$\vdash_L ((B \longrightarrow C)\longrightarrow B) \longrightarrow B$ Using the Deduction Theorem

Asked Apr 06 '22 at 22:26

Active Apr 07 '22 at 06:10

Viewed 114 times

Trying to deduce $((B \longrightarrow C)\longrightarrow B) \longrightarrow B$ with the the Axioms of Hilbert-Ackerman formal system, but cannot achieve so.

It is only permited to use Deduction Theorem or proven Theorems.

Edit. The tree Axioms are:

$\displaystyle A \longrightarrow (B \longrightarrow A)$
$\displaystyle (A\longrightarrow (B \longrightarrow C)) \longrightarrow ((A\longrightarrow B) \longrightarrow (A \longrightarrow C))$
$\displaystyle (\neg B \longrightarrow \neg A) \longrightarrow ((\neg B \longrightarrow A) \longrightarrow B)$

edited Apr 07 '22 at 06:10

José Carlos Santos

440,053

asked Apr 06 '22 at 22:26

Blue Tomato

4

What are the exact axioms of the system that you are working with? There are many ‘Hilbert’ systems, but the exact rules make a difference. Can you please add those to your post? – Bram28 Apr 06 '22 at 23:44
2

This is called Pierce's Law, and many people have asked how to derive it on this site. See, for instance, here, here, or here. – Chris Grossack Apr 07 '22 at 00:18
@HallaSurvivor that's what i was looking for!!! that was not trivial for me – Blue Tomato Apr 07 '22 at 06:25
It is (I seem to recall) the simplest tautology without any negations which needs an axiom with negations to prove. – ancient mathematician Apr 07 '22 at 07:16

$\vdash_L ((B \longrightarrow C)\longrightarrow B) \longrightarrow B$ Using the Deduction Theorem

0 Answers0