Hi agramza ,
I think you need to change the OR highlighted below to "AND".
Then you'll need to add conditions within this set to let them pass if no Other selection is made, and Other text entry is empty.

Thank you so much for the tip. I was able to figure it out with your help!

Hi again Matthew,
I think I spoke too soon. Although, my logic has now allowed someone to skip choosing an importance response if they don't write-in an "other" in the write-in box AND forces them to choose an importance response if they DO write something in the "other" box, it does not trigger a validation if someone chooses an importance choice if they don't write something in the "other" box.
Do you know how to do this? Basically, respondents shouldn't be choosing an importance level for a blank fill-in response.
Here is my amended "if" statement.

Okay, it seems like it only triggers a valid response if there is no text entry in the Other category and not at all important is chosen. If there is no text entry and any of the other 3 importance categories is chosen, it allows people to move forward.
Any thoughts on that?

I think you need to move the Other (text entry) Is Not Empty line in the condition to the end, and use "AND" before it. The three previous lines should have "OR" after you move the first line.

Thanks. I tried what you suggest and I'm still able to move through if I click a importance response without writing anything in the text box. Any other suggestions?

Assuming that the unseen logic above is OK, try inserting "AND Other (text entry) Is Not Empty" in between each OR condition, as shown below.

Hi Again,
After trying it that way, I still couldn't get it to work. Therefore, I decided to get simpler and just pull the other: fill in response question into a new matrix-type question so I could just force responses to the fixed reasons.
I also thought this would be easier to create a custom validation logic statement to allow people to pass if they leave the box empty and don't rate the importance, BUT make sure they write a response in the text box if they rate it as important (or vice versa, force a rating if they write something in the box).
Anyways, I used two logic statements that we discussed extensively and I can pass if I don't write anything in the box or choose a importance rating. However, I can choose an importance rating but not write anything in the box.
Am I missing something simple now?

Sorry I can't seem to get closer to a solution. In the first example, though, you need to change the highlighted "ORs" to "AND".
If that still doesn't fix it, break each "Is Selected" AND "Is not empty" pair into four separate conditions.

I just wanted to follow back up to say:
1) Thank you for your help!
2) I FINALLY got it to work after working with a customer support person.
Below is the final logic.

Oh good, glad you were able to get more qualified support! That looks like my final suggestion. Sorry I didn't arrive at it earlier. These are always challenging to look at and evaluate; I usually have to do them hands on to get them just right.