WHERE WE ARE AT

In Part 1 and Part 2 of this series, I have shown that Dr. Erasmus’ objection to my skeptical reasoning (a) attacks a STRAW MAN, and (b) is based on an INVALID INFERENCE.  In doing so, I also argued that Dr. Erasmus does not have a good understanding of probability calculations, especially concerning the use of Bayes’ Theorem.

In this post I will further defend my skeptical reasoning about the resurrection of Jesus found in the post that has been criticized by Dr. Erasmus. I will do so by presenting this skeptical reasoning in more detail, with more logical and mathematical rigor, and with greater clarity.

SOME BASIC CONCEPTS AND PRINCIPLES OF PROBABILITY

If we know for certain that statement (S) is TRUE, then the probability of (S) is ONE:

P(S)  =  1.0

If we know for certain that statement (S) is FALSE, then the probability of (S) is ZERO:

P(S)   =   0

If it is as likely as not that (S) is TRUE, then the probability of (S) is ONE-HALF:

P(S)   =   .5

The probability of each and every statement is greater than or equal to ZERO, and is less than or equal to ONE:

0  ≤  P(S)  ≤  1.0

Because a statement is either TRUE or it is not TRUE, the probability that a particular statement is TRUE when combined with the probability that it is not TRUE is equal to ONE:

P(S) + P(~S)  =  1.0

If two statements have the same truth value in any circumstance, then they are logically equivalent:

(S)  ≡  (Q)

If two statements are logically equivalent, then the probabilities of those statements are EQUAL:

IF (S)  ≡  (Q), THEN P(S)  =  P(Q)

If one statement logically implies another statement, then the probability of the second statement is greater than or equal to the probability of the first statement:

IF (S)  É  (Q), THEN  P(Q)  ≥  P(S)

The probability of one statement (S) GIVEN THAT another statement (Q) is the case is a conditional probability:

P(S|Q)

If the statement (Q) logically implies the statement (S), then the conditional probability of (S) GIVEN THAT (Q) is ONE:

IF (Q)  É  (S), THEN  P(S|Q)  =  1.0

If the statement (Q) logically implies that statement (S) is not the case, then the conditional probability of (S) GIVEN THAT (Q) is ZERO:

IF (Q)  É  (~S), THEN P(S|Q)  =  0

We can define the probability of a CONJUNCTION of two statements in terms of a CONDITIONAL PROBABILITY:

P(S & Q)  =  P(S|Q)  x  P(Q)

Note that the probability of (S & Q) is EQUAL TO the probability of (Q & S), in other words the order of the conjuncts makes no difference to the probability of the conjunction:

P(S & Q)  =  P(Q & S)

So we can infer another equation concerning the probability of a CONJUNCTION:

P(Q & S)  =  P(S|Q)  x  P(Q)

A basic principle of CONDITIONAL PROBABILITY is that the probability of one statement GIVEN THAT another statement is the case is EQUAL TO the probability that both statements are true DIVIDED BY the probability that the second statement is true:

P(S|Q)  =  P(S & Q) / P(Q)

Note that the probabilities on the right side of the above equation are UNCONDITIONAL PROBABILITIES, so this is a definition of CONDITIONAL PROBABILITY in terms of UNCONDITIONAL PROBABILITIES.   Because division by zero is undefined, the value of P(Q) must be greater than zero for this formula to work.

BAYES’ THEOREM can also be derived from the above basic principle of CONDITIONAL PROBABILITY:

P(S|Q)  =  [P(Q|S) x P(S)]  /  P(Q)

If the UNCONDITIONAL PROBABILITY of a statement EQUALS the probability of that statement GIVEN THAT another statement is the case, then those statements are INDEPENDENT of each other, in other words, the truth value of one statement has no impact on the probability of the other statement:

IF P(S)  =  P(S|Q), THEN (S) and (Q) are INDEPENDENT statements.

The probability of the CONJUNCTION of two INDEPENDENT statements EQUALS the product of the probabilities of those statements:

IF (S) and (Q) are INDEPENDENT statements, THEN P(S&Q)  =  P(S)  x  P(Q)

INITIAL ANALYSIS OF “GOD RAISED JESUS FROM THE DEAD”

(GRJ) God raised Jesus from the dead.

(GPM) God has performed at least one miracle.

(JRD) Jesus rose from the dead.

What does the statement “God raised Jesus from the dead.” mean?  The meaning of a statement consists of the logical implications of that statement.  Here are some important logical implications of (GRJ):

• (GRJ) implies (GPM)
• (GRJ) implies (JRD)

Recall this basic principle of probability:

IF (S) implies (Q), THEN  P(Q)  ≥  P(S)

Based on this principle, we can draw some important inferences:

• P(GPM)  ≥  P(GRJ)
• P(JRD)   ≥  P(GRJ)

Furthermore, since (GRJ) logically implies both (GPM) and (JRD), it implies the conjunction of those two claims:

(GRJ)   É   [(GPM) & (JRD)]

This allows us to draw another inference using the above principle of probability:

P[(GPM) & (JRD)]   ≥  P(GRJ)

In English: The probability of it being the case both that God has performed at least one miracle and that Jesus rose from the dead is GREATER THAN OR EQUAL TO the probability that God raised Jesus from the dead.  So if we can determine the probability of it being the case that both (GPM) and (JRD) are true, that will establish an upper limit on the probability that (GRJ) is true.

We can use the general multiplication rule to draw this inference:

P[(GPM) & (JRD)]  =  P(GPM|JRD) x P(JRD)

And then we can use the substitution of equivalents to draw this inference:

P(GPM|JRD) x  P(JRD)   ≥   P(GRJ)

INITIAL ANALYSIS OF “JESUS ROSE FROM THE DEAD”

Let’s consider one element of this equation, the probability that Jesus rose from the dead:

P(JRD)

The claim that Jesus rose from the dead can be analyzed into two basic components:

(DOC) Jesus died on the cross on the same day he was crucified.

(JAW) Jesus was alive and walking around in Jerusalem about 48 hours after he was crucified.

If these two claims are both TRUE, then so is the claim that Jesus rose from the dead:

[(DOC) & (JAW)]   É   (JRD)

What Dr. Erasmus fails to notice, and fails to understand, is that the reverse is also the case:

 (JRD)    É   [(DOC) & (JAW)]

If the conjunction of (DOC) and (JAW) implies (JRD, and (JRD) implies the conjunction of (DOC) & (JAW), then we have a logical equivalence:

(JRD)   ≡   [(DOC) & (JAW)]

Recall the probability relationship between logically equivalent statements:

IF (S)  ≡  (Q), THEN P(S)  =  P(Q)

So, we can infer the following important probability relationship:

P(JRD)  =   P[(DOC) & (JAW)]

There are some plausible objections to my claim that (JRD) implies (DOC) and (JAW).  There appear to be various logically possible scenarios where we would be inclined to say that it is TRUE that Jesus rose from the dead, even if (DOC) was FALSE.  There also appear to be various logically possible scenarios where we would be inclined to say that it is TRUE that Jesus rose from the dead, even if (JAW) was FALSE.

Here are a few such counterexamples:

Alternative Location:  Jesus was crucified in Rome and he died on the cross on the same day he was crucified, and he was entombed in Rome and was alive and walking around in Rome about 48 hours after he was crucified.

Alternative Death:  Jesus was NOT crucified, but he was killed by being be-headed.  He then was entombed, and he was alive and walking around in Jerusalem about 48 hours later.

Long Crucifixion: Jesus died on the cross, but only after hanging on the cross for a week.  He then was entombed, and he was alive and walking around in Jerusalem about 48 hours later.

Long Entombment: Jesus died on the cross on the same day he was crucified, but after his body was placed in a tomb, he stayed dead for a week and then came back to life and walked around in Jerusalem.

Non-walking Jesus:  Jesus died on the cross on the same day he was crucified, and he was alive in Jerusalem about 48 hours later, but he never walked again, but was instead carried around everywhere by his disciples.

In all of these scenarios we are inclined to say that it is TRUE that Jesus rose from the dead, but that either (DOC) or (JAW) is FALSE.  However, there are a couple of considerations that mitigate the force of these counterexamples.

If Jesus was NOT crucified, but was actually be-headed, then the Gospel accounts of the death of Jesus are works of fiction that have little connection to reality and actual history.  Similarly, if Jesus was crucified, but the crucifixion took place in Rome rather than in Jerusalem, then the Gospel accounts of Jesus’ death are works of fiction that have little connection to reality and history.  In other words, in the case of most such counterexamples we imagine scenarios that are completely contrary to what the Gospel accounts of Jesus’ death and burial and resurrection state, and so if such a scenario was TRUE, then that would largely or completely destroy the credibility of the Gospel accounts of Jesus’ death, and burial, and resurrection.  But if the credibility of the Gospel accounts is largely or completely destroyed, then there is NO HOPE of establishing the resurrection of Jesus.

So, although I admit that we can imagine scenarios in which it appears that (JRD) is TRUE but (DOC) or (JAW) was FALSE, such scenarios would at the same time destroy the credibility of the Gospel accounts of Jesus’ death, burial, and resurrection, thus destroying the possibility of showing that it is PROBABLE that Jesus rose from the dead.  Such counterexamples, therefore, can be ignored, because they are NOT compatible with the aim of building a strong case for the PROBABILITY of the resurrection of Jesus.  Therefore, it is reasonable to treat (JRD) as being logically equivalent to the conjunction of (DOC) and (JAW), even though there are some logically possible scenarios in which (JRD) appears to be TRUE while (DOC) or (JAW) are FALSE.

A second mitigating consideration concerning these counterexamples is that what Christians mean by “Jesus rose from the dead” is something MORE than just the literal meaning of the words in this sentence.  We need to take into account the CONTEXT in which this statement is typically asserted.  In making this claim, most Christian believers have in mind the Gospel stories about the death, burial, and resurrection of Jesus.  In asserting that “Jesus rose from the dead” they have in mind various historical claims and details about how these events allegedly unfolded.  They are, in effect, asserting that the Gospel accounts of the death, burial, and resurrection of Jesus are basically correct, that those accounts are true for the most part.  I take it that the truth of their claim depends to a large degree on the truth of various key historical claims and details provided in the Gospel accounts.

Now, it would obviously be UNFAIR and UNREASONABLE to insist that the claim “Jesus rose from the dead” was FALSE unless EVERY last detail in all four canonical Gospel passion narratives was TRUE.  It is only reasonable to allow for some of the details in those accounts to be FALSE and yet to still admit that “Jesus rose from the dead” if enough of the key historical claims and details were TRUE.

If the Gospel accounts of the crucifixion of Jesus are true for the most part, and if the burial stories are true for the most part, and if the resurrection stories are true for the most part, then that could be enough to make it the case that the claim “Jesus rose from the dead” is TRUE, even if some of the details were incorrect or FALSE.  Therefore, the meaning of the claim “Jesus rose from the dead” when this claim is asserted by Christian believers is tied up with historical claims and details found in the Gospel accounts of Jesus’ death, burial, and resurrection.  The meaning of this claim, what it implies, is more than just the literal meaning of the sentence “Jesus rose from the dead”.  The additional details implied in this claim rule out many or most of the counterexamples to the logical equivalence of (JRD) with the conjunction of (DOC) and (JAW).

To Be Continued…