Hey Tommy,
Good questions. Examples 1, 2, and 3 at the bottom of page 207 illustrate possibilities consistent with the H
G symbol. In other words, none of these three scenarios violates H
G.
No, you do not need to write out these possibilities. When symbolizing, just recognize that the only scenario that contradicts H
G is that in which both H and G are selected together—this is impossible, a contradiction.
With respect to your latter question: the double arrow, a "material equivalence," you can think of as two conditionals expressed simultaneously: as described in the LRB both sides are sufficient AND necessary for the other. A implies B, and B implies A. The syntax "if and only if" and its equivalents make explicit these coincident relationships.
I sometimes use the example of dictionary definitions to clarify this concept:
- cat – noun — A member of a domesticated subspecies of feline animal, commonly kept as a house pet.
If you have a cat, then you have a member of a domesticated subspecies of feline animal, commonly kept as a house pet.
AND
If you have a member of a domesticated subspecies of feline animal, commonly kept as a house pet, then you have a cat.
AND
If you don't have a member of a domesticated subspecies of feline animal, commonly kept as a house pet, then you don't have a cat.
AND
If you don't have a cat, then you don't have a member of a domesticated subspecies of feline animal, commonly kept as a house pet.
The list of ways to express "if and only if" is a terrific resource, but make sure you are solid on this underlying concept.
