in the public sphere, as proposes, we should see not only unusual but still lawful events—such as rare medical recoveries—but also unmistakable impossibilities, clear violations of physical law that leave no room for natural explanation. By contrast, if no such God intervenes, as states, then all miracle reports should cluster at the edge of possibility: rare coincidences, psychosomatic healings, and emotionally vivid experiences, but no genuine law-breaking.
The actual distribution of miracle claims, , shows exactly this fringe clustering. Reports highlight improbable yet lawful outcomes, while well-documented, public impossibilities are absent. This pattern is surprising on , since a clarity-seeking God would have strong reason to produce some unmistakable demonstrations. But it is precisely what predicts, given the ubiquity of coincidence, placebo, and bias in human life.
As a result, the probability of observing is high under and low under . In likelihoodist terms, 
. By the Law of Likelihood, this means that the data favor over . Even in Bayesian terms, unless the prior odds against are extraordinarily steep, the strong likelihood advantage shifts the balance toward naturalism.
Attempts to preserve often involve auxiliary assumptions: perhaps God avoids coercing belief, prefers subtlety, or intervenes only rarely. But these modifications dilute predictive content. They do not explain the absence of impossibilities so much as redefine the hypothesis to accommodate ambiguity. That move reduces discriminability, leaving as the simpler and more testable account.
In sum, the observed pattern of miracle claims—concentrated at the fringe of possibility and devoid of impossible-class events—fits seamlessly with naturalistic expectations while resisting the predictions of a clarity-seeking deity. Contemporary miracle testimony, taken on its own, is therefore stronger evidence for than for .
Free of Faith 
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