Skepticism and the Multiplication of Probabilities – Part 1

K-Dog said…
How many premises are there in your argument, jeesh? Are you aware that even if there are only 5 premises in your argument, and we grant them an .8 likelihood, that your conclusion is only .33 likely to be true! I am guessing that your argument is even longer though which makes it all the more improbable.
January 19, 2012 3:24:00 PM CST
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I agree with the underlying principle in K-Dog’s comment: skepticism is a two-edged sword that cuts away at both the beliefs with which a skeptic disagrees and also at the skeptics own or favored beliefs. If one uses a strong or strict criterion for determining whether the claims of one’s opponents are ‘known’ or ‘probable’, then the same criterion should be applied to one’s own claims and beliefs. Double-standards are contrary to the aims of critical thinking.

One question at issue is whether the sort of skeptical reasoning I use about specific historical points (e.g. ‘Did the apostle John write the Fourth gospel?’) can be applied to my own argument against the resurrection of Jesus to show that my argument is weak. But I think there are probably some interesting points of logic and epistemology lurking in the background here, so I don’t want to focus exclusively on the objection to my argument against the resurrection of Jesus.

One important point of logic is that premises can support conclusions in different ways. Specifically, sometimes premises provide independent support for a conclusion, while in other cases, two or more premises work together to provide support for a conclusion. In deductive reasoning, premises often work together to support a conclusion:
1. Socrates is a man.
2. All men are mortal.
Thus:
3. Socrates is mortal.

If (1) is false, then (2) does not by itself provide support for the conclusion, and if (2) is false, then (1) does not by itself provide support for the conclusion.

But premises in deductive arguments can also provide independent support for a conclusion:
4. Socrates is a man, and all men are mortal.
5. Socrates died, and anyone who has died is mortal.
Thus
6. Socrates is mortal.


Premise (4) provides support for the conclusion all by itself, and premise (5) also provides support for the conclusion all by itself. If premise (4) is false, premise (5) still supports the conclusion, and if premise (5) is false, premise (4) still supports the conclusion.

Inductive arguments often involve premises that provide independent support for the conclusion:
7. John had pancakes for breakfast on Monday.
8. John had pancakes for breakfast on Tuesday.
9. John had pancakes for breakfast on Wednesday.
10. John had pancakes for breakfast on Thursday.
11. John had pancakes for breakfast on Friday.
Thus:
12. John had pancakes for breakfast on Saturday.

Each premise in this argument provides some independent support for the conclusion. Premise (7) provides some support for the conclusion, even if all the other premises are false. The same goes for premise (8). The cumulative force of all the premises being true is, in this case, greater than the force of just one or two premises being true, but each premise provides a part of that force.

The use of the multiplication of probabilities in skeptical critiques of arguments works best on deductive arguments in which the premises work together to support the conclusion, such as the first deductive argument about Socrates above. If the probability that Socrates is a man is .9. and the probability that all men are moral is .9. then we can conclude that the probability that Socrates is mortal is .81 or (sticking to one significant digit) about .8, because .9 x .9 = .81.

In the case of the inductive argument about John eating pancakes for breakfast, we cannot simply multiply the probabilities of the premises together. Suppose that each premise had a probability of .8 of being true. In that case it is likely that most of the premises are in fact true, and the conclusion would still be made probable (more probable than not) in that case. But if you simply multiplied the probabilities of the premises, that would yield a probability of .32768 or (sticking to one significant digit) about .3, which would mean the conclusion was improbable. Multiplying probabilities does not work in this case, because each premise provides some support for the conclusion, independent of the other premises.

However, there can be more than one argument for the same concusion. For example:
13. Socrates died.
14. Anyone who has died is mortal.
Thus:
15. Socrates is mortal.

This is an argument for the same conclusion as the first deductive argument above for the mortality of Socrates. Suppose that the probability of (13) was .8, and the probability of (14) was 1.0 (because it is an analytic truth), so the probability of the conclusion (based on this argument) would be .8.

But now we have two separate arguments for the same conclusion. So, intuitively, although each argument by itself makes the probability of the conclusion .8, the combination of these two arguments would make the probability of the conclusion something greater than .8.

Thus, an important question to ask, when multiplying probabilities of premises is:

Are there other important arguments that also need to be considered and weighed along with the argument currently under consideration?

To be continued…

Is the Religious Right Finished?
Why not a Fast from Religion?
Jesus: True Prophet or False Prophet? - Part 3
Jesus: True Prophet or False Prophet? - Part 2
About Bradley Bowen

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