Your Title postulates no rules at all, but the first sentence posits (meerly) different rules.

There are different formalisms for reducing math to a few primitives; e.g. ZFC.

When it comes to using the principles to build up something, there are different kinds of logic with differing levels of power.

Is it possible for logic itself to be based on totally different rules? Well, you'll have a hard tim getting anywhere without Modus Ponens. But in the most abstract terms any number of different formulations exist with the same expressive power.

I suggest you read GEB, post haste! You seem to have the appetite for it right now, even without knowing it.

Say hi to the Tortoise for me.

Your Title postulates no rules at all, but the first sentence posits (merely) different rules.

There are different formalisms for reducing math to a few primitives; e.g. ZFC.

When it comes to using the principles to build up something, there are different kinds of logic with differing levels of power.

Is it possible for logic itself to be based on totally different rules? Well, you'll have a hard time getting anywhere without Modus Ponens. But in the most abstract terms any number of different formulations exist with the same expressive power.

I suggest you read GEB, post haste! You seem to have the appetite for it right now, even without knowing it.

Say hi to the Tortoise for me.