There are many logics. For example, there are modal logics, deontic logics, fuzzy logics, first-order prepositional logics, first-order predicate logic, higher-order predicate logic, inclusive predicate logic, monadic predicate logic, polyadic predicate logic, etc. etc.
If you take a first-order prepositional equational logic (used often with mathematics), for example, the axioms for one of those logics is as follows:
Associativity of ==: ((p == q) == r) == (p == (q == r))
Symmetry of ==: p == q == q == p
Identity of ==: true == q == q
Definition of false: false == ~true
Distributivity of not: ~(p == q) == ~p == q
Definition of /==: (p /== q) == ~(p == q)
Associativity of \/: (p \/ q) \/ r == p \/ (q \/ r)
Symmetry of \/: p \/ q == q \/ p
Idempotency of \/: p \/ p == p
Distributivity of \/: p \/ (q == r) == p \/ q == p \/ r
Excluded Middle: p \/ ~p
Golden rule: p /\ q == p == q == p \/ q
Implication: p => q == p \/ q == q
Consequence: p p
Anti-implication: p /=> q == ~(p => q)
Anti-consequence: p / |- E[x:= P] = E[x:= Q]
Transitivity: If P = Q and Q = R are theorems, then so is P = R.
|- P = Q, |- Q = R ---> |- P = R
Equanimity: If P and P == Q are theorems, then so is Q. |- P, |- P == Q ---> |- Q
If you notice, each axiom and rule refers to an undefined term (P, Q, p, q, r, =, V, etc). If you want to say that this particular logic 'exists', then you have to ask what those variables mean. They are simply left undefined in the formal language, hence to translate this into ontology you would be saying that reality is also undefined.
Warm regards, Harv