EDIT: someone just showed my the typo in the lecture notes. instead of:
{¬¬φ,φ}├ φ
{¬¬φ,φ}├ ¬φ
it should be:
{¬¬φ,¬φ}├ ¬¬φ
{¬¬φ,¬φ}├ ¬φ
than by the third form of proof by contradiction we get that:
{¬¬φ}├ φ
and by deduction:
├ (¬¬φ→φ)
ORIGINAL POST: Given Hilbert system and:
Deduction:
K∪{ψ}├ φ iff K├ (ψ⇒φ)
Proof by contradiction:
For every φ,ψ – {ψ,¬ψ}├ φ
Inconsistent set proves every verse
If K∪{¬φ} is inconsistent then K ├ φ
How can it be proven that:
├ (¬¬φ⇒φ)
├ (φ⇒¬¬φ)
I am studying from a lecture note that claims the set {¬¬φ,φ} is inconsistent because:
{¬¬φ,φ}├ φ
{¬¬φ,φ}├ ¬φ
But I can’t prove {¬¬φ,φ}├ ¬φ and frankly don’t see why it should be true given that ¬¬φ is supposed to be equivalent to φ – that is what we are trying to prove after all, so why would two equivalent verses prove their negation? I will be grateful for every proof of the above mentioned, but if it is possible to confirm or deny the proof from my lecture notes (not mine, actually. I should hope I wouldn’t need help understanding my own notes. In fact they were written by a teacher, but are known to contain some typos) I will be especially grateful.
I do not study in English so I may have created some bad or non-conventional
translations, for which I apologize.
EDIT: as per @GitGud's request, here is a screenshot of the notes (the text is in Hebrew but I think the math notation is clear even without the text):
