Notes On The Model
So, a working objection may begin to look something like the following.
I state that the empty essence lemma may possibly be stated as the middle Ω \ φ ∧ φ of a disjunction:
So φ(y) can be stated as (Ω ∧ Ω \ φ) ∨ φ. As ∧ is an associative relation, and the relation here between possible worlds (compossible sets in the partition of Ω) transitive or possibly only symmetric, I can write the middle as Ω ∧ (Ω \ φ ∧ φ) or equivalent to the application of the empty essence lemma, so that I hold Ω ∧ φ(x).
Then, I find in the disjunction of (Ω ∧ Ω \ φ) ∨ φ the equivalence to (Ω ∨ φ) ∧ (Ω ∨ φ) by De Morgans law, leaving me only with the simple disjunction Ω ∨ φ, extant the left side only.
Note that if (Ω ∧ φ) ∨ (Ω \ φ) holds then φ holds but one side and on the left, else there is simple antinomy if φ holds on both.
But then, if P(Ω ∧ Ω \ φ) => P(N(Ω ∧ Ω \ φ)) and that P(N(φ(y)))=>∃x:NE(x) as by the definition of NE, then if φ(y) never holds in the middle x∧y, then as x ∨ φ(y) holds in the model and x is non-empty (there is no empty world), if the empty essence is to be considered to hold across the middle then N(φ(y)) => N(∃(x)) =>N(¬φ(x)) => "N(⊤)" in contradiction: for N(¬φ(x)) holds on that side of the disjunction within the disjoint and separate ultrafilter in all "x" with its own principal element of NE (as if always another individual). That results in the empty essence only ever applying to one side as predicated, for it does not hold upon the other in any such disjunction: i.e. it only holds on that side which is never extant. If the otherwise empty middle x ∧ y is ever found it is negated to inaction.
If φ holds across sides in the disjunction, the **necessarily completed ** middle (as with that empty essence) of all Ω (which can never be of that predicate φ) is always exemplified and positive for, colloquially, Pos(x)<=>¬Pos(¬x) and all of Ω on the left side of Ω ∨ φ is positive and exemplified into action as always chosen extant. Then, as every real (non-trivial) middle is simply found to be "not positive", it is positively unexemplified rather than ever contradictory or ever "false". The middle may exist, but by God it is never exemplified in GF(8).
The result that the octal's elements present (e_{p}, e_{r}, e_{s} etc.) are only exeunt one side simply makes the predicate φ to appear to hold over the middle as if also that one side extant and with x = y (when the two are incompossible): yet it is only ever found to be empty as a consequence of that free decidability of positive predicates, for any non-trivial middle is not found a predicate ever represented in or of the essence(s) of the ultrafilters, the predicate(s) made of NE. It is true the schema of the octal in the sets of the disjunction leaves the middle always "negative": the octal simply leaves the resulting middle completely empty or positively "inactive", and the singletons in GF(8) themselves are always found to be positive.
Then if the empty essence holds, it does not appear to do so in the middle (as empty both sides, the self difference cannot equal itself as empty without contradiction, all empty "worlds" are identical.) but must only be predicated to the side never extant
(as God has already "decided" the disjunction and the world has moved with Him), that side then found as impossible as ⊥.
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