Question Regarding Possibility

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

You've misapplied two logical rules.

First, a conditional statement is not logically equivalent to its inverse. 

Consider the claim, "It rained, so my grass is wet." Suppose it is true. The inverse is, "It did not rain, so my grass is not wet." This is not necessarily true, since the sprinkler could have gone off. You may be looking for the contrapositive.

Second, you are not correctly "distributing" the negation throughout the OR statement. That is,

~(A OR B) == ~A AND ~B. 

To see why this is true, let A be "it is raining" and B be "it is summer". If we have (A OR B) we are claiming that it is either raining or it is summer (or both). If we have ~(A OR B) we are claiming that is not the case that it is raining or it is summer. That is, we are claiming that it is not raining AND it is not summer. 

To summarize:

A → B is not equivalent to ~A → ~B. [inverse]

A → B is equivalent to ~B → ~A. [contrapositive]

~(A OR B) == (~A AND ~B) [De Morgan's Law]

To your point, the following would be a correct logical sequence:

P(x) → (∃xP(x) ∨ ¬∃xP(x))

¬(∃xP(x) ∨ ¬∃xP(x)) → ¬P(x)

(¬∃xP(x) ∧ ∃xP(x)) → ¬P(x)

The final statement is vacuously true, since its antecedent is apparently always false. (And any conditional with a false antecedent is defined to be true.)

thanks, i appreciate the feedback! although, a couple follow-up questions: When you say,

P(x) → (∃xP(x) ∨ ¬∃xP(x))

do you mean to exclude the possibility quantifier? As well, would adding it change the dynamics of the conditionals at all?

Obviously it is rational - it follows from the natural meaning of the words.

You appear to have invented a symbol ⋄  for 'possibility'. You may not have noticed (∃xP(x) ∨ ¬∃xP(x)) is (A ∨ ¬A) and so reduces to simply 'true',

so ⋄Q = true, whatever Q is!

The ⋄ operator has the effect of making anything true, which is ok, but not very useful!

It's like putting 'possibly' in a sentence; it becomes true. 'It is possible i will win the lottery', 'it is possible the moon is made of cheese'.

As ⋄Q is always true,  ⋄Q=⋄R(=true) for any Q and R, so you are right that ⋄P(x) and ⋄¬P(x) are equivalent (both are true).  That makes sense as 'it is possible the moon is made of cheese' is equivalent to 'it is possible the moon is not made of cheese'.

(¬⋄)P → ¬∃P ?

(¬⋄)P → ¬∃P

"If P is not possible then no P exists".

"not possible" aka inviolate ergo, a finite set of limited freedoms.

Philosopy considers what is possible and what is not possible.

Science considers what is possible and derives from that info, what is not possible.

I was just playing with the ⋄ notation.