In Boolean Logic, the answer is no. For any statement (A) in a set of possible statements (or propositions), there is exactly one unique opposite statement (¬A).
There can be multiple distinct propositions B,C,D,....such that A ∧ B = FALSE, A ∧ C = FALSE, A ∧ D = FALSE. And there can be multiple distinct propositions I,J,K,...such that A ∨ I = TRUE, A ∨ J = TRUE, A ∨ K = TRUE,...
But there can be only one unique proposition that satisfies the conditions:
- A ∧ ¬A = FALSE
- A ∨ ¬A = TRUE.
- ¬¬A = A
This is because the rules of Boolean Logic lead to a mathematical structure called an orthocomplemented lattice.
Consider the set of days of the week:
The proposition "Today is Monday" is not the opposite of "Today is Tuesday" because even though the statement ["Today is Monday" and "Today is Tuesday"] is always False, the statement ["Today is Monday" or "Today is Tuesday"] is not always True.
The proposition "Today is Tuesday" has one unique opposite proposition which is "Today is either Monday, or Wednesday, or Thursday, or Friday or Saturday, or Sunday".
However, in other forms of logic, such as some - but not all - Fuzzy Logics, or Quantum Logics, it is possible for a single proposition to have more than one opposite. This is because such logics lead to different structures which are described by orthomodular lattices instead of orthocompelmented lattices.
In particular, in Quantum Logic, the reason why one proposition can have more than one opposite or complement is because just as in your initial example, statements are represented as vectors in multiple dimensional Hilbert Spaces, instead of being represented as subsets of a Boolean Algebra.