Double Arrow Question

[Emma]

Hi! I have the 2023 LG Bible book and I'm having trouble understanding the double arrow concept discussed on pgs. 292-293. We have the statement "A will be selected if and only if B is selected." The argument the book makes is that this is split into two conditional statements. 1. B if A and 2. A if B. this creates a double arrow/block of AB in a grouping game.

I am confused on why B necessitates A. I understand that if there is A there must be B. However, I am reading the claim to say 'If there is a situation where A, there must be B also.' Meaning, if A was to be selected, B must also be there. Why couldn't there be a situation where B is present but A was not selected?

My first guess that the operative words are 'will be'. Does 'will' always directly imply a closed absolute instead of a open 'can be'? My second is that I'm confusing the difference between A if B and A only if B. How does only change the meaning? I'm reading those rules as restatements of the same.

Thank you!

[Dave Killoran]

Hi Emma,

No there's no situation there where B will be present without A--they are ALWAYS together.

For clarity, the B A part of the double=arrow relationship comes from the bolded portion of this statement: "A will be selected if and only if B is selected." That is the same "if B is selected A is selected," which yields that arrow.

Does that help? Please let me know. Thanks!