TITLE: Law of Clavius explained
QUESTION [7 upvotes]: Law of Clavius states  $ \sim P \Rightarrow  P \vdash P$  
And the only explanation I sort of understand is  
"If we ought to philosophise, then we ought to philosophise; and if we ought not to philosophise, then we ought to philosophise (i.e. in order to justify this view); in any case, therefore, we ought to philosophise."
which to me reads as $(P \Rightarrow  P)\wedge (\sim P \Rightarrow P ) \vdash P   $
where P is "we ought to philosophise"
Which is not quite the same.  
I am aware of the following relationship
$(\sim P \Rightarrow P ) \Leftrightarrow \; (\sim(\sim P)\; \vee \: P)$
and combining this with the above would basically be a form of "Or Elimination" or proof by cases. But this is just an explanation of how its proof would be worked out.
Anyway the above is more of a digression, but perhaps it may have some weight on my question. Really what I am interested in is an intuitive understanding of Clavius' law. What are some common examples in everyday logic?

REPLY [2 votes]: Clavius Law aka Consequentia Mirabilis is an instance of the more common Proof by Cases. Proof by Cases is the axiom schema:
/* Proof by Cases */
(¬ B → A) → ((B → A) → A)
If you now instantiate B = A, you get:
/* Clavius Law aka Consequentia Mirabilis */
(¬ A → A) → A
Interestingly you can go backwards and derive Proof by Cases from Consequentia Mirabilis as well. The interderivability
holds over minimal logic. The instantiation B = A is also mentioned here:
Aristotle and the Consequentia Mirabilis
William Kneale - 1957
https://www.jstor.org/stable/628635?seq=1