TITLE: Can we define definitions as axioms in logic?
QUESTION [6 upvotes]: In a proof or deduction system, there are some axioms and some inference rules. To have a small proof system, axioms are defined for the primitives (for example for negation and implication in propositional logic). Now, we want to have a proof system that proves all provable formulae including the formulae contain non-primitive logical connectors (such as OR in propositional logic). The question is should we add definitions for non-primitives as axioms to the proof system, or we should do something else? In other words, do logicians accept definitions as axioms?

REPLY [3 votes]: Yes, absolutely you can introduce definitions as axioms.  That's what Metamath does:
http://us.metamath.org/mpegif/mmset.html#definitions
See also
http://www.hss.cmu.edu/philosophy/techreports/181_Avigad.pdf
which defines a formal system with definitions:

We extend the foundational framework in two ways. First, we allow for explicit deﬁnitions of new predicates and functions on the universe of sets. And,
  second, we allow function symbols to denote functions that are only partially de-
  ﬁned, using a logic of partial terms. We call the resulting formal system DZFC .