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        riddles >> medium >> Discursive Judges
        
(Message started by: BMAD on Jun 4^th, 2014, 5:05am)

Title: Discursive Judges
Post by BMAD on Jun 4^th, 2014, 5:05am

The accused is brought forth in front of the tribunal. The three judges must hear the accusations and the defense to make a determination of innocent or guilt. In this particular case, the accused is found guilty ( C ) if the action in question is defined as illegal (P) and the accused committed the action (Q) or in other words C=P^Q. Judge number 1 found both P and Q true, Judge number 2 found only Q true, and Judge number 3 found only P true. Should the accused be found guilty?

Title: Re: Discursive Judges
Post by Grimbal on Jun 4^th, 2014, 7:11am

I'd say if the juges can't aggree on either point, you can't condemn the accused.

Another way to see it is that assuming the juge's opinions reflects the probability of C and P being true, then there is only a 4/9 chance that the accused deserves prison.

Title: Re: Discursive Judges
Post by dudiobugtron on Jun 4^th, 2014, 1:23pm

If Judge 2 found 'only Q true', does that mean that she found P to be not true? (Similarly for Judge 3 - did he find Q to be not true?)

Title: Re: Discursive Judges
Post by BMAD on Jun 4^th, 2014, 1:26pm

If Judge 2 found 'only Q true', does that mean that she found P to be not true? (Similarly for Judge 3 - did he find Q to be not true?)

Yes.

Title: Re: Discursive Judges
Post by dudiobugtron on Jun 4^th, 2014, 1:47pm

In my country's legal system, the defendant's guilt needs to be established 'beyond reasonable doubt'.

I think we should assume that the judges are all reasonable, and therefore the fact that Judge 3 thinks the defendant did not commit the action means that there is reasonable doubt. Therefore, even if all judges agreed P were true, I think the defendant should go free.

Of course, the judges have access to more information than we do, so perhaps they are better placed to decide whether Judge 3's doubt is 'reasonable'.

Title: Re: Discursive Judges
Post by BMAD on Jun 4^th, 2014, 1:52pm

In your country do you use 1 judge or 3?

Title: Re: Discursive Judges
Post by dudiobugtron on Jun 4^th, 2014, 1:59pm

on 06/04/14 at 13:52:02, BMAD wrote:

In your country do you use 1 judge or 3?

Usually 1 judge per case in the smaller courts, but once you get up to higher courts, the number of judges per case increases. I think the supreme court has 5 who hear each case.

-----------------------

I think I see what you are getting at, though. You have 2/3 judges who think P is true, so by that vote the court should find P true. You also have 2/3 judges who find Q true, so by that vote the court should find Q true. However, you only have 1/3 judges who think C is true. So the court should find C false. But that's a contradiction!

Title: Re: Discursive Judges
Post by BMAD on Jun 4^th, 2014, 2:05pm

Yes. This is a famous paradox problem in game theory.

Title: Re: Discursive Judges
Post by rmsgrey on Jun 5^th, 2014, 2:45am

on 06/04/14 at 14:05:51, BMAD wrote:

Yes. This is a famous paradox problem in game theory.

It's as much a paradox as when I get my (fully functional and correctly operating) calculator, show that the screen displays "2", press "=" without altering the display (showing that that is indeed the current running total) and then press "*", "2", and "=" in that order, making the number displayed change to "5"

Title: Re: Discursive Judges
Post by dudiobugtron on Jun 5^th, 2014, 3:22am

on 06/05/14 at 02:45:09, rmsgrey wrote:

I don't see yet how that relates to the OP, but it seems like an interesting puzzle in its own right! Perhaps it deserves its own thread?

Also, does the * button on your calculator correspond to the 'multiply' operation?

Title: Re: Discursive Judges
Post by rloginunix on Jun 7^th, 2014, 2:14pm

on 06/05/14 at 02:45:09, rmsgrey wrote:

I get my (fully functional and correctly operating) calculator, show that the screen displays "2", press "=" without altering the display (showing that that is indeed the current running total) and then press "*", "2", and "=" in that order, making the number displayed change to "5"

Don't want to pollute the current topic so may be it should be spawned into its own thread.

I remember implementing in C (and Java) Dijkstra's Shunting Yard algorithm for numeric arithmetic and string Boolean expressions. Two phases. Phase 1 - parse the arbitrarily parenthesised input infix expression into either postfix or prefix form. Phase 2 - "calculate" the result: if the current item is an operand then push it onto the stack otherwise (it's an operator) pop the required number of operands off the stack (1 for unary, 2 for binary), "do" the operator, push the result back onto the stack.

By the end of this process there should be only one item left on the stack - the "answer". A number for an arithmetic or "true" or "false" value for a Boolean string expression. Any other case means an error somewhere.

So what we have here is 2 "displayed": int 2 at the top of the operands stack. "=" means do the above processing: int 2 at the top of the stack still. "*" I take means multiply - a binary operator. "2" is pushed onto the operands stack. "=" again do the processing. Pop two ints off the stack, 2 and 2, multiply, 4. Push 4 back onto the stack. End. Display 4. So by this account the answer should be 4, not 5.

Any other calculator's capabilities involved here?

Title: Re: Discursive Judges
Post by rmsgrey on Jun 9^th, 2014, 7:37am

The calculator allows me to set how many digits its display rounds to, while still using the most exact values in calculation - so I enter 2.4 into the calculator, set it to round to either 1sf or 0dp depending on the machine's precise capabilities, show you the display reading "2", double it to 4.8, which displays as "5" (if the calculator truncates rather than rounding, start with 2.6 rather than 2.4)

The link with this puzzle's apparent paradox is that both rely on suppressing precision - in the calculator's case, it's rounding the hidden digits; for the judges, it's rounding the 2/3 majority opinions up to appear unanimous.