When glossing a logical proposition with English words, we are not really using the English language system; it's just a way to represent the unambiguous mathematical concepts in a mnemonic and easy-to-pronounce manner. For this reason, mathematicians and logicians like to present specialized definitions of the words they use. That is what is being done in this passage. You'll only confuse yourself if you try to use the ordinary meaning of English words in this context.
I find this a rather poor definition for an introductory text; I'm not a mathematician, so it may in fact not be unambiguous, but I find myself unsure if the author means for this to represent exclusive disjunction or inclusive disjunction. It seems to me to technically be a definition of inclusive disjunction. Usually in logic, the phrase "Either A or B" is taken to mean that exactly one of the following 3 propositions is true:
- A is true and B is true
- A is true and B is false
- A is false and B is true
However, in ordinary English, "A, unless B" seems to additionally imply "if B is true, A is not true". Clearly the author means instead for "A, unless B" to represent "if B is true, A may or may not be true", but that is an extremely unfortunate definition of the word "unless" in my opinion.
I'm afraid I can't say much about the linguistic aspect of this question.
Evidence from the 2002 edition:
In the 2002 edition, page 56, accessed through Google Books, Gensler uses the notation "D ∨ B" as equivalent to "D unless B". The ∨ sign is used for inclusive disjunction, so that does indeed appear to be what he means with this supposed translation of logic into ordinary English words. In other words, by "A unless B" he really means "Either A, or B, or both".