Undergrad Need help solving this Existence Algorithm for truth

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The discussion revolves around a complex equation involving existential quantifiers and logical operations. The user is uncertain about the solvability of the expression (x ¬ | ∃x) and its implications regarding the existence of x independent of its reference. The equation presented is (x ¬ | ∃x) = ∅ ⊕ {∅}) ⊕ ∅. The user expresses confusion about whether there are scenarios where x can exist independently. The thread is currently closed for mentor review.
ollieha
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Need help solving algorithm for truth
I have an equation that I need some serious help with. I’m using a “not such that”, and I don’t know if the critical component (x ¬ | ∃x) is solvable!

Well here it is:

(x ¬ | ∃x) = ∅ ⊕ {∅}) ⊕ ∅

So if x exists independently from the reference of x, the first bit is true, but is there ever a time when that is the case?

So confused-
Oliver
 
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Thread is closed temporarily for Mentor review...
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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