[Note: I found one essay, about Aristotle, that used the word "demiset," although at a glance it seemed like they might've been substituting this terminology for a counterpart to the subset/superset relation.]
Let a Peirce cut be C. S. Peirce's use of enclosed spaces on the Blank Page of Truth to indicate generic negation/complementation. On the other hand, I can only make this question "visually" go through by having the orientation of the cuts reversed in value: the One True Page is actually, when utterly empty, the False, and it is only by enclosing spaces that the Truth manifests.
I know that sounds overdramatic, but anyway, I did find out eventually that the use of parentheses in logic notation is not absolutely trivial (I mean: whether to design one's notation with or without them is not without some consequence). So, while dying inside trying to understand the concept of antisets, my soul on another plane of existence was also dying because it wanted to make sense of the concept of antiset-minus, a point in the universe of infinite sets where nontrivial negation operations vanished into nothingness. This mutated, eventually, into trying to impose a transfinite ordinal limit on the negative hyperoperations that could be used to indicate demi-negation, which mutated into something even more difficult to prove the meaningfulness of, but:
I have always been uneasy about using the bare inset notation {} to mark out an empty set and, then, the "essence of the number zero." Note that if one starts with the concept of a set generically, there is no default fixture of a first-considered set as well-founded. However, if one starts with a pure ur-element, which has no essence but to be an element of some singleton, then that singleton is by direct character well-founded; the set of the ur-element is not defined as well-founded "from the outside" but this factoid is given from "within" it.
It is an endless theme in mathematics that various procedures either share or lack various properties that are spoken of as "left-directed" or "right-directed." Adjoint functors, associativity and commutativity, and so on and on all involve such a comparison/contrast. Is it necessary to have the inset operation, the adjunction of the {}=notation to some x, be "all at once"? We do have, in interval notation, that one can flip through (), (], [), and [] to indicate opening and closure. In the {}-case, I'm asking whether we could usefully refer to a demi-set operation, one that only encloses some x from either the right or the left (whichever is designated for the relevant reason).
Or, then, start from set {x} and relate it to a demiset-theoretic antiset. Let there be an antielement -e such that {x} ∪ {-e} = x, -e}, i.e. the antielement "deletes" one side of x's initial enclosure. Then let there be some -e2 such that {x} ∪ {-e, -e2} = x. What is negated is not the element x itself but its (local) property of elementhood. So though antielements occur initially, their "purpose" is not to create an empty set for that set's own sake but just to remove the set relation on some element entirely.
My assumption is to let {x} → 1 (for 1 = True) and x to -1 {for False). What about {x or x}? Assume the imaginary-numerical semantics for demi-negation and the turnstile notation overall. Off the top of my head, the easiest way to write it out would be:
- For T(S) = the truth-value function on sentences S, for sentences of the form, "x ∈ {x," or, "x ∈ x}," T(x ∈ {x) = i, i.e. is demi-negated.
This doesn't seem as if it really should work just like that, though. Even with fixing i: ⊣ and -i: ⊢, it fits the theme more to say:
- Starting with x, if there is some Z such that Z ⊢ x, this gives us Z(x) = {x or x}. Then for some Z' it should be that that Z'(x}) = {x}, i.e. a full proof encloses x at (or as) True. By contrast, repeated demi-negation starts from {x} and has some F, F' such that (F&F') ⊣⊣ {x} = F'(x}) = x.
However, I can't quite get the i-metaphor(?) to work properly. Offhand, ⊣⊣ and ⊢⊢ would be i2x and (-i)2x = -1x, which would make a double proof into a disproof. I do get the right result by demi-negating a demi-proof, though (i times -i = -1(i2) = -1(-1) = 1).
The relevance of Peirce cuts, mirrored, is then that fully closing x is equivalent in meaning to making a Peirce cut for x, except that when x is unhoused is when x maps with the False.