5.3 Advanced Boolean Translations
There are a few tricky translations into BOOL to watch out for.
Okay, that wasn't too difficult. But a translation students often struggle with this neither/nor sentences, and one of the translations you just did can help us understand how neither/nor works.
That's because, even though it looks different, one of those two sentences says the same thing as "Neither Pia nor Quinn is guilty."
Neither/nor can be one of the trickiest Boolean translations, but you've already figured it one way to do it.
"Neither X nor Y" means not X and not Y. That might feel counter intuitive because "nor" has "or" in it, but we translated it with a wide-scope & not v.
Another reason why people find neither/nor hard is that there's a second equally good way of translating it. See if you can figure it out.
That means you can translate neither/nor two ways. One way uses & and the other uses v, but notice that the scoping is different.
Both are good because they are equivalent: they really come to the same thing. In the next chapter we'll see how BOOL can prove that they are equivalent.
Before moving on, we should apply your new knowledge to a harder example.
A second difficult translation is this: "Pia is guilty and Quinn is guilty or Raquel is guilty."
But now we need to clarify a fact about the rule to group from left to right.
When you learned that rule, we just considered sentences with one type of connection: all conjunctions or all disjunctions.
When the connectives are the same type, like "Pia and Quinn and Raquel are guilty", we can make up an arbitrary rule because (P&Q)&R and P&(Q&R) are equivalent. They come to the same thing: whenever one is true, so is the other.
It's like the associativity property of addition: (1+2)+3 = 1+(2+3)
But the sentence we are translating in this section has mixed connectives: it has & and v. Now consider this:
Since these two sentences are different, we cannot make an arbitrary choice to group from left to right.
That's because the author of the sentence could have been saying two different things, and we need to figure out what they said before we can translate it.
Consequently, we will amend our old rule: when English doesn't group connectives of the same type, group from left to right.
And we will also need a new rule: when you have mixed binary connectives (& and v), and English doesn't group them, you must give both translations until you know which is correct.
The problem is that the English sentence is ambiguous: it can mean two different things.
We want you to be able to recognize ambiguity when you encounter it, and be able to identify all the different translations.