I see that Plantinga’s skeptical argument refers to “Dwindling Probabilities” rather than “Dwindling Probability”.  Sorry about my failure to get the name of this topic quite right.

I should mention that I did not learn about this sort of skeptical argument from the Christian philosopher Alvin Plantinga.  I learned about the Multiplication Rule of probablity in high school math, and then again in one of many courses on logic and critical thinking that I took in college and as a graduate student of philosophy.

Although I enjoyed learning about basic probability calculations in a Critical Thinking class at UCSB (esp. from The Elements of Logic by Stephen Barker, Chapter 7, 5th edition), the significance of the Multiplication Rule did not fully register with me until (I think) I read a skeptical argument by a Christian bible scholar: Robert Stein.

In his book Jesus the Messiah, Stein makes a skeptical argument about scholarly attempts to reconstruct the historical development of Q, a hypothetical source that most N.T. scholars believe was used by the authors of the Gospel of Luke and of the Gospel of Matthew.  Stein notes eight different hypotheses required in order to arrive at such a reconstruction of the history of Q.  Then Stein suggests estimated probabilities for each of the first five of the eight hypotheses, and argues that the probability that all five of those hypotheses is true is equal to the multiplication of the probabilities of those five hypotheses:

In other words, if the probability of the first five hypotheses were (1) 90 percent, (2) 80 percent, (3) 60 percent, (4) 50 percent, (5) 40 percent, the possibility of the fifth being true is .90 x .80 x .60 x .50 x .40, or a little more than 8 percent!  (Jesus the Messiah, p. 40)

Stein is a little sloppy here, and he appears to contradict himself.  He seems to be saying that the probability of the fifth hypothesis being true is 40 percent and also saying that the probability of this hypothesis being true is a little more than 8 percent.  But I think what he means is that the probabilty of the fifth hypothesis being true GIVEN the relevant facts AND the truth of the previous four hypotheses is 40 percent, and I think what he means is that the probability of the fifth hypothesis being true GIVEN only the relevant factual data is a little more than 8 percent (because the truth of the conjunction of the previous four hypotheses is NOT certain, but is actually somewhat improbable).

In any case, this skeptical argument presented by Stein inspired me to make use of the Multiplication Rule of probability in constructing skeptical arguments.

Richard Swinburne has raised some objections to Plantinga’s “Dwindling Probabilities” argument, and I am going to state and clarify those objections, and respond to each objection in relation to my example of “Dwindling Probabilities” presented in Part 1 of this series of posts.

Swinburne presents one primary objection, and then presents two more objections.  Swinburne’s primary objection is stated early in his essay on this issue:

Now, strictly speaking – as Plantinga acknowledges, but takes no further – P(G/K) is the sum of the probabilities of the different routes to it.   G might be true without some of these intermediate propositions being true.  

First, let me explain the meaning of P(G/K).   Read this as “The probability of G given K.”

G means:

The central elements of  Christian doctrine are true.

(e.g. God exists; Jesus rose from the dead; Jesus’ death on the cross atoned for our sins; etc.).

K refers to:

The totality of what we know apart from theism.

So P(G/K) means:

The probability that the central elements of Christian doctrine are true GIVEN the totality of what we know apart from theism.

One “route” to G is to establish the authority of the teachings of Jesus, and the reliability of the Gospel accounts of the teachings of Jesus.  If one could show that the teachings of Jesus are a reliable source of theological truths, and that  the Gospel accounts of the teachings of Jesus are accurate and reliable, then one could establish the probable truth of many or most Christian doctrines on the basis of the teachings of Jesus as presented in the Gospels.

So, one could break this line of reasoning down into various components, assign probabilities to each of the components, and then multiply the probabilities to arrive at a probability for G, for it being the case that the central elements of Christian doctrine are true:

1.  God exists.

2. Jesus existed.

3. Jesus was crucified in Jerusalem about 30 CE, assuming that Jesus existed.

4. Jesus rose from the dead, assuming that God exists, and that Jesus existed, and that Jesus was crucified in Jerusalem about 30 CE.

5.  God showed approval for Jesus’ claims about himself by raising Jesus from the dead, assuming that God exists and that Jesus rose from the dead.

6.  The Gospel accounts of the words and teachings of Jesus are accurate and reliable accounts, assuming that Jesus existed.

7.  Jesus claimed to be a prophet who was a reliable source of truth about God and theological matters, assuming that Jesus existed and assuming that the Gospel accounts of the words and teachings of Jesus are accurate and reliable accounts.

8.  Jesus’ teachings about God and theological matters are a reliable source of truth, assuming that God showed approval for Jesus’ claims about himself by raising Jesus from the dead and assuming that Jesus claimed to be a prophet who was a reliable source of truth about God and theological matters.

9.  The central elements of Christian doctrine are true, assuming that Jesus’ teachings about God and theological matters are a reliable source of truth and assuming that the Gospel accounts of the words and teachings of Jesus are accurate and reliable accounts.

None of these claims is certain.

A careful and rational evaluation of this line of reasoning would require assigning probabilities to each of these claims.  There is some probability that God exists, and some probability that Jesus existed, and some probability that Jesus was crucified (given that he existed), and some probability that Jesus rose from the dead (given that he existed and was crucified), etc.

Even if we assign a high probability to each of these claims (such as .8 or .9), when we use the Multiplication Rule of probability to determine the probability of G, the claim that the central elements of Christian doctrine are true, the probability will be fairly low.  For example, suppose that we assign a probability of .9 to each of the first four claims.  In that case the probability of the conjunction of these four claims would be: .9 x .9 x .9 x .9 =  .81 x .81 = .6561  or about .7  which is not exactly a high probability.

If we assigned a probability of .8 to each of the first four claims, then the probability of the conjunction of those claims would be:

.8 x .8 x .8 x .8 =  .64 x .64 = .4096 or about .4 which is clearly NOT a high probability.

Swinburne’s objection is that there may be other “routes” to the ultimate conclusion that G is the case, and if this is so, then we have to add the probability of arriving at G from other routes to the probabilty of G based on the particular route described above.

Let’s consider a simpler example to make Swinburne’s point more clearly:

1.  It will (probably) rain this afternoon.

2. If it rains this afternoon, then your lawn will (probably) be wet this evening.

Therefore:

3.  Your lawn will (probably) be wet this evening.

Neither premise of this argument is certain.  We coud assign a probability to each premise and use that to calculate the probability of the conclusion.  Supose that there is an 80% chance of rain this afternoon, and if it rains this afternoon, there is a 90% chance that your lawn will be wet this evening.  We could calculate the probability of the conclusion (3) by multiplying .8 x .9  to get:  .72.  Thus, the probability of (3) appears to be about .7 based on these assumptions about the probability of the premises.

However, there could be other “routes” or ways that your lawn could become wet:

4. Your lawn sprinkler system will (probably) turn on and water the lawn for an hour this afternoon.

5. If your lawn sprinkler system turns on and waters the lawn for an hour this afternoon, then your lawn will (probably) be wet this evening.

Therefore:

3.  Your lawn will (probably) be wet this evening.

We could assign probabilities to each of the premises in this argument to arrive at a probability for the conclusion.  Suppose that the sprinkler system is fairly reliable, and has been set to water the lawn for an hour each afternoon.  In that case, we might assign a high probabililty of .9 to premise (4), and a probability of .9 to premise (5).  We could calculate the probability of conclusion (3) by multiplying .9 x .9 to get:  .81.  Thus, the probabilty of (3) appears to be about .8 based on these assumptions.  But this is a different probability than what we arrived at based on the previous argument.  Which probability is correct?  .7 or .8?

If both arguments apply on the same day to the same lawn, then NEITHER estimate is correct, because the probabilty that (3) will be true would be higher than either estimate, since there are TWO DIFFERENT WAYS, each of which has a significant probability, that your lawn could become wet this afternoon.

Presumably the operation of the sprinkler system would NOT affect the weather, and thus NOT affect the chance of rain.  However, if it rains, that could affect the operation of the sprinkler system.  Some sprinkler systems can detect rain or detect moisture in the soil and adjust the watering schedule based on that data.  A sprinkler system might be designed to cancel the scheduled watering for the afternoon if it starts to rain early in the afternoon.   So, with some sprinkler systems, rain in the early afternoon would reduce the probability of the scheduled afternoon watering to nearly ZERO.  But if the scheduled watering begins early in the afternoon, that would have no impact on whether it would rain later that afternoon.

But suppose the sprinkler system has a simple timer and no mechanism for detecting rain.  In this case the sprinkler system which is set to water the lawn each afternoon, will turn the sprinklers on whether it rains that afternoon or not.  In that case, we could reasonably assume that these two different ways of making your lawn wet, operate INDEPENDENTLY of each other, and thus both of the above calculations of the probability of (3) would be too low, because each calculation assumes that there is only ONE WAY for your lawn to become wet, when there are actually (at least) TWO WAYS for this to occur.  Rain is one ROUTE for making your lawn wet, but a sprinkler system is another different ROUTE for making your lawn wet.

One ROUTE for showing central Christian doctrines to be true, is through the resurrection of Jesus as evidence for the authority (reliability) of the teachings of Jesus about God and theological matters.  But other ROUTES are possible,  as Swinburne points out, so the probability of the truth of central Christian doctrines does NOT rest exclusively on the ROUTE through the resurrection of Jesus as evidence for the authority (reliability) of the teachings of Jesus.  In order to arrive at an accurate probabilty of G, one must take into account any and every ROUTE that contributes some degree of probability to G.

My response to this objection in relation to my example of dwindling probabilities in the previous post is that there is ONLY ONE ROUTE (that has a probability higher than ZERO) to the claim “Jesus died on the same day he was crucified” in my probability tree diagram.  So, although I agree with Swinburne’s point and his logic, this point has NO RELEVANCE in relation to my particular example of dwindling probabilities.

There is some relevance to Swinburne’s point, however, if one uses my example probability tree diagram as part of one’s thinking about the resurrection of Jesus. The claim “Jesus died on the same day he was crucified” reflects the standard Christian view or scenario about the death of Jesus. According to the Gospels, Jesus died on the cross on the same day that he was crucified (which is somewhat unusual – crucifixion was intended to be a slow, long, drawn-out, and painful death).  But it is possible that Jesus rose from the dead, even if he did not die on the day that he was crucified.

Jesus might have been barely alive when removed from the cross, the soldiers mistakenly believing that he was already dead, and Jesus might have been placed in a nearby tomb, again by someone who mistakenly believed he was already dead, and then Jesus might have survived that Friday night and died in the cold, dark tomb early on Saturday morning, but came back to life on Sunday morning about 24 hours later.

This would still count as rising from the dead, and would still be more-or-less in line with Christian belief and doctrine. Therefore, it is not absolutely required that “Jesus died on the same day he was crucified” in order for it to be the case that “Jesus rose from the dead”. So, there is this alternative ROUTE or WAY that the resurrection could have occured, and in order to accurately assess the probability of the resurrection of Jesus, the probability of this alternative ROUTE must be added to the probability of the standard ROUTE, where Jesus dies on the same day that he was crucified.

To be continued…