"Not both" and "neither nor"

Two common English phrases that can sometimes cause confusion are "not both" and "neither nor". These two phrases have different meanings and thus are translated with different symbolic logic sentences. Let's look at an example of each.

Carla will not have both cake and ice cream.

Carla will have neither cake nor ice cream.

The first sentence uses the phrase "not both" and the second "neither nor". One way of figuring out what a sentence means (and thus how to translate it) is by asking the question: What scenarios does this sentence rule out? Let's apply this to the "not both" statement (which we first saw back in the beginning of section 2.4). There are four possible scenarios, and the statement would be true in every one except the first scenario: 


Carla has cake Carla has ice cream False
Carla has cake Carla does not have ice cream True
Carla does not have cake Carla has ice cream True
Carla does not have cake Carla does not have ice cream True

To say that Carla will not have both cake and ice cream allows that she can have one or the other (just not both). It also allows that she can have neither (as in the fourth scenario). So the way to think about the "not both" locution is as a negation of a conjunction, since the conjunction is the only scenario that cannot be true if the statement is true. If we use the constant "C" to represent the atomic sentence, "Carla has cake," and "I" to represent "Carla has ice cream," then the resulting symbolic translation would be:

~(C ⋅ I)

Thus, in general, statements of the form "not both p and q" will be translated as the negation of a conjunction: ~(p ⋅ q) Note that the main operator of the statement is the negation. The negation applies to everything inside the parentheses - i.e., to the conjunction. This is very different from the following sentence (without parentheses):

~p ⋅ q

The main operator of this statement is the conjunction and the left conjunct of the conjunction is a negation. In contrast with the "not both" form, this statement asserts that p is not true, while q is true. For example, using our previous example of Carla and the cake, the sentence

~C ⋅ I

would assert that Carla will not have cake and will have ice cream. This is a very different statement from ~(C ⋅ I) which, as we have seen, allows the possibility that Carla will have cake but not ice cream. Thus, again we see the importance of parentheses in our symbolic language.

Earlier (in section 2.3) we made the distinction between what I called an "exclusive or" and an "inclusive or" and I claimed that although we interpret the wedge (v) as an inclusive or, we can represent the exclusive or symbolically as well. Since we now know how to translate the "not both," I can show you how to translate a statement that contains an exclusive or. Recall our example:

Bob placed either first or second in the race.

As we saw, this disjunction contains the two disjuncts, "Bob placed first in the race" (F) and "Bob placed second in the race" (S). Using the wedge, we get:

F v S

However, since the wedge is interpreted as an inclusive or, this statement would allow that Bob got both first and second in the race, which is not possible. So we need to be able to say that although Bob placed either first or second, he did not place both first and second. But that is just the "not both" locution. So, to be absolutely clear, we are asserting two things:

Bob placed either first or second.

and

Bob did not place both first and second.

We have already seen that the first sentence is translated: "F v S". The second sentence is simply a "not both F and S" statement:

~(F ⋅ S)

Now all we have to do is conjoin the two sentences using the dot:

(F v S) ⋅ ~(F ⋅ S)

That is the correct translation of an exclusive or. Notice that when conjoining the "F v S" to the "~(F ⋅ S)" I needed to put parentheses around the "F v S" to show that it was grouped together. Thus, it would have been incorrect to write:

F v S ⋅ ~(F ⋅ S)

since that is not a well-formed formula. The problem, as before, is that this sentence is ambiguous between two sentences that have different meanings:

F v (S ⋅ ~(F ⋅ S))

(F v S) ⋅ ~(F ⋅ S)

While both of these sentences are well-formed, only the latter is the correct translation of the exclusive or.

Let's move on to the English locution "neither...nor" as in:

Carla will eat neither cake nor ice cream.

This statement might be true if, for example, Carla was on a diet (and was sticking to her diet). Using the same method I introduced earlier, we can ask under what conditions the statement would be true or false. As before, there are only four possibilities, which I represent symbolically this time:

C I False
C ~I False
~C I False
 ~C  ~I  True

There is only one circumstance in which this statement is true and that is the one in which it is false that Carla eats cake and false that Carla eats ice cream. That should be obvious from the meaning of the "neither nor" locution. Thus, the correct translation of a "neither nor" statement is as a conjunction of two negations:

~C ⋅ ~I

The main operator of this statement is the dot, which is conjoining the ~C with the ~I. Thus, the form of any "neither nor" statement can always be translated as a conjunction of two negations:

~p ⋅ ~q

As we will see in a later section (where we will prove it), this statement is also equivalent to a negation of a disjunction:

~(p v q)

Thus, the English locution "neither nor" can also be translated using this statement form.