Logic, the description of relations between propositions, operates on a few basic principles. While there are some additional qualifications for particular systems of logic (i.e. modal, deontic), what is universal to logical systems and makes them be capable of describing any substantial forms of relations are the postulation (aka just stating something to be the case), negation, and, and or. The other two common operators, like if… then... and equivalence are themselves capable of being stated in terms of just the other four. So they aren’t basic, but rather more complex versions of logical statements which we use for convenience.
So why do we have these four basic operators?
I think this has to do with being. Being, the original postulation, is in-itself. However, it is naturally opposed to that which it isn’t (in this case, to that which just isn’t), meaning that in postulation we find the necessary reflection, negation, posited. If there weren’t in some occasions negation then postulation would lose its meaning.
This gives us two objects of description, i.e. x and ¬x. Because there are two different propositions, we need some way of describing their ability to be related. Our way of understanding the possible relations of these two is what gives us the other two basic operators, the or and the and. The two ways of understanding the necessary relations between x and ¬x are;
x v ¬x
¬(x • ¬x)
So that is my theory of why the four basic logical operators are as they are. They are borne intrinsically of being in opposition to non-being.
