Quantifier negation in n-ary relations: Philosophy Forums

Quantifier negation in n-ary relations
Can a quantifier be negated, when its var. is used as arg. in n-ary relations?

Quantifier negation in n-ary relations

moonlight
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Posted 02/26/08 - 09:30 PM: Subject: Quantifier negation in n-ary relations	#1
Hi all, Quick question. I use 'A' and 'E' to denote respectively 'for all' and 'there exists' quantifiers. By definition we know that: ~Ax.~P(x) = Ex.P(x). My question is: how about when the binded variable is used in n-ary relations: how can the quantifier be negated to obtain an equivalence. For example: Ax.Ey.R(x,y). That is not equivalent to ~Ex.Ey ~R(x,y) if I'm not mistaken. So is there any transformation that can be done when the variable occurs in a relation of arity higher than 1? Thanks in advance for any input about this. Cordially, moonlight.
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keda
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Posted 02/26/08 - 11:35 PM:	#2
~Ex.Ay ~R(x,y) Given the defintion: 1. ~Ax.~P(x) = Ex.P(x). negate both sides 2. ~~Ax.P(x) = ~Ex.P(x). doube negataion 3. Ax.~P(x) = ~Ex.P(x). doube negataion 4. Ax.~P(x) = ~Ex.~~P(x). let Q be the negation of P 5. Ax.Q(x) = ~Ex.~Q(x). Ax.Ey.R(x,y). can thus according to 5 be written as ~Ex.~Ey.R(x,y). with 3 you get ~Ex.Ay.~R(x,y).
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muxol
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Posted 02/29/08 - 02:41 PM:	#3
The arity of the formula following the quantifier (in this case either P or R) is irrelevant.

the.yangist
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Posted 03/01/08 - 05:29 PM:	#4
People always beat me to the fun ones.
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moonlight
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Posted 03/13/08 - 10:20 AM:	#5
Hi keda, Thanks for the reply. I read your answer over and over again but I can't see how: ~(Ax.Ey. R(x,y)) = ~Ex.Ay. ~R(x,y) I think there is a mistake there. After doing some reading I'd suggest the following method to negate quantified statements in PNF: move the negation to the inside, up until the matrix R, and switch quantifiers every time the negation passes one. For example: ~(Ax.Ey. R(x,y)) = ~Ax.Ey. R(x,y) = Ex.~Ey. R(x,y) = Ex.Ay.~R(x,y). Indeed suppose Ax.Ey. R(x,y) means "every guy is taller than a girl". Then the negation Ex.Ay.~R(x,y) means "some guy is shorter than all girls". Which makes sense. I just thought I'd write this here, because I struggled quite a bit with this question for a moment, and it might help others. Take care & thanks for helping me out, moonlight. Edited by moonlight on 03/13/08 - 10:24 AM
All are lunatics, but he who can analyze his delusion is called a philosopher. - Ambrose Bierce -

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