Thread: What's the notion of Cancellation of hypotheses in natural deduction?

Hi everybody,

I am a bit confused with the notion of Cancellation of hypotheses in natural deduction.What does it mean?
When we derive psi from phi for example, and by this we derive psi->phi , then how is the hypothesis cancelled? What does even that mean?
Note that my reference is VanDallen's "Logic and structure".
Thanks

Originally Posted by Mathelogician

I am a bit confused with the notion of Cancellation of hypotheses in natural deduction.What does it mean? When we derive psi from phi for example, and by this we derive psi->phi , then how is the hypothesis cancelled? What does even that mean?

I think that there is problem with translation here. That said, I will make a guess.
It seems that you are asking about proof by contradiction .
See this link.

No,
There is no translation problem!
My question is clear and as i mention before, you can find out about the notion in VanDallen's book.
Indeed i am asking about meaning of cancellation of hypothesis in (Gentzen's - as the author says ) natural deduction.
[Although it also happens in proof by contradiction-But my question mostly refers to (-> introduction) rule of deduction.]

I would try to restate more clear:

What does it mean when we derive (phi->psi) from a derivation with hypothesis phi and conclusion psi?
Does it mean that for phi->psi to be hold, we don't need always to have phi to hold?
I am really confused with the notion and need clarification as soon as possible.
Thanks...

I can prove any mathematical statement. Take Riemann hypothesis, for example, and denote it R. I assume R, and from this assumption I trivially conclude R. Did I prove R? Yes. The problem is that I have not canceled (or closed, or discharged) all of my assumptions. I can cancel the assumption, but then the statement that I proved will no longer be R but R -> R, which is trivial, of course. What would earn me tenure in this case is actually proving the assumption, which brings be back to proving R. Oh well.

So, derivations have value if they are closed, i.e., have no open assumptions. The natural deduction rule of implication introduction takes a derivation with an open assumption A and a conclusion B and allows one to cancel, or close, the assumption A. However, the price for this is that the conclusion is changed from B to A -> B. Indeed, if I can prove B under the assumption that A holds, I can prove A -> B without making any assumptions.

Originally Posted by Mathelogician

What does it mean when we derive (phi->psi) from a derivation with hypothesis phi and conclusion psi?

That's precisely what it means: there is a rule of inference that allows us to derive (phi->psi) from a derivation with hypothesis phi and conclusion psi.

Originally Posted by Mathelogician

Does it mean that for phi->psi to be hold, we don't need always to have phi to hold?

Yes, if 2 + 2 = 5, then the Moon is made of green cheese. This statement is true, and it does not require 2 + 2 = 5 to hold.

A perfect answer like always.