We will conclude this discussion with several examples illustrating the issues we have discussed. First, a case that is entirely straightforward.
It isn’t warm out
¬ it’s warm out
¬ W
not W
[W: it’s warm out]
The next shows that uncovering even a simple form can require some thought and a paraphrase.
No one saw anyone enter the building
¬ someone saw someone enter the building
¬ S
not S
[S: someone saw someone enter the building]
Care is needed in distinguishing not-both forms from not-and-not forms. Everyone understands the distinction quite well intuitively, but it is easy to get tripped up when you are first learning to make this understanding explicit. Compare the following.
|
Britain and France won’t both vote
¬ Britain and France will both vote ¬ (Britain will vote ∧ France will vote)
¬ (B ∧ F)
|
Britain and France both won’t vote
Britain won’t vote ∧ France won’t vote ¬ Britain will vote ∧ ¬ France will vote
¬ B ∧ ¬ F
|
[B: Britain will vote; F: France will vote]
The negation of a conjunction is not the same as a conjunction of negations. The second form is also the way we would analyze Neither Britain nor France will vote.
The scope of negation is one respect in which English sentences are often ambiguous, and it is not hard to find examples that people will interpret differently. For example, you may find it possible to understand the second sentence above as a denial of Britain and France will both vote—i.e., as equivalent to the first. The first seems unambiguous, but other sentences in which not appears before both are less clear. For example, it might be possible to understand Tom didn’t like both the service and the price to say that he liked neither (if you have trouble understanding it to say anything but that, try reading it with an emphasis on both).
Finally, here is a somewhat longer example.
¬ ((T ∧ P) ∧ ¬ (¬ G ∧ ¬ V))
not both T and P and not both not G and not V
[G: Al got to the game; P: Al got to the party; T: Al got to the meeting; V: Al got to the movie]
The final step of analyzing X missed Y as contradictory to X got to Y is not crucial at this point in the course. While it is important to exhibit as much logical structure as possible, we end up with four logically independent sentences whether carry out the final step or not. However, when we go on to press analyses below the level of sentences, this sort of step will be of value since it leads us to four components that differ only in the object of the preposition to and therefore can be analyzed in a way that re-uses vocabulary.