Fixed/critical points of a nonexistence quantifier/function

Let j(∃0) = 1, and j(∃1) = 1, for a justification function j on ∃-sentences. So far, 0 is the initial critical point of the composite quantifier-function, and 1 is the initial fixed point.

So let there be a difference for j(∄0) = -1, i.e. we are not merely lacking in justification for a belief in the nonexistence of 0, but we are substantively contrary-to-justified if we think that 0 doesn't exist. Furthermore, j(∄1) = -1. Now, we would have to also go back, then, and say j(∃-1) = 1, but then j(∄-1) = -1, so that -1 would be the initial fixed point of a nonexistence quantifier function.

Note that T(not(S)) = not(T(S)), or that, "There does not exist..." = "There does exist not..." alongside, "There is not actually..." and, "There is actually not..." Yet so in modal logic, ◊¬ ≠ ¬◊. Would assuming that j(Ex)-sentences involve 0, 1, and -1 as stated above, be justifiable by varying reference to such modal symmetries and asymmetries?

MOTIVATION FOR THE SYMMETRY/BREAKING QUESTION:

In this IEP article on contemporary modal logic, the author says of one operator apiece from temporal and deontic logics:

For instance, ‘always’ is ‘not sometimes not’, or ‘ought’ is ‘not permitted that not’. Such algebraic duality patterns are so ubiquitous that in the 1950s, it was even proposed to include them in broadcasts into outer space announcing our presence to other galactic civilizations – a truly fitting endeavor for possible worlds theorists.

• OP sounds like degree of credence for free logic with existence predicate applying to non-objectual intensional constants in your second version of system thus existence predicate ensures bivalence normalizing your range to be the normal 0-1 aligning with Cox theorem. Then nonexistence 0 is the fixed point in the second kind. Yet Shurangama (literally meaning fixed point) sutra warned: this person perceives both existence and nonexistence. Experiencing this branching, his mind becomes confused...distorted, false theories that are total speculation based on the sophistry of immortality... Commented Oct 17, 2023 at 5:48