Distributing 'Not' over a Conjunction

Date: 09/18/2003 at 11:22:31
From: Sarah
Subject: logic

If 

  p = today is sunny 

and 

  q = tomorrow is Friday 

then which of the following means "It is not true that today is sunny
and tomorrow is Friday"? 

  1. not (p^q) or 
  2. not p ^ q

I'm not sure whether "it is not true" in front of a conjuction applies
to both conjuncts or only the first one.

Date: 09/18/2003 at 23:22:09
From: Doctor Achilles
Subject: Re: logic

Hi Sarah,

Translations are always tricky.  However, I would say that "it is not 
true that ..." means that the negation applies to everything that 
follows.

Contrast 

  It is not true that today is sunny and tomorrow is Friday

with 

  Today is not sunny and tomorrow is Friday

In the sentence I just made up, the 'not' is only in the first clause,
and so only applies to 'p'.  In your sentence, the 'not' governs the
whole thing, so should apply to 'p' and 'q'.

So the correct translation is

  not (p^q)

I hope this helps.  If you have other questions or you'd like to talk 
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/