Thread: question on notion of 'proves'...

Is it possibe to have
A proves B
and
not A proves B

I'm tending towards thinking you can't, but not sure why. Any ideas?

Originally Posted by NYC

Is it possibe to have
A proves B
and
not A proves B

I'm tending towards thinking you can't, but not sure why. Any ideas?

I'm not sure what you mean by "proves". If you mean implies then the answer is yes.

p q p-->q

T T T
T F F
F T T
F F T

Originally Posted by NYC

Is it possibe to have
A proves B
and
not A proves B

I'm tending towards thinking you can't, but not sure why. Any ideas?

No (at least as I read your question).

But here's a valid sequent you might consider.

A->B, ~A->B |- B or alternatively,

|- (A->B) -> ((~A->B) -> B),

or getting conjunction into the "equation",

|- ((A->B) & (~A->B)) -> B

Is it possibe to have
A proves B
and
not A proves B

This is possible iff B is provable by itself.

Originally Posted by emakarov

This is possible iff B is provable by itself.

Thanks
In this case is B a tautology?

Originally Posted by NYC

Thanks
In this case is B a tautology?

You betcha!

Originally Posted by oldguynewstudent

You betcha!

Thanks for the quick response!
How would you show that B is a tautology?

Originally Posted by NYC

Thanks for the quick response!
How would you show that B is a tautology?

That would depend on B. Do you have the example of your actual problem?