# Find Theorems of a Formal Theory

Going through a book on formal logic, I have encountered the following problem. Since I am somewhat new to formal logic, I am confused about how to approach it.

A certain formal theory has exactly two axioms:

(a) 2 + 2 = 4 -> (2 + 2 = 4 -> 2 + 3 = 6)
(b) 2 + 2 = 4


and has modus ponens, i.e., (P->Q, P) -> Q

Find all theorems of this theory.

I understand that the axioms themselves are theorems. How can I find the others?

• Hint: think about it like like a self-assembly kit. You have been given some basic parts (the axioms) and some tools for combining existing parts to make new ones (the inference rules). In your case you only have one inference rule (modus ponens) and a very small supply of axioms. What can you build? Mar 28, 2017 at 21:17
• What confuses me is the double implication in axiom (a). How does one understand it? Mar 28, 2017 at 23:45