Happy to try and help —
One thing is that I’m not 100% sure I understand your notations correctly, so if I missed something you were saying please let me know —
A lot of this will be a repeat of what’s in the Trainer but just to have it all here —
1) I believe that the easiest and most accurate way to think of conditional statements is in terms of guarantees (specifically, unless we are dealing w/biconditionals, one way guarantees).
If A then B means A guarantees B. (And it does not mean that B guarantees A.)
2) This guarantee will be stated either in a positive or negative way —
So, imagine that, in order to get into an amusement park, you have to have a ticket.
Get in -> have ticket.
You can either say —
“If you want to get in, then you must have a ticket.” (and the guarantee here is that if one gets in they have a ticket).
Or you can say —
“You can’t get in unless you have a ticket.” (and again, the guarantee here is that if one gets in they have a ticket).
3) When you are stuck about which way the guarantee goes, what I recommend is mentally walking through all options and seeing which one is actually the one relationship you know must absolutely be guaranteed and correct.
So, for example, if you weren’t sure if the statement “You can’t get in unless you have a ticket” means
“Have ticket, guaranteed to get in” vs
“Got in, guaranteed to have had a ticket”
You can think to yourself — “Per this statement, is there any way you can have a ticket and not be able to get in?” — and maybe some come to mind — maybe you aren’t wearing the right clothes, or you are drunk, etc.
And you think to yourself – “Per this statement, is there any way you can get in without a ticket?” and there isn’t — so, you know the second interpretation is the correct one.
4) If and when all the above doesn’t work (and again, this is just my suggestion and I realize other companies teach this differently), that’s when, if you have a strong sense of sufficiency/necessary indicators and such, you go to those rules. However, I think what you might very well find is that if you practice 1- 3 enough you will need to do this less and less.
Okay — so, applying all this to the specific phrases you asked about —
“You can’t get into the park without a ticket.”
What’s the strongest guarantee you can take out of that? — that if you got in, you must have had a ticket —
“Got into the park -> Had a ticket.”
“You can never get into the park without a ticket.”
What’s the strongest guarantee you can take out of that?
Does it mean that if you have a ticket you will definitely get in? No — maybe there are other problems.
So again, here, the certainty is that if you got in, you must have had a ticket.
“Got into the park -> Had a ticket.”
Either/or is first and foremost an indication of just that — information about an “or” situation, and, secondarily, you can often (especially in Logic Games) make conditional inferences based off of this “or” information.
To illustrate with an example —
If an LR stimulus gives you the statement: “Tom either has a ticket or a season pass.” I wouldn’t think of it conditionally, but, rather, I’d focus on the fact that he has one, the other, or both.
If it gives us “To get into the park, one must have a ticket or a season pass.” then the “or” is not directly related to the main conditional, but, rather, is just part of the conclusion:
Got in -> Have a ticket or season pass (one, other, or both).
If it gives you “To get into the park, one must have a ticket or a season pass. Tom got into the park.” — my main thought is that he must have a ticket or season pass (or both) — the secondary conditional inference is that if he doesn’t have a ticket, he must have had a season pass, and vice versa.
No ticket -> Yes season pass
No season pass -> Yes ticket
Again, or statements tend to yield more significant conditional inferences on Logic Games (especially when they are included in games that are designed to test your ability to link conditional information together).
Most commonly, they will appear in “In/Out” games and the three “variations” of rules you need to be comfortable with (and be able to distinguish from one another) are:
“Either X or Y is in.” (this yields a conditional inference whenever we also know that either X is out or Y is out: -X -> Y and -Y -> X.)
“Either X or Y is out.” (this yields a conditional inference whenever we also know that either X is in or Y is in: Y -> – X, X -> -Y)
“Either X or Y, but not both, is in.” (this yields conditional inferences whenever we know either is in or out: Y -> – X, X -> – Y, -Y -> X, -X -> Y).
Wow the length of that got away from me — too much coffee!
And again, I understand that what I recommend above isn’t how everyone teaches this stuff, and certainly isn’t the only way to think about it, but it’s what I think is most helpful —
Feel free to use what you want and ignore the rest, and let me know if you have any follow-up —
Take care —