As I know all humans share a unique logic and even less-educated people uses same common sense in their statements. For example, it is rational for anybody that if P is correct then P or Q is correct too.
Inspired by this question about God and logic, is there any other axiom of logic?
Does it really makes sense in any way? (If yes please explain by an example)
Is our current logic the result of evolution of our brain or the logic is an abstract concept no matter how our brain is developed?
@vanden's answer gives a good overview of different "logics". You can also read more information at Wikipedia. If you want to get deeper into this topic, Stanford Encyclopedia of Philosophy is a great resource with several articles:
Regarding your question, "Is there any other axiom of logic?", I would like to recommend an article by Lewis Carroll entitled What the Tortoise Said to Achilles. I had the same question, and when we read this in my theory of knowledge class, it cleared up a lot of things.
In short, it is generally accepted in the field of analytic philosophy that any attempts to find an underlying axiom of logic will result in an infinite regress.
However, I have found that some areas of continental philosophy, (specifically phenomenology/intentionality and in some of Nietzsche's works). These works are considerably more difficult to read and to comprehend than most analytic philosophy, but if you put in the effort you will encounter interesting alternative paradigms for logic and philosophy.
I am going to answer entirely on the logic side, disregarding the connection to religious issues.
There are a wide range of logics available.
Call Classical First-Order Logic (CFOL) the logic that is usually taught in contemporary introductions to formal logic. It has several properties of relevance here. For example: from P & ~P, everything follows, P v ~P is a logical truth, and quantification (statements like 'There is an x such that Fx' and 'For all x ....) range only over objects (hence 'FO').
There are various extensions of CFOL.
Higher order logics allow one to quantify over properties of objects (Second Order Logic), or properties of properties (Third Order Logic), etc. SOL is needed to state the full generality of the principle of mathematical induction in arithmetic, to articulate a formal theory of arithmetic such that the natural numbers are (up to isomorphism) the unique model of the theory, and to state in full generality Leibniz's laws of identity. For each order above first, there are alternative semantics available which induce different rules of inference.
There are various modal logics, which add operators for notions like metaphysical possibility and necessity, temporal relations, epistemic notions, and notions of permissibility and obligatoriness. Even for metaphysical necessity (likely the best studied modal logic), there are multiple logics available, depending on the underlying view about the relationships between possibility and necessity.)
There are also many restrictions of CFOL.
Intuitionistic logic gives up on the general truth of P v ~P, doesn't in general allow you to conclude P from Q & ~Q (though ~P still follows), and treats quantification somewhat differently.
Relevant logics require more for the truth of P --> Q than the CL condition that either P is false or Q is true in order to avoid the paradoxes of material implication. (Along the way, it looses the principle of Disjunctive Argument: From P v Q and ~P, Q.)
Paraconsistent logics drop the principle that everything follows from a contradiction (in a fashion rather different than does Intuitionistic logic); it is motivated in part by the idea that there may be true contradictions.
There are multi-valued logics which assign, in their semantics, more than 2 truth-values (even for 3 values, there is more than one way to proceed) and adopt corresponding principles of proof. These are not in general classifiable as restrictions or extensions of CFOL; it depends upon the details of the particular multi-valued logic.
That is just a very quick (hand-wavy, really) survey of some of the better know alternatives to CFOL. It is, however, sufficient to establish that not only are alternative logics possible, they are actual. However, none of these (including CFOL) are universally accepted as logic; the demarcation of the bounds of what really counts as logic is one of the central problems in the philosophy of logic.
Does God know what he's going to do tomorrow? If so, could he do something else?" If God knows what will happen, and does something else, he's not omniscient. If he knows and can't change it, he's not omnipotent. "
if God knows what will happen and he does something doesn't makes him not omniscient. He still has infinite knowlege and he just choose to let the world goes on in another direction. Both the original and altered way of developement of the world is within the scope of God's knowledge. It's like you know where you will be if you choose which way at a cross road with the knowledge of wihch road leads to where, and your choice does not change your knowledge.
"Can 'an omnipotent being' create a stone so heavy that it cannot lift it?"
Things such as " a stone so heavy that it cannot lift it? " does not exist in this scenario. God is omnipotent does not mean everything has to exist in world, for example, there can't be a number that is both 0 and 1. So if you admit god is omnipotent, there's no such thing that God cannot lift it. The two things are like two sides of a coin, and you can only choose one.
For your last question, I think the current logic can only be the evolution of our brain. Because every thing we think of and talk about is product of our brain, including the concept "abstract concept". In other words, you can not talk about anything out of the product of our brain, unless you have a different brain ...
