The Argument from Silence, Part 2: Peter Kirby’s First Argument from Silence Against the Empty Tomb

Now that I have provided a Bayesian interpretation of arguments from silence, I want to evaluate my friend Peter Kirby’s argument from silence against the historicity of the empty tomb of Jesus (hereafter, “Kirby’s argument”). To be precise, in his essay, Kirby considers two arguments from silence. The first is based upon the silence of various writers regarding the empty tomb. The second is based upon the lack of veneration of the empty tomb. In this essay, I will be referring only to the former when I refer to “Kirby’s argument.”

To his credit, Kirby correctly notes that “there are better arguments from silence and worse arguments from silence.” This is because arguments from silence are best understood as inductive arguments. Like all inductive arguments, arguments from silence have degrees of strength; some can be stronger than others.

Kirby’s Criteria for Evaluating the Strength of Arguments from Silence
Allow me to quickly rehearse the logical form of the argument from silence as an explanatory argument before returning to Kirby’s interesting essay. Let B be the relevant background evidence. Let S be some proposition about the silence of a potential source of evidence regarding some hypothesis. Let H1 and H2 be rival explanatory hypotheses. Then the explanatory version of the argument from silence looks like this.

(1) S is known to be true, i.e., Pr(S | B) is close to 1.
(2) H1 is not intrinsically much more probable than H2, i.e., Pr(H1 | B) is not much more probable than Pr(H2 | B).
(3) S is antecedently more probable on H2 than on H1, i.e., Pr(S | H2 & B) > Pr(S | H1 & B).
(4) Therefore, other evidence held equal, H1 is probably false, i.e., Pr(H1 | B & S) < 0.5.

From a Bayesian perspective, the key question for evaluating arguments from silence is premise (3). Kirby proposes three interesting criteria which can be used to evaluate (3).

The first criterion is the presumption of knowledge. This criterion asks, how likely is it that a particular writer knew of an event if it had happened? The second criterion is the presumption of relevance. This criterion asks, how likely is it that the writer would mention this event in this document? The third criterion is applied after we have a number of different writers and documents that have been evaluated through the first two. The third one asks, how likely is it that all these documents fail to mention this event? While perhaps it would be understandable if any particular one failed to make a note of the event, the argument is strengthened by several silences when it would seem a strange coincidence for every one to happen not to mention the event.

I agree with Kirby that all of these criteria are relevant. Before evaluating Kirby’s argument, I want to provide a Bayesian interpretation of his third criterion.

To keep things manageable, suppose we have three potential sources of evidence about an event and all three are silent about the event. If we want to consider the cumulative effect of their silence, then we can consider the silence of each source as a distinct item of evidence as follows.

S1: the silence of source 1 regarding the event
S2: the silence of source 2 regarding the event
S3: the silence of source 3 regarding the event
S: S1 & S2 & S3

As I explained here (skip down to the section on “cumulative cases”), we can apply the chain rule to give a formal, mathematical definition of the probability relations which must obtain in order for an argument from silence to be logically correct. Using the chain rule it follows that:

Pr(S | H & B) = Pr(S1 & S2 & S3 | H & B) = Pr(S1 | H & B) x Pr(S2 | S1 & H & B) x Pr(S3 | S1 & S2 & H & B).

Therefore, S1, S2, and S3 are a cumulative case against H1 and for H2 just in case:

Pr(S | H1 & B) = Pr(S1 & S2 & S3 | H1 & B) = Pr(S1 | H1 & B) x Pr(S2 | S1 & H1 & B) x Pr(S3 | S1 & S2 & H1 & B)

is less than:

Pr(S | H2 & B) = Pr(S1 & S2 & S3 | H2 & B) = Pr(S1 | H2 & B) x Pr(S2 | S1 & H2 & B) x Pr(S3 | S1 & S2 & H2 & B).

Using that result, we can thus define the following schema for a cumulative case based upon the silence of multiple authors.

1. Pr(S1 | H2 & B) > Pr(S1 | H1 & B).
2. Pr(S2 | S1 & H2 & B) > Pr(S2 | S1 & H1 & B).
3. Pr(S3 | S1 & S2 & H2 & B) > Pr(S3 | S1 & S2 & H1 & B).
4. S is known to be true, i.e., Pr(S | B) is close to 1.
5. H1 is not intrinsically more probable than H2, i.e., Pr(H1 | B )=< Pr(H2 | B).
6. Other evidence held equal, H1 is probably false, i.e., Pr(H1 | B & S) < 0.5.

This schema can obviously be expanded as needed in order to accommodate the desired number of (silent) sources of evidence.

Kirby’s Argument Formulated (as an Explanatory Argument)

We are now in a position to formally state Kirby’s argument. 

B: The Relevant Background Evidence

B1.The historicity of Jesus, i.e., the New Testament Jesus is based upon a real historical individual. (Note: this proposition should not be interpreted as making any claims about the various deeds attributed to Jesus.)
B2. Jesus died by crucifixion.

B3. The canonical gospels, i.e., the gospels of Matthew, Mark, Luke, and John, all mention the empty tomb.
B4. The non-canonical Gospel of Peter is dependent on the canonical gospels.
B5. If the empty tomb were historical, then it is epistemically probable that any particular Christian writer would know that fact.

E: The Evidence to be Explained

E. Outside of the four canonical gospels, there is no independent source before Justin Martyr that mentions the tomb of Joseph of Arimathea or the discovery of the empty tomb.

H: The Proposed Explanatory Hypothesis and Its Alternatives

H: Jesus’ tomb was empty after his death and burial.
~H: Jesus’ tomb was not empty after his death and burial.

Kirby’s Argument Formulated

(1) E is known to be true, i.e., Pr(E | B) is close to 1.
(2) H is not intrinsically much more probable than ~H, i.e., Pr(H | B) is not much more probable than Pr(H | B). [Note: Kirby doesn't address the intrinsic probability of H at all, but I am adding this to be maximally charitable.]
(3) E is antecedently more probable on ~H than on H, i.e., Pr(E | ~H & B) > Pr(E | H & B).
(4) Therefore, other evidence held equal, H is probably false, i.e., Pr(H | B & E) < 0.5.

Kirby’s Defense of (3)

In defense of (3), Kirby writes this.

Although there may have been no particular reason for any one of these writers to mention the story, it could be argued that, if they all accepted the story, perhaps one of them would have entered a discussion that would mention the empty tomb story. For example, if there were a polemic going around that the disciples had stolen the body, one of these early writers may have written to refute such accusations. In any case, it is necessary to mention these documents if only to note that there is no conflicting evidence that would show that the empty tomb story was an early or widespread tradition since the argument from silence would be shown false if there were. (italics mine)

The problem with this paragraph, as emphasized by my italics, is the word “perhaps.” Possibility is not probability; in order to defend (3), what we need is an argument that E, i.e., the collective silence of the numerous writers listed by Kirby, is antecedently more probable on ~H than on H. But Kirby isn’t finished.

The writers of the foregoing documents would be likely to have been aware of the empty tomb story if it were true as opposed to a late legend or gospel fiction. If all these writers were aware of the empty tomb story, there is some reason to think that one of them would have mentioned the empty tomb story. Because none of them did, the argument from silence provides a reason to think that the empty tomb story is false. This does fall short of proof, but this should be given consideration as admissable [sic] historical evidence. If this were the only count against the empty tomb and there were very strong evidence for the empty tomb, then the judgment would fall in favor of the empty tomb. Nonetheless, this argument should be placed on the scales so that a complete assessment of the evidence is made.

I agree with Kirby that, on the assumption H is true, it is antecedently probable that the writers would have been aware of H. Thus, I think Kirby’s first criterion can be satisfied. But what about the second? Here I think Kirby’s argument is, at best, incomplete. Here is the relevant sentence from the above paragraph:

If all these writers were aware of the empty tomb story, there is some reason to think that one of them would have mentioned the empty tomb story. (italics mine)

Again, as my italics should make clear, this sentence merely states the needed conclusion without providing the argument to back it up. If all these writers were aware of the empty tomb, why is it epistemically probable that at least one of them would have mentioned it, i.e., Pr(E | H & B) > 0.5? Indeed, in light of B3, one obvious reply to Kirby’s argument is that these other writers were silent about the empty tomb because it had already been reported by the writers of the canonical gospels; these other writers did not believe that further reports of the empty tomb were needed. Kirby, however, says nothing about this. Therefore, as it stands, Kirby’s essay does not contain a defense of the claim that his argument passed his second criterion. Similarly, even without the benefit of a Bayesian analysis of the cumulative effects of the 24 documents which are (allegedly) silent about the empty tomb, I think it is obvious that Kirby has not yet provided a defense that his argument passes his third criterion. Therefore, Kirby’s defense of (3) is, at best, incomplete.

ISIS Violence IS Religious
The Theistic Arguments: A Brief Critique
“A Leap of Faith”
Lessing’s Broad Ditch and Brad’s Lesser Ditch
About Jeffery Jay Lowder

Jeffery Jay Lowder is President Emeritus of Internet Infidels, Inc., which he co-founded in 1995. He is also co-editor of the book, The Empty Tomb: Jesus Beyond the Grave.

  • Peter Kirby

    I believe you have represented the argument quite fairly as it stands
    alone. It is a relatively weak argument in the essay, which was
    intended to present many arguments, some of which had little or no
    value. When I rated that argument in terms of its value as evidence,
    in the conclusion of the essay, I assigned it only a 1 point value
    ("The Silence of Early Christians"). Of course you may argue that it
    has no value, but I want to point that out.

    I just noticed that it happens to fall first in the outline. Perhaps I
    should have arranged my essay to indicate more clearly my evaluation
    of the merit of the arguments. It's been a long time since I looked at

    I agree that the essay didn't establish that there is good reason to
    presume that any one of these writers would have mentioned the empty
    tomb (or that at least one of the lot would, for that matter). In that
    regard, your emphasis on the word "perhaps" is sound.

    I like the fact that you have framed the argument in Bayesian terms.
    One of the criticisms of my essay (not from you) is that it used terms
    such as probability without adopting any formalism that would allow it
    to be evaluated. And I think I would adopt an approach similar to
    yours if I did revisit it in such terms.

    While I would emphasize again the meager value of the fact as
    evidence, I am still left with the conclusion that it has meager
    value, if I understand correctly the following.

    If the writer didn't know about it, he didn't write about it. Let "K"
    be that he knew about it and "W" be that his writing mentioned it. P(
    W | ~K ) = 0 and P( ~W | ~K ) = 1.

    Now perhaps he wrote about it if he knew about it. So P( W | K ) > 0.
    And P( ~W | K ) < 1. There is some value for P( W | K ) that may be
    small, such as 0.01.

    What I want to say is how a result of ~W affects the posterior
    probability of K. That is,

    P( K | ~W ) = P( ~W | K ) * P( K ) / P( ~W )

    Where P( ~W | K ) is some value less than 1, such as 0.99, as stated
    above. P( K ) is the anterior probability of K, and P( ~W ) is the
    anterior probability of ~W.

    P( ~K | ~W ) = P( ~W | ~K ) * P( ~K ) / P( ~W ) = P( ~K ) / P( ~W )

    Because P( ~W | ~K ) is said to be 1, above.

    I said "if I understand correctly the following" because my
    understanding of using Bayesian concepts is not especially strong and
    gets weaker when trying to apply them to a non-repeatable experiment.
    But if I understood correctly, then I think you could say, when
    putting this back into English, that the chance of K, of the writer
    knowing about the event, is ever so slightly lower, even if there is
    just the small probability that that they *perhaps* would have
    mentioned it had they known about it.

    And yes it is a little embarrassing not to be completely comfortable
    with using these concepts when I am going back to school to get a
    degree in Math. :)

    Thank you again for giving me a chance to reply concerning your
    article. I would put this on a blog now except that I tend to write a
    few posts and then nothing for several years and delete them. :)

    best regards,
    Peter Kirby

  • Jeffery Jay Lowder

    If the writer didn't know about it, he didn't write about it. Let "K" be that he knew about it and "W" be that his writing mentioned it. P( W | ~K ) = 0 and P( ~W | ~K ) = 1.

    I cannot agree that Pr(W | ~K) = 0, since I can't rule out the possibility that a writer would mention something he had no knowledge of. Likewise, I cannot agree that Pr(~W | ~K) = 1 and for the same reason. It seems to me that Pr(W | ~K) > 0 and Pr(~W | ~K) <1. Maybe, for the sake of argument, Pr(W | ~K) = .001 or even 10^-20, but not zero.

  • Peter Kirby

    I can agree to that too.

    So long as it is less than the amount assigned Pr( W | K ), the
    outcome of the argument would be the same.

    I understand the concern for not using the numbers 0 and 1. I only
    used them because of the arbitrary nature of the assignments in
    general, and I thought it a bit more straightforward at the time than
    assigning an arbitrary very small number. I did not want to give the
    impression of presuming to have information that would allow the
    probability to be determined more exactly.

  • Pulse

    Shouldn't a fourth criterion involve the survival of documents to the present day? I'm asking because there are stories of early Christians destroying documents that criticized them. This would have created an artificial silence.

  • Jeffery Jay Lowder

    Pulse: Yes. In fact, I applied pretty much that exact criterion to William Lane Craig's argument from silence regarding the (alleged) lack of competing burial traditions, which is part 5 of my series on arguments from silence.

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