Just a small question regarding the logical negation of a statement:

$∀x (∃y (∀z A(x,y,z)))$ where a is $A$ formula dependant on $x,y,z$.

I know that when negating statements quantifiers are reversed and then the statemnent is negated, but I just want to check the order of statements is correct.

Based on above I worked out:

$∃x (∀y (∃z ¬A(x,y,z))$

But it doesn't quite seem right just thinking about the wording of the statement.

Any help would be greatly appreciated!

  • 2
    $\begingroup$ Why do you think it is not right? $\endgroup$ – Ashot Mar 23 '13 at 6:26
  • $\begingroup$ The only reason it's not quite right is that it is missing a parenthesis :) $\endgroup$ – Trevor Wilson Mar 23 '13 at 6:38

Test the result with a sample situation where that statement might arise. Like, say...

For all houses, there exists a plug shape such that, for all power sockets, the plug will fit into the socket.

(that is to say, each and every house has all of its power sockets shaped in such a way that at least one specific plug shape will fit into all of them within that house)

The negation becomes:

There exists a house such that, for all plug shapes, there exists a power socket for which the plug will not fit into the socket.

(that is, in at least one house in which, for any particular plug shape, there's at least one socket that won't accept that plug)

Seems to make sense to me - if the first statement is false, then the second one is true, and vice versa.


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