I'm not certain of this, so don't take it to the bank, and it's actually something that has been actively researched recently...
...but I think the latest thinking is that a Big Red Or should be interpreted like parentheses around the preceding and succeeding clauses.
For example, on a dilemma, suppose you had the following text:
"To get past, must have ENGINEER + Leadership or Geology + Anthropology"
That's ambiguous, because you could read it two ways. It could mean:
Option A wrote:To get past, must have (ENGINEER + Leadership) or (Geology + Anthropology)"
or it could mean:
Option B wrote:"To get past, must have ENGINEER + (Leadership or Geology) + Anthropology."
Turning that little "or" into a Big Red Or resolves the ambiguity:
"To get past, must have ENGINEER + Leadership OR Geology + Anthropology"
You know immediately, without ever thinking about clauses or parentheses, that the parentheses go around the entire preceding clause ("ENGINEER + Leadership") and the entire succeeding clause ("Geology + Anthropology"). Option A is the correct reading.
On cards like The Final Frontier, there's no real ambiguity, but I think the Big Red Or serves the purely stylistic purpose you mention. Big Red Ors often make cards easier to read, even when they serve no syntactic purpose.
One interesting thing about Big Red Ors is that you can only use them for one "level" of clause per card. For example, if you use a Big Red Or to separate entire functions on a card, you can't use it to further separate sub-clauses within those functions. Imagine if Handshake read like this:
Plays once each turn (for free) if you have played no cards this turn and have fewer in hand than opponent; they must choose to discard three cards OR allow you to draw three. OR Plays to look at top five cards in any draw deck OR discard pile for ten seconds; rearrange as desired. OR Plays if you have at least three other cards in hand; discard entire hand and draw seven cards. Discard incident after any use.
Lunacy! You can't do it! It's unreadable! For these cases, you have to use normal "ors" in the subclauses or -- where appropriate -- actual parentheses. This happens on missions pretty routinely.
Yet you could absolutely make a card that said:
hypothetical incident wrote:Plays once each turn (for free) if you have played no cards this turn and have fewer in hand than opponent; they must choose to discard three cards OR allow you to draw three.
...it'd actually be a little weird if you DIDN'T do a Big Red Or there.
So the presence of Big Red Ors at one level of a card has a huge influence on how the Ors at other levels of the card can/should be presented.
Again, this is a question that is being actively researched, so consider this answer provisional. It includes a fair bit of my own personal speculation. (And feel free to present counter-examples that blow up this unofficial theory.)
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