Proof of p ^ !q |- !(p v q)

[JBL]

Can anyone help me out with this sequent ?

(notation: ^ is conjunction, ! negation)

Thanks in advance.

[Peter_Smith]

Er .... it's not a correct sequent (suppose p is true and q false, then
the premiss is true, and conclusion false). Is the "v" supposed to be
another conjunction???

[Owen]

Post by Peter_Smith
Er .... it's not a correct sequent (suppose p is true and q false, then
the premiss is true, and conclusion false). Is the "v" supposed to be
another conjunction???

?? (p & ~q) -> ~(p v q), is tautologous.

If p is true and q is false, then, the premise (p & ~q) is false.

[Robert J. Kolker]

Post by Owen
?? (p & ~q) -> ~(p v q), is tautologous.
If p is true and q is false, then, the premise (p & ~q) is false.

What????!!!!!. That is absurd.

Bob Kolker

[Owen]

Post by Robert J. Kolker

Post by Owen
?? (p & ~q) -> ~(p v q), is tautologous.
If p is true and q is false, then, the premise (p & ~q) is false.

What????!!!!!. That is absurd.

Yes, my mistate. I tried to delete it but I don't know how. Do you?

Post by Robert J. Kolker
Bob Kolker

[Robert J. Kolker]

Post by Owen

Post by Robert J. Kolker

Post by Owen
?? (p & ~q) -> ~(p v q), is tautologous.
If p is true and q is false, then, the premise (p & ~q) is false.

What????!!!!!. That is absurd.

Yes, my mistate. I tried to delete it but I don't know how. Do you?

This depends on what software you are using to get as the User Group. I
am using the software that comes with Netscape (tm). It has a mode for
deleting one's own positing.

Your recension is duly noted. I am happy that it was just a typo.


Bob Kolker

[Robert Low]

Post by Robert J. Kolker

Post by Owen
Yes, my mistate. I tried to delete it but I don't know how. Do you?

This depends on what software you are using to get as the User Group. I
am using the software that comes with Netscape (tm). It has a mode for
deleting one's own positing.

But bear in mind that cancelling a message is not universally
successful. I think that google groups continues to show
cancelled messages, for example.

[Owen]

Post by Owen

Post by Robert J. Kolker

Post by Owen
?? (p & ~q) -> ~(p v q), is tautologous.
If p is true and q is false, then, the premise (p & ~q) is false.

What????!!!!!. That is absurd.

Yes, my mistate. I tried to delete it but I don't know how. Do you?

This depends on what software you are using to get as the User Group. I am
using the software that comes with Netscape (tm). It has a mode for
deleting one's own positing.
Your recension is duly noted. I am happy that it was just a typo.

Truth value analysis is sufficient in these cases.

Bob Kolker

[George Dance]

Post by Owen

Post by Peter_Smith
Er .... it's not a correct sequent (suppose p is true and q false, then
the premiss is true, and conclusion false). Is the "v" supposed to be
another conjunction???

?? (p & ~q) -> ~(p v q), is tautologous.
If p is true and q is false, then, the premise (p & ~q) is false.

Would you mind showing the proof of that?

[Owen]

Post by George Dance

Post by Owen

Post by Peter_Smith
Er .... it's not a correct sequent (suppose p is true and q false, then
the premiss is true, and conclusion false). Is the "v" supposed to be
another conjunction???

?? (p & ~q) -> ~(p v q), is tautologous.
If p is true and q is false, then, the premise (p & ~q) is false.

Would you mind showing the proof of that?

You are advised to read subsequent posts before you make such silly remarks!

[George Dance]

Post by Owen

Post by George Dance

Post by Owen

Post by Peter_Smith
Er .... it's not a correct sequent (suppose p is true and q false, then
the premiss is true, and conclusion false). Is the "v" supposed to be
another conjunction???

?? (p & ~q) -> ~(p v q), is tautologous.
If p is true and q is false, then, the premise (p & ~q) is false.

Would you mind showing the proof of that?

You are advised to read subsequent posts before you make such silly remarks!

That's obviously advice you don't follow yourself, since you've done
the exact same thing to me more than once.

[Owen]

Post by George Dance

Post by Owen

Post by George Dance

Post by Owen

Post by Peter_Smith
Er .... it's not a correct sequent (suppose p is true and q false, then
the premiss is true, and conclusion false). Is the "v" supposed to be
another conjunction???

?? (p & ~q) -> ~(p v q), is tautologous.
If p is true and q is false, then, the premise (p & ~q) is false.

Would you mind showing the proof of that?

You are advised to read subsequent posts before you make such silly remarks!

That's obviously advice you don't follow yourself, since you've done
the exact same thing to me more than once.

If so, I apologise.
Where did I do these things?

[Robert J. Kolker]

Post by Peter_Smith
Er .... it's not a correct sequent (suppose p is true and q false, then
the premiss is true, and conclusion false). Is the "v" supposed to be
another conjunction???

Yep. That makes sense. As it is written, it is wrong. If one of the
items is false then the conjunction of the items is false.

Bob Kolker

[William Elliot]

I've not the time nor the inclination for the clerical
chore of back tracking the thread to reconstruct the
line of thought. If you want intelligent and well thought
answers, then include the context pertinent to your reply
and to whom your are talking.

Please learn and use better math group manners as demonstrated
by other participants of this newsgroup and as described at
http://oakroadsystems.com/genl/unice.htm#quote

Post by Peter_Smith
Er .... it's not a correct sequent (suppose p is true and q false, then
the premiss is true, and conclusion false). Is the "v" supposed to be
another conjunction???

Jerk, you removed OP's definitions of notation.

[Peter_Smith]

If "jerk" is addressed to me, then I do take some real exception.

We were presented with a sequent p ^ -q |- -(p v q) and I took the
request for "help" with it to be a plea for a proof. I pointed out that
as it reads, it isn't a valid sequent, so can't be proved.

I then wondered if the vel sign was a typo for another hat for
conjunction (which would give a valid sequent). For example, maybe
there was a typo in the original exercise sheet that the original
poster was struggling with. That seems a perfectly sensible and
unjerkish supposition to me. As someone who in his time has given out
too many exercise sheets with embarrassing typos in them I know how
easy it is to do.

[William of Ockham]

And at least one internet-posted paper, eh Peter?

Best wishes

"Bill"

[William Elliot]

Post by Peter_Smith
If "jerk" is addressed to me, then I do take some real exception.

What are you talking about? But as you've removed the context ...

Post by Peter_Smith
We were presented with a sequent p ^ -q |- -(p v q) and I took the
request for "help" with it to be a plea for a proof. I pointed out that
as it reads, it isn't a valid sequent, so can't be proved.

I agree.

Post by Peter_Smith
I then wondered if the vel sign was a typo for another hat for
conjunction (which would give a valid sequent). For example, maybe
there was a typo in the original exercise sheet that the original
poster was struggling with. That seems a perfectly sensible and
unjerkish supposition to me. As someone who in his time has given out
too many exercise sheets with embarrassing typos in them I know how
easy it is to do.

Did OP mean to ask
p & ~q |- ~(p & q) ?

That I leave OP to inform us if perchance he's perturbed by being told his
problem (as stated) is impossible. Yes the problem you present is valid
and your supposition plausible. Where's OP in this thread to set things
straight?

[David C. Ullrich]

On Fri, 2 Sep 2005 18:32:25 -0700, William Elliot

Post by William Elliot
I've not the time nor the inclination for the clerical
chore of back tracking the thread to reconstruct the
line of thought. If you want intelligent and well thought
answers, then include the context pertinent to your reply
and to whom your are talking.
Please learn and use better math group manners as demonstrated
by other participants of this newsgroup and as described at
http://oakroadsystems.com/genl/unice.htm#quote

Post by Peter_Smith
Er .... it's not a correct sequent (suppose p is true and q false, then
the premiss is true, and conclusion false). Is the "v" supposed to be
another conjunction???

Jerk, you removed OP's definitions of notation.

Exactly where did the OP define "v"?

Jerk.


************************

David C. Ullrich

[George Dance]

Post by Peter_Smith
Er .... it's not a correct sequent (suppose p is true and q false, then
the premiss is true, and conclusion false). Is the "v" supposed to be
another conjunction???

Most likely v is disjunction, and the left hand side should read !p ^
!q.

However, there's no need to assume it is and give the OP a proof; since
he wasn't interested enough to clear that up, he can't be all that
interested in seeing a proof either.

[William Elliot]

Post by JBL
Can anyone help me out with this sequent ?

No it's helpless as explained below.

Post by JBL
(notation: ^ is conjunction, ! negation)

! is ~

and v is disjunction. By DeMorgan rules
~(p v q) <-> ~p & ~q
Note how the premise
p ^ ~q
conflicts with the conclusion ~p & ~q.

That is when assuming the premise, were the conclusion true,
you'd have p & ~p.

[Paul Holbach]

Post by JBL
Can anyone help me out with this sequent ?

p & ~q |- ~(p v q)

I don't think this is provable because the following is provable:

(01) 1 p & ~q P
(02) 1 p 1 &E
(03) 1 p v q 2 vI
(04) (p & ~q) -> (p v q) 1,3 CP
q.e.d.


Regards
PH

[G. Frege]

Can anyone help me out with this sequent?

I guess you meant

~P & ~Q |- ~(P v Q).

There's a rather short proof for this sequent in Colin Allen and Michael
Hand's System of ND:

1 (1) ~P & ~Q A
2 (2) P v Q A
1 (3) ~P 1 &E
1,2 (4) Q 2,3 vE
1 (5) ~Q 1 &E
1 (6) ~(P v Q) 4,5 RAA(2)


F.


--
"I do tend to feel Hughes & Cresswell is a more authoritative
source than you." (D. Ullrich)