Are -(pvq) and -pv-q the same?

Question

I'm having some trouble with my logic homework. In a truth-functional table do -(pvq) and -pv-q have the same final answer? Because I'm not sure if I did mine right...

Answer 1

Hi



No, '~(P v Q)' and '~P v ~Q' do not have the same truth table. '~(P v Q)' says that neither 'P' nor 'Q' is true; 'P' and 'Q' are both false. '~P v ~Q' says that either 'P' or 'Q' is false. It is perfectly possible that '~P v ~Q' should be true yet '~(P v Q)' false. For example, imagine that 'P' is true and 'Q' is false. Because 'P' is true, 'P v Q' is true, and so '~(P v Q)'' is false. But because 'Q' is false, '~Q' is true, and so '~P v ~Q' is true. Thus, when 'P' is true and 'Q' is false, '~P v ~Q' is true but '~(P v Q)' is false; '~P v ~Q' and '~(P v Q)' do not have the same truth table.



An example might make things clearer. Suppose that 'P' reads 'Jessica has a smelly butt' and 'Q' reads 'Chelsea is short'. Then '~(P v Q)' says that Jessica does not have a smelly butt and Chelsea is not short, and '~P v ~Q' says that either Jessica does not have a smelly butt, or Chelsea is not short.



Imagine that you go into the bathroom and find that it's really stinky. You ask a friend why it's so smelly and learn that Jessica has just got off the potty :-) From this you infer that Jessica has a smelly butt; you infer that P. Now, if Jessica does indeed have a smelly butt, then ~~(P v Q); if P, then either P v Q (i.e., if Jessica has a smelly butt, then either Jessica has a smelly butt, or Chelsea is short). Hence, if 'P' is true, then '~(P v Q)' is false, regardless of whether 'Q' is true or false.



Let's suppose that Chelsea is 5 feet 8; while we may be reluctant to say that she's tall, she is certainly not short. Thus, 'Q' is false - ~Q - and, hence, ~P v ~Q; if Chelsea is not short, then either Jessica does not have a smelly butt, or Chelsea is not short.



To summarize, if P, then '~(P v Q)' is false; and if ~Q, then '~P v ~Q' is true. Hence, if P and ~Q - i.e., if Jessica has a smelly butt and Chelsea is not short - then '~P v ~Q' is true but '~(P v Q)' is false. Thus, '~(P v Q)' and '~P v ~Q' do not have the same truth table.



'~(P v Q)' is false when 'P' and 'Q' are both true, false when 'P' is true and 'Q' is false, false when 'P' is false and 'Q' is true, and true when 'P' and 'Q' are both false.



'~P v ~Q' is false when 'P' and 'Q' are both true, true when 'P' is true and 'Q' is false, true when 'P' is false and 'Q' is true, and true when 'P' and 'Q' are both false.



Thus, the truth tables for '~(P v Q)' and '(~P v ~Q)' differ on the second and third lines.



I really hope that this has helped a little. Good luck, and feel free to email me if you have any more questions :-)

Answer 2

There are several ways to prove that they are not the same. The post above mine gives a reduction to an absurdity method of proof, also called resolution.



You can also use what is called GPS to arrive at ~(~pv~q), the opposite of what you have.



A third way is to use truth tables and compare the truth values for the two statements and notice that they are not the same.

Answer 3

One way to do this is to assume that the conclusion is false. If you arrive at a contradiction, then you know that they are the same.





1. ~(pvq) Premise

2. ~(~pv~q) Assumption

3. ~~p&~~q 2, demorgan's law

4. p & q 3, double negation

5. ~p&~q 1, demorgan's law

6. p 4, simplification

7. ~p 5, simplification

8. p&~p 6,7 conjunction

9. ~pv~q 2-8, reductio ad absurdum (reduction to absurdity)