By "entire", do you mean "complete"? If so, then, towards proving B, assume that . Then is a tautology and hence an axiom. By repeatedly using Modus Ponens, one can derive from .
For C, similarly, use the fact that is a tautology.
I'll be delighted to get some help in the following:
Let D be the following system of inference: The axioms are all the tautologies except the ones that is their main symbol and the only rule of inference if the rule:
A. Prove that D is entire in the weak meaning, but not entire (I've proved it).
B. Let D' be the system of inference that derives from D by changing the rule by :
. Prove that D' is entire.
C. Sketch a proof tree for : from the set .
I've proved A... I realy need help in B and C.
Here is what I know:
Thanks !in part B: if we can prove somehow that D' is entire in the weak meaning, we will be able to deduce what we need (because of the special rule of inference we have) ... But how can we prove it?
In C-I know how to prove that is true in D'... But can't figure out how to prove the full statement...
I remember that when I was listening to a lecture on logic as a fourth-year student and this subject came up, many students also did not understand this at first. However, using nested implications is very convenient.
Let n = 2 for now. The formulas P1 -> (P2 -> Q)) and (P1 /\ P2) -> Q are logically equivalent. The first one says, "If P1, then if P2, then Q", which is the same as "If P1 and P2, then Q". Formally, their equivalence can be shown using truth tables.