Some Logic in Swinburne’s Cosmological Argument

I have been struggling for the past week or two to make clear the logic behind one premise of Swinburne’s cosmological argument. Perhaps those readers of The Secular Outpost who have an interest in logic or in Swinburne’s arguments will be able to help me with this task.

Actually, his inductive cosmological argument is very simple:

1. A complex physical universe exists.
Therefore:
2. God exists.

It is not this argument that I am struggling to understand and clarify, but rather Swinburne’s critical argument for the claim that the above cosmological argument is a good inductive argument, that the empirical claim (1) provides evidence which increases the probability that (2) is the case.

Here is the premise of Swinburne’s critical argument (CPU = complex physical universe):

(TCA15) The probability that a CPU exists given that God does not exist is approximately equal to the probability that a CPU exists without an explanation.

Swinburne spells out his reasoning in support of (TCA15) in just a few sentences in The Existence of God, 2nd edition (hereafter EOG):

Let h be the hypothesis of theism, and k be mere tautological evidence. Let e be the existence over time of a complex physical universe. e could not, as we have seen, have a scientific explanation. Either e occurs unexplained, or it is due to the action of a person, the most likely being God. It is now reasonable to ignore the alternative that we have shown to be a priori much less probable, that e was brought about by a person or persons of very large but finite power, very considerable but limited knowledge, etc. Hence we may regard P(e|~h & k) as the probability that there be a physical universe without anything having brought it about. (EOG, p.149)

I think that Swinburne’s logic is correct here, but I want to spell out the reasoning in a careful and rigorous way, to verify that his logic is in fact OK. But it appears that there are many little inferences involved here, perhaps a dozen or more steps to get from premises to the desired conclusion, and I have had difficulty figuring out just how to get from the premises to the conclusion. I think I’m getting close to being able to spell out the logic behind Swinburne’s verbal statement of the argument, but I’m not quite there yet.

Some abbreviations that I will use (CPU = complex physical universe):

e: A CPU exists.
s: A CPU exists and there is a scientific explanation for this fact.
p: A CPU exists and there is a personal explanation for this fact.
n: A CPU exists and there is no explanation for this fact.
t: A CPU exists and the explanation of this fact is that God brought about the existence of a CPU.
f: A CPU exists and the explanation of this fact is that one or more finite gods (who don’t owe their existence to an infinite person/God) brought about the existence of a CPU.
g: God exists.
k: [tautological truths - all the truths of logic, math, and analytic conceptual truths]

The conclusion that we are trying to get to is (TCA15), which can be symbolized this way:

P(e|~g & k) is approximately equal to P(n|k).

The term “approximately equal to” is a bit vague, but in relation to an inductive cosmological argument, I think it is reasonable to think of probability values in terms of tenths: 0, .1, .2, .3, .4, .5, .6, .7, .8, .9, and 1.0.

A probability of 1.0 means certainty, certain truth. A probability of 0 means certain falsehood. Given this general scheme, it is reasonable to interpret the phrase “approximately equal to” as meaning something like “plus-or-minus .1″. For example, a probability of .5 is “approximately equal to” a probability of .4, but NOT “approximately equal to” a probability of .3, and is “approximately equal to” a probability of .6, but NOT to a probability of .7. So, let’s clarify the conclusion accordingly:

Conclusion: P(e|~g & k) is equal to P(n|k), plus-or-minus .1

NOTE: Since probability values do not go below 0 or above 1.0, P(e|~g & k) cannot be equal to -.1, even if P(n|k) is equal to 0. Similarly, P(e|~g & k) cannot be equal to 1.1 even if P(n|k) is equal to 1.0.

Let’s put some of the premises on the table.

e could not, as we have seen, have a scientific explanation.

This claim might be represented as ~s. But Swinburne’s claim is actually a bit stronger. His claim is that it is logically impossible that the existence of a complex universe has a scientific explanation. This could be symbolized using a conditional probability claim:

(P1) P(s|k) = 0

This says that we know for certain that s is false given only tautological truths (truths of logic, mathematics, and analytic conceptual truths). In other words, we know that s is false based strictly on logic.

From (P1) we can infer the falsehood of s:

(P2) ~s

Either e occurs unexplained, or it is due to the action of a person

This claim is based on a previous claim made by Swinburne. The previous claim was that there are only three possibilities concerning an explanation of the existence of a CPU: (1) it has a scientific explanation, (2) it has a personal explanation, or (3) it has no explanation at all. We can represent this as follows:

(P3) IF e, THEN either s or p or n.

It is also obvious that the s implies e, and p implies e, and n implies e, so the implication runs both directions:

(P4) e IF AND ONLY IF either s or p or n.

From (P4) we see that there are only three logical possibilities for explaining e, and from (P2) we see that one of those possibilities has been eliminated, so we can infer that there are only two possibilities concerning the existence of a CPU:

(P5) e IF AND ONLY IF either p or n.

This inference would require a few steps in a logic proof, but it is fairly obvious that if you eliminate one of a total of three logical possibilities, then you are left with only the two remaining logical possibilities.

…the most likely [personal explanation of the existence of a CPU] being God. …the alternative [personal explanation] that we have shown to be a priori much less probable, that e was brought about by a person or persons of very large but finite power, very considerable but limited knowledge, etc.

This claim could be represented as an inequality. The probability that a CPU exists and was brought about by God given only tautological background knowledge is GREATER THAN the probability that a CPU exists and was brought about by one or more finite gods (who don’t owe their existence to an infinite person/God) given only tautological background knowledge:

P(t|k) > P(f|k).

However, because of the phrase “much less probable” and because of the requirement to show that some probability value is equal to another plus-or-minus .1, I believe that Swinburne needs a stronger premise than this. I think what his argument requires, to be successful, is the claim that the probability that a CPU exists and was brought about by God given only tautological background knowledge is GREATER THAN TEN TIMES the probability that a CPU exists and that it was brought about by one or more finite gods (who don’t owe their existence to an infinite person/God) given only tautological background knowledge:

(P6) P(t|k) > 10 x P(f|k)

Swinburne does not explicitly assert (P6) but it seems fairly clear to me that he needs to make a claim along the lines that P(t|k) is an order of magnitude greater than P(f|k) in order for the logic of his argument to work (based on my understanding of “is approximately equal to X” as meaning “equals X, plus-or-minus .1″).

From Swinburne’s point of view the claim that God exists is less than completely certain, so the probability that God exists is less than 1.0. This implies that the probability that a CPU exists and that God is the explanation for that fact given only tautological knowledge is also less than 1.0:

(P7) P(t|k) < 1.0

Considering (P7) in the light of (P6), we can draw a further inference:

(P9) P(f|k) < .1

It is now reasonable to ignore the alternative that we have shown to be a priori much less probable, that e was brought about by a person or persons of very large but finite power, very considerable but limited knowledge, etc.

I think Swinburne’s reasoning here is probably correct, but a number of steps of logic are involved, so there could be a problem hidden in the details. That is why I want to work out the details.

Another premise that is not explicit in the quoted paragraph is that personal explanations of the existence of a CPU can be divided into two possibilities: either God (an infinite person) brought about a CPU, or else one or more finite gods brought about a CPU:

(P10) IF p, THEN t or f.

Since t implies p, and f implies p, the implication works both directions:

(P11) p IF AND ONLY IF t or f.

Hence we may regard P(e|~h & k) as the probability that there be a physical universe without anything having brought it about.

The words “we may regard” X “as the probability” I have interpreted to mean “X is approximately equal to the probability”. Also, I’m using ‘g’ to represent the claim ‘God exists’ instead of using ‘h’. So, the conclusion that we are attempting to derive from the premises is this:

Conclusion: P(e|~g & k) is equal to P(n|k), plus-or-minus .1

================================
UPDATE:
There are a couple of tempting errors that Swinburne might have made in reasoning to get to the conclusion.

One tempting error is to assume this:

g IF AND ONLY IF t.

It is true that t implies g, and it is also true that ~g implies ~t. However, the logical relationship does not work in the reverse direction. The assumption that g implies t is FALSE, and the assumption that ~t implies ~g is also FALSE.

This is because in both Christian theology and in Swinburne’s concept of God, it is not necessary that God brings about a CPU. For Swinburne, it is somewhat likely that God would bring about a CPU because it is somewhat likely that God would bring about humanly free agents, and humanly free agents require the existence of a CPU. But this means that God could also have chosen NOT to bring about humanly free agents, and NOT to bring about a CPU. Therefore, there is a chance that God exists but that there is no CPU, given only tautological background knowledge.

We know, of course that there is a CPU, so this is not what actually happened, but our knowledge that there is a CPU is empirical knowledge, knowledge based on experience, not knowledge based purely on logic or tautological truths. If we imagine ourselves to be in a state of ignorance concerning empirical matters, and knowing only tautological truths, then the fact that God exists would NOT by itself logically imply that there is a CPU, for God might well have chosen not to bring about a CPU. Thus, the assumption that g implies t is false, because t implies e (the existence of a CPU), but g does NOT imply e.

Another possible error in Swinburne’s reasoning is to treat “is approximately equal to” as a transitive relation. The relation “is equal to” is a transitive relation, meaning that equality transfers through a chain:

IF x equals y and y equals z, THEN x equals z.

But “is approximately equal to” is NOT a transitive relation, so the following assumption would be FALSE:

IF x is approximately equal to y and y is approximately equal to z, THEN x is approximately equal to z.

To be continued…


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