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
===============


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…

About Bradley Bowen
  • http://www.blogger.com/profile/16641266062186767500 Keith Parsons

    Bradley,

    Allow me to second your excellent reply. The idea that you can evaluate an argument just by counting the number of premises is simply fatuous. Consider:

    1) All whales are mammals.
    2) All mammals are air-breathers.

    This, of course, entails

    3) All whales are air-breathers.

    Likewise,
    1) All whales are mammals.
    2) All mammals are air-breathers.
    3) Haggis is delicious.
    4) Bill Belichick is a swell guy.

    Also entails

    5) All mammals are air-breathers.

    The fact that the second argument has more premises than the first and that two of those premises are, at least, dubious, does not make the second argument one bit less sound than the first.

    With inductive arguments, on the other hand, it is often the case that the more premises the merrier, even if each individual premise is quite unlikely to be true. Suppose that many poker hands are dealt. We might now make this argument:

    1) Poker player #1 was dealt a hand, and it was a straight flush.
    2) Poker player #2 was dealt a hand, and it was a straight flush.
    3) Poker player #3 was dealt a hand, and it was a straight flush.
    .
    .
    .
    n) Poker player #n was dealt a hand, and it was a straight flush.
    Conclusion: Some player was dealt a straight flush.

    Here each individual premise is very likely false, but if the number of premises, n, is large enough, the conclusion can be quite strongly supported by those premises. Further, the more such premises we add, the more strongly supported the conclusion will be.

    It is always a pleasure to see a good philosopher like Bradley give a patient, well-reasoned answer to a sophomoric critique.

  • http://www.blogger.com/profile/16641266062186767500 Keith Parsons

    (oops!) Small correction to previous post: the conclusion labeled (5) should be:
    (5) All whales are air-breathers.

    Sorry for any confusion!

  • http://www.blogger.com/profile/01786844757672182664 K-Dog

    I am pleased to see Keith Parsons once again act like an ass and this time I have caught him red handed. I already knew this critique was 'sophmoric' but I am merely regurgitating here what one of your own members (Stephen Law) tried to use as a 'sophmoric critique' against Glenn Peoples on his own blog concering the moral argument. I sincerely hope you will be a good colleague and e-mail Stephen Law immediately Mr. Parsons too let him know that despite the fact that he is a professional philosopher, you find his objection sophmoric.

  • http://www.blogger.com/profile/01786844757672182664 K-Dog

    I would also like to thank Brad for his patient and thoughtful response.

  • http://www.blogger.com/profile/16641266062186767500 Keith Parsons

    So, let's see: "K-Dog" admits that his comments were sophomoric, indeed, he says, intentionally so. Yet he gets in high dudgeon because I point this out. Uh huh.

    Of course, I have my own opinions about who is and who is not being an "ass." But I very much want to be fair in my comments on Secular Outpost. So, in the future, should "K-Dog" say something intelligent, informed, or even civil, I promise to note it.

  • http://www.blogger.com/profile/09534596842153546368 evodevo

    Points 7 through 12 are used by stock brokerages to lure in new customers – "return over 1 year – 10%; 10-year return – 25%"
    Except that the last year has been anomalous, and the 10-year return included two stock market crashes which are ignored !!
    There is a sucker born every minute.

  • http://www.blogger.com/profile/05211466026535549638 Bradley Bowen

    I have not yet tried to reply to K-Dog's objection. I'm just working on laying out appropriate constraints and qualifications for the use of multiplying probabilities of premises. So far, I only have a couple of qualifications:

    1. Applies when premises work together to support a conclusion, but not when they provide independent support for the conclusion.

    2. Even when premises work together to support the conclusion, there may be other arguments that need to be considered (to arrive at an 'all things considered' probability).

    There are more constraints and qualifications to consider, as well as some interesting implications and issues about the nature of arguments and reasoning.

  • http://www.blogger.com/profile/01786844757672182664 Truth Seeker

    The point Mr. Parsons (and you didn't deny it) is that my comments weren't my comments at all. Had I quoted Stephen Law up front, you most likely would have been much more respectful of 'his' objection. Unfortunately, you have a history of showing me not only no respect, but disrespect. So, in order to expose your character, all I had to do was use Dr. Law's objection. I think your objections are usually important and worth thinking about, and they would no doubt be more effective if you improved your tone. But, what do I care, 'I chose to call myself K-dog,' and 'Do I even have a job?', etc.; of course these are just some of things you have felt compelled to say as a grown ass man for some reason.


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