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locqust
Unfettered

Joined: 14 Oct 2005
Posts: 545
Location: Gloucestershire UK

[Black maze set] #216 Formal logic

Card worth 54 points

Puzzle by Andrew McCarthy

Text says.....

Key
'(upside down capital A)' Corresponds roughly to the English word 'every'
'(back to front capital E)' Corresponds roughly to the English word 'some'
The variables x,y,z range over objects.
'_ -> _ ' Corresponds roughly to the English phrase 'if_then_'.

Question one
Which of the following formalises the claim that every person has a common ancestor (where 'P' = '_is a person' and 'A' = 'is an ancestor of _ _').

(i) ((back to front capital E)x)((upside down capital A)y)(P(y) -> A(xy))
(ii) ((upside down capital A)x)((back to front capital E)y)(P(x) -> A(yx))
(iii) ((upside down capital A)x)(P(x) -> A(xx))

Question 2
What is the name of the paradox that is represented here in the following set-theoretic notation.

R={x : x (euro sign) x}

end of text.
 formallogic.jpg Description Filesize 140.37KB Viewed 248 Time(s)
 formallogicback.jpg Description Filesize 160.79KB Viewed 187 Time(s)
_________________
"If you'd been listening you would know that nintendos pass through everything." Col. Jack O'Neill

Posted: Fri Nov 25, 2005 4:06 am
Last edited by locqust on Fri Nov 25, 2005 1:53 pm; edited 2 times in total
mupdan
Boot

Joined: 04 Aug 2005
Posts: 22
Location: Bath, UK

 Spoiler (Rollover to View): (i) In English, this reads "There exists an x such that for all y, y is a person implies x is an ancestor of y." In case anyone's interested, (ii) is the theory that "Every person has an ancestor", and (iii) is that "Every person is their own ancestor"

Can't remember my set theory of the top of my head, I'm at work now so can't look it up!

Posted: Fri Nov 25, 2005 4:23 am
Lazarus
Boot

Joined: 29 Oct 2004
Posts: 19
Location: Weymouth, U.K.

For question 2:

 Spoiler (Rollover to View): I'm pretty sure it's Cantor's Paradox. The logic says that for a given x, x is a subset of x. There's more info on it at http://en.wikipedia.org/wiki/Cantor%27s_paradox

Posted: Fri Nov 25, 2005 6:03 am
chichiri
Decorated

Joined: 31 Oct 2005
Posts: 207

 Spoiler (Rollover to View): (i) and (iii)

just confirmed

no da

Posted: Fri Nov 25, 2005 6:34 am
locqust
Unfettered

Joined: 14 Oct 2005
Posts: 545
Location: Gloucestershire UK

At least give me chance to put the scan up!
_________________
"If you'd been listening you would know that nintendos pass through everything." Col. Jack O'Neill

Posted: Fri Nov 25, 2005 11:42 am
myf
Guest

Erm..... Is the colour of the box at the top of the card significant in any way? It seems to fade from yellow across to white. On most of cards, the bit with the title, etc. seems to be either white, or a continuation of the background design on the rest of the card. There are a few others that seem to be coloured in like this one, but I'd not really noticed it on the others. This one just seemed to really stick out, so I thought I'd mention it.

Apologies if this has been mentioned elsewhere. Had a brief look and couldn't see anything.

Posted: Thu Dec 29, 2005 5:09 pm
Boot

Joined: 31 Dec 2005
Posts: 12
Location: Northants, UK

This one would make more sense to me if 'A' = '_ is an ancestor of _' rather than the way it has been written on the card. Or is this just me?

Posted: Sun Jan 01, 2006 7:03 am
Magma
Veteran

Joined: 10 Jan 2006
Posts: 119

It should be noted that the second question has a symbol that looks like a euro symbol with a line through it. That is important, because it changes the statement from "is a set of" to "is not a set of". This makes it:

 Spoiler (Rollover to View): Russell's Paradox. http://www.steve.edwards.freeuk.com/odl122_answer_2.htm This page explains that it considers the set of all sets that are not members of themselves. It also handily includes the same equation as printed on the card, and a real-world example of the paradox: A village is divided into two seperate groups of people - Those that shave themselves, and those that are shaved by the barber. Anyone that is shaved by the barber cannot be in the group that shaves themselves, and anyone that shaves themselves cannot be in the group who the barber shaves. The barber shaves himself - He is both a man who shaves himself, and a man who is shaved by the barber. He can fit into neither group, thus the paradox.

Posted: Tue Feb 14, 2006 7:07 pm
kookaburrah
Greenhorn

Joined: 29 Mar 2006
Posts: 6
Location: Berkshire, UK

[delusional paranoia]
Okay, two questions:
1 - Why is the equation in Question 2 blurred?
2 - What on earth is the thing peeking from the lower right hand corner? Can you recognise it?
[/delusional paranoia]

Kooka

Posted: Wed Mar 29, 2006 4:22 pm
Magma
Veteran

Joined: 10 Jan 2006
Posts: 119

If you look carefully at the euro-looking symbol, it looks a bit pixellated. It probably was a low resolution picture, or a non-truetype font, so they blurred it to get rid of the horrible edges. As for the object, I don't know. There is a Staedtler pencil on there too so perhaps it is an item of stationery?

Posted: Thu Mar 30, 2006 10:51 am
zaeil
Decorated

Joined: 31 Aug 2004
Posts: 233
Location: NC, US

 kookaburrah wrote: 1 - Why is the equation in Question 2 blurred?

Because the equation was an image (instead of text, like in Question 1) that was probably saved in a lossy format like JPG and then inserted into a text document in word processor and printed. Basically, this is a graphic design issue and really doesn't mean anything.

 Quote: 2 - What on earth is the thing peeking from the lower right hand corner? Can you recognise it?

Calculator maybe, considering the subject matter. Probably just there to fill white space, give the card more visual interest.

 Quote: [delusional paranoia]

Indeed! Relax--not every little thing is something important.
_________________

Posted: Thu Mar 30, 2006 10:55 am
Scott
Entrenched

Joined: 11 Sep 2004
Posts: 1139
Location: 390 Chestnut Ridge Rd, Rochester NY, 14624, USA

it seems to me -- and mind, I'm just a meager student of the subject -- that option (i) is setup wrong. the english translation appears to be:

 Code: (fFor some X) and (fFor all Y), (if Y are people, then X is an ancestor of Y)

That is to say: some people_X are ancestors of all people_Y. But that seems to be almost the opposite of what the question is asking us.

I'm not saying the correctly accepted answer is wrong. I'm saying I just don't understand the answer and what it means. Somehow I'm interpretting it the wrong way. Can anyone explain it some?
_________________
Perplex City is a game whose only rule is: There must be a party.
Balance of Powers is a game whose only rule is: There must be a political party.

Posted: Mon Sep 04, 2006 3:17 am
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