How did Richard Swinburne and Stephen Davis respond to Hume's argument using Bayesian probability?

The response of Richard Swinburne and Stephen Davis to David Hume's argument against miracles represents one of the most important developments in contemporary philosophy of religion, where they employed Bayesian probability theory to reformulate the debate in precise mathematical terms, thereby revealing logical gaps in Hume's classical argument.

Inadequate Responses to Avoid

From some believers:

"Hume is a biased atheist against religion, so his arguments are rejected a priori." This is a lazy response. Hume presented a systematic philosophical argument that deserves a systematic reply, not emotional rejection. Even if he were biased, the argument should be evaluated by its logical merits, not by the motivations of its proponent.

"Miracles are true because the sacred texts say so." Circular reasoning. The question is: how do we know these texts are truthful? One of the alleged evidences is miracles. So we cannot prove miracles by the texts and then prove the texts by miracles.

"Mathematics and probabilities do not apply to divine matters." This evades the discussion. If miracles are events in the natural world (healing, resurrection of the dead, parting of seas), they are amenable to evaluation by logical and mathematical tools.

From some critics:

"Swinburne's and Davis's responses are mathematically complex to hide their weakness." An accusation without evidence. Mathematical complexity is not a flaw if it serves methodological precision. Physics is mathematically complex—does this invalidate it?

"Bayesian probability is subjective, so it doesn't solve the problem." A half-truth. True, Bayesianism involves subjective priors, but its strength lies in how it updates these probabilities with evidence. The discussion shifts from "are miracles possible?" to "what strength of evidence is required?"

Why These Responses Are Inadequate

They avoid the core of the discussion: how do we methodologically evaluate miracle testimonies? Hume posed a logical challenge that requires a logical response, and this is what Swinburne and Davis provided.

Hume's Original Argument: Quick Reminder

In "Of Miracles" (1748), Hume presented a dual argument:

The principled argument: A miracle is "a violation of natural law." The uniform experience of humanity supports natural laws, so the prior probability of a miracle is nearly zero. Any human testimony is subject to error or deception, so the probability of false or mistaken testimony is always greater than the probability of a miracle occurring.

The applied argument: Historically, miracle testimonies come from "primitive" societies, or from religiously biased individuals, or in contexts lacking scientific documentation.

Swinburne's Response: Bayesian Reformulation

In "The Concept of Miracle" (1970) and "The Existence of God" (2004), Swinburne reformulated the discussion:

Basic Bayesian Formulation:
```
P(M|E) = P(E|M) × P(M) / P(E)
```

Where:
- P(M|E) = probability of miracle after testimony
- P(E|M) = probability of testimony if miracle occurred
- P(M) = prior probability of miracle
- P(E) = overall probability of testimony

Swinburne's Critique of Hume:

First, Hume conflates "low prior probability" with "impossibility." Yes, P(M) is low, but not zero. If God exists, miracles are logically possible.

Second, Hume ignores P(E|M). If a miracle actually occurred, what is the probability that reliable persons would testify to it? It might be very high. For example, if someone is healed from an incurable disease before doctors, the probability of their testimony is high.

Third, Hume assumes that P(E|¬M) (probability of testimony without miracle) is always high. But in some cases, it's difficult to explain the testimony without a miracle. For instance, testimony from early Christianity's enemies about Christ's resurrection requires explanation.

Swinburne's Applied Example:

Suppose:
- P(M) = 0.001 (low prior probability)
- P(E|M) = 0.9 (if miracle occurred, reliable witnesses will testify)
- P(E|¬M) = 0.01 (difficult to explain reliable testimony without miracle)

By Bayesian calculation:
```
P(M|E) ≈ 0.083
```

The testimony raised the probability from 0.1% to 8.3%—a significant increase!

Stephen Davis's Response: Development and Deepening

In "God and the Ethics of Belief" (2005) and "Christian Philosophical Theology" (2006), Davis developed the Bayesian response:

Distinguishing Types of Miracles:

Davis distinguished between:
- "Direct intervention" miracles (like parting the sea)
- "Divine timing" miracles (storm saving a believing army)
- "Personal transformation" miracles (radical change in personality)

Each type has different Bayesian probabilities. Timing miracles are more acceptable than direct intervention miracles.

The "Theistic Background" Argument:

Davis emphasized that P(M) is not constant but depends on background knowledge:
- In an atheistic background: P(M) ≈ 0
- In an agnostic background: P(M) is low but present
- In a theistic background: P(M) is reasonable

This means that evaluating miracles cannot be separated from the broader discussion about God's existence.

The "Multiple Independent Testimony" Argument:

If several independent persons testify to the same miracle, the probabilities multiply Bayesianly. Five independent testimonies with 10% error probability each give only 0.001% collective error probability.

Contemporary Applications

The Lourdes Miracles Case:

Precise medical examination of 70 "miraculous healing" cases recognized among 7000 claims. Bayesian application:
- P(natural healing) very low for selected cases
- P(medical testimony|actual healing) high
- Result: reasonable probability of non-natural intervention

Resurrection Testimonies Case:

Applying the Bayesian method to Christ's resurrection testimonies (as in works by Richard Swinburne and Timothy McGrew):
- Multiple testimonies from disadvantaged parties (disciples risked their lives)
- Difficulty explaining Christianity's emergence without extraordinary event
- The debate remains open, but the Bayesian framework clarifies strengths and weaknesses

Counter-Criticism and Responses

"Bayesian Circularity" Criticism:

Some critics (Jordan Howard Sobel in "Logic and Theism") claimed that Bayesianism leads to circularity: we need P(God) to calculate P(miracle), and we need P(miracle) as evidence for God.

Response: It's not circularity but interaction. Different evidences (cosmological, moral, religious) interact Bayesianly. Each evidence raises the probability slightly, and the total is cumulative.

"Arbitrary Prior Probabilities" Criticism:

Prior probabilities in Bayesianism are partially subjective. A critic and a theist will start with very different probabilities.

Response: Bayesianism doesn't claim absolute objectivity but clarifies how evidence should affect beliefs. Even with different priors, strong evidence can approximate results.

Contemporary Positions

The "Modified Bayesian" current (Lydia McGrew, Timothy McGrew) develops more precise Bayesian tools for evaluating historical testimonies.

The "Bayesian Criticism in Religion" current (Herman Philipse, Graham Oppy) sees limits to Bayesian application in metaphysical matters.

The "Integration with Cognitive Psychology" current (Justin Barrett, Kelly James Clark) studies how humans psychologically evaluate miracles and its effect on Bayesian calculations.

Where We Stand in This Debate Today

Swinburne's and Davis's response changed the nature of the debate. The question is no longer "can miracles be believed?" (Hume's answer: no), but "what strength of evidence is required?" and "how do we evaluate this evidence methodologically?"

Bayesianism doesn't "prove" miracles, but it provides a methodological framework for discussion. It shows that Hume's absolute rejection is logically unjustified, and that evaluating miracles requires careful analysis of each case.

For Advanced Reading

- Advanced level: Lydia McGrew's Bayesian applications to New Testament testimonies
- Richard Swinburne, The Concept of Miracle (Macmillan, 1970)
- Stephen T. Davis, "Hume on Miracles" in God, Reason and Theistic Proofs (Edinburgh UP, 1997)