Re: [math-fun] Smullyan paradox about belief
What's the mathematical version of this question? Because anyone can be mistaken about anything, including their own beliefs, it would seem. —Dan Jim Propp wrote: ----- I recall encountering in one of Raymond Smullyan’s books a thought experiment that convinced me that it is possible to be mistaken about what one believes. That is, one can say “I believe X” and be speaking falsely even though one is not intentionally lying. Does anyone know what thought experiment I’m dimly recalling, and which of his books it appears in? -----
I want somebody to wade through the oh-so-arch Smullyanese and give an executive summary of the punchline. I sometimes lose patience with Smullyan, when he indulges his sense of the dramatic by wrapping up some fairly prosaic logical observation in a complicated and disturbing story. (Daniel Dennett does this too.) On Thu, May 2, 2019 at 2:11 PM Dan Asimov <dasimov@earthlink.net> wrote:
What's the mathematical version of this question? Because anyone can be mistaken about anything, including their own beliefs, it would seem.
—Dan
Jim Propp wrote: ----- I recall encountering in one of Raymond Smullyan’s books a thought experiment that convinced me that it is possible to be mistaken about what one believes. That is, one can say “I believe X” and be speaking falsely even though one is not intentionally lying. Does anyone know what thought experiment I’m dimly recalling, and which of his books it appears in? -----
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Let x=y be a doubtable proposition P and introduce function B(P)=1 for belief in P, and B(P)=0 for doubt. These are again doubtable propositions. Let x’=y’ be an undoubtable proposition P’, such that B(P’)=0 by definition. This is again a doubtable proposition, but an honest person will say B(B(P’)=0)=1 and B(B(P’)=1)=0. An abuse victim is made to say that B(P’)=1. The abuse suffered is so bad that victim reports B(B(P’)=1)=1, and positive belief for all recursions. An oracle machine is sensitive to abuse, and can tell that the victim has been coerced into lying at every juncture. The operator of the machine then confronts the victim with this opinion. The victim does not attain enlightenment, but suffers more from abuse at the hands of the epistemologist, end scene 2. *** Smullyan is very fun about birds and Combinators, but on this sort of thing, I also lose patience. Nagarjuna’s four propositions are more to my taste, and usually shorter to read. Cheers, Brad
On May 2, 2019, at 1:17 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I sometimes lose patience ...
On 5/2/2019 5:56 PM, bradklee@gmail.com wrote:
Let x=y be a doubtable proposition P and introduce function B(P)=1 for belief in P, and B(P)=0 for doubt. These are again doubtable propositions.
So does B(P)=1 => B(~P)=-1 ?
Let x’=y’ be an undoubtable proposition P’, such that B(P’)=0 by definition.
? If P' is undoubtable then B(P')=1, or I don't understand what "belief in P" means. Brent
This is again a doubtable proposition, but an honest person will say B(B(P’)=0)=1 and B(B(P’)=1)=0.
An abuse victim is made to say that B(P’)=1. The abuse suffered is so bad that victim reports B(B(P’)=1)=1, and positive belief for all recursions.
An oracle machine is sensitive to abuse, and can tell that the victim has been coerced into lying at every juncture. The operator of the machine then confronts the victim with this opinion.
The victim does not attain enlightenment, but suffers more from abuse at the hands of the epistemologist, end scene 2.
***
Smullyan is very fun about birds and Combinators, but on this sort of thing, I also lose patience.
Nagarjuna’s four propositions are more to my taste, and usually shorter to read.
Cheers,
Brad
On May 2, 2019, at 1:17 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I sometimes lose patience ...
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So does B(P)=1 => B(~P)=-1 ?
In Nagarjuna’s system, not entirely.
? If P' is undoubtable then B(P')=1, or I don't understand what "belief in P" means.
B(P’)=0 is an axiom that means “P’ does not require a belief”. To make a more definite distinction between classes P and P’, we could require (without loss of applicability) that the P’ are all trivially decidable statements such as 1+1=2 or “red”=“color of book” and that the P are all statements about belief of the form B(x)=y. This actually makes a little bit of sense because any statement involving belief should be doubtable. —Brad
Maybe you'd like to try Harry Frankfurt's essay On Bullshit. Seems to reach a more general conclusion, and it's a quick read. On 5/2/19 11:17 , Allan Wechsler wrote:
I want somebody to wade through the oh-so-arch Smullyanese and give an executive summary of the punchline. I sometimes lose patience with Smullyan, when he indulges his sense of the dramatic by wrapping up some fairly prosaic logical observation in a complicated and disturbing story. (Daniel Dennett does this too.)
On Thu, May 2, 2019 at 2:11 PM Dan Asimov <dasimov@earthlink.net> wrote:
What's the mathematical version of this question? Because anyone can be mistaken about anything, including their own beliefs, it would seem.
—Dan
Jim Propp wrote: ----- I recall encountering in one of Raymond Smullyan’s books a thought experiment that convinced me that it is possible to be mistaken about what one believes. That is, one can say “I believe X” and be speaking falsely even though one is not intentionally lying. Does anyone know what thought experiment I’m dimly recalling, and which of his books it appears in? -----
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participants (5)
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Allan Wechsler -
Andres Valloud -
bradklee＠gmail.com -
Brent Meeker -
Dan Asimov