The-Voice-of-Truth

if one is to assert that some property P(x) is possibly exemplified, then is it rational to say that P could or could not obtain? i suppose i'd characterise this as ⋄P(x) → (∃xP(x) ∨ ¬∃xP(x))

my intutition would seem to suggest that this is the case, and, if indeed it is, then would not the inverse ⋄¬P(x) → (¬∃xP(x) ∨ ∃xP(x)) also be true?

If both instances are true, then wouldn't antecedents ⋄P(x) and ⋄¬P(x) be equivalent, as the order of the terms in the consequent doesn't partcularly matter, since (∃xP(x) ∨ ¬∃xP(x)) ≡ (¬∃xP(x) ∨ ∃xP(x))?

still a bit new to this so any insight is valued

∃ ¬ ◻ ⋄

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