“Not Both” and “Neither Nor”

1. Not Both

Two common English phrases that can sometimes cause confusion when translating into formal logic are “not both” and “neither nor.” These two phrases have different meanings and thus are translated in different ways. 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 statement 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. Recall two tutorials ago, where Bob had to choose between going to class and playing video games. As you might remember, there were four scenarios, and only one was false: if Bob did both. That’s true here as well.

Carla has cake.	Carla has ice cream.	Carla will not have both cake and ice cream.
True	True	False
True	False	True
False	True	True
False	False	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 might have neither (as in the fourth scenario). So, the way to think about the “not both” statement is as the negation of a conjunction, since the conjunction is the only scenario that cannot be true. We can now run through our usual steps to translate.

Steps 1 & 2	Carla has cake. (C) Carla has ice cream. (I)
Step 3	(C ∧ I)
Step 4	¬(C ∧ I)

You might notice this is the same formal structure as our translated statement about Bob:

¬(C ∧ G)

That’s because any “not both” statement will translate as ¬(p ∧ q), regardless of the meaning of p and q. The parentheses are crucial, because without them, our translation would say something else entirely. For example, if we forgot the parentheses above, we would have:

¬C ∧ I

This would mean Carla will definitely not have cake and will definitely have ice cream, which is not the meaning of the original sentence.

watch

View a demonstration of this example.

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2. The Exclusive Or

When we learned the disjunction symbol, or wedge, we learned that it is an “inclusive or,” meaning that while at least one of the disjuncts are true, both can also be true. For example, using the constants above, we could write this disjunction:

C ∨ I

This states that Carla will have at least one of the two options, cake or ice cream, but can also decide to have both. We also covered how to write that Carla will definitely not have both but may have neither. What if we wanted to write that Carla will definitely have one, and exactly one, of the options? In other words, she will have either cake or ice cream, but not both.

Since we know how to write the “either or” sentence and the “not both” sentence, you might already see the answer. But let’s work through the steps. Here is our statement:

Carla will have cake or ice cream, but not both.

Steps 1 & 2 are to spot the atomic statements and assign constants.

Carla will have cake. (C)

Carla will have ice cream. (I)

Step 3 is to determine which atomic sentences are grouped and assign the logical operators that connect them. We already know these from our work above.

Carla will have cake or ice cream.	(C ∨ I)
Carla will not have both cake and ice cream.	¬(C ∧ I)

Step 4 is to connect the groups with the main operator. The word “but” is a logical equivalent of “and,” a conjunction.

Carla will have cake or ice cream but not both.

(C ∨ I) ∧ ¬(C ∧ I)

For step 5, instead of translating back into English to test our formulation, we will use a truth table. You can see that the column for “Carla has cake or ice cream but not both” is true when and only when the two preceding columns are true, “Carla has cake or ice cream,” and “Carla doesn’t have both cake and ice cream,” as we wrote above.

Carla has cake.	Carla has ice cream.	Carla has cake or ice cream.	Carla doesn’t have both cake and ice cream.	Carla has cake or ice cream, but not both.
True	True	True	False	False
True	False	True	True	True
False	True	True	True	True
False	False	False	True	False

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3. Neither Nor

Let’s move on to one more sentence type: “neither nor.”

Carla will have neither cake nor ice cream.

If we walk through the steps, we will arrive at a correct translation.

Step 1 & 2	Carla will have cake. (C) Carla will have ice cream. (I)
Step 3	¬C ¬I
Step 4	¬C ∧ ¬I

However, there is another way to write this that is logically consistent. Note that we have a disjunction here (or), not a conjunction (and).

¬(C ∨ I)

If you translate it back into English, you can see it is the same. “It is not the case that Carla will have either cake or ice cream.” We can understand this as saying that the disjunction isn’t true; in order for that to be the case, both disjuncts must be false. We can also use a truth table to show they are consistent across the possibilities.

Carla will have cake.	Carla will have Ice cream.	Carla will have either cake or ice cream.	Carla will have cake and ice cream.	Carla will not have cake or ice cream.	Carla will not have cake and will not have ice cream.
True	True	True	True	False	False
True	False	True	False	False	False
False	True	True	False	False	False
False	False	False	False	True	True

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4. Logical Equivalence

There is an important result in this table. Notice that ¬(C ∨ I) and ¬C ∧ ¬I have the same truth values. They reveal a logical law: we can push a negation through a conjunction or disjunction and “flip the sign” to get a statement with the same truth values. This means they are logically equivalent.

Let’s work through this more slowly. Take ¬(C ∨ I): first, we push the negation through the parentheses and assign it to each of the disjuncts, (¬C ∨ ¬I). Then, we “flip the sign,” that is, we flip the disjunction upright, making it a conjunction. The result is a sentence equivalent to ¬(C ∨ I): ¬C ∧ ¬I. (We no longer need the parentheses since the entire statement is within them.)

This rule also works when the sign is a conjunction. If we start with ¬(C ∧ I), we can follow the same steps to end up with a disjunction with the same truth value assignments. First, we push the negation through the parentheses and assign it to each of the conjuncts: (¬C ∧ ¬I). Then, we flip the sign, changing the conjunction to a disjunction: ¬C ∨ ¬I. (As before, we no longer need the parentheses).

Finally, the rule works in reverse. By this, we mean that we can push the negation from inside to outside the parentheses and flip the sign as follows. Start with the sentence: ¬C ∧ ¬I. First, we add the implied parentheses: (¬C ∧ ¬I). Second, we push the negation outside of the parentheses: ¬(C ∧ I). Last, we flip the sign, changing the conjunction to a disjunction: ¬(C ∨ I). This rule is formally called DeMorgan’s Law.

try it

Repeat this process with ¬C ∨ ¬I.

terms to know

Logically Equivalent

Two ways of writing a statement that have the same truth values in all possible circumstances.

DeMorgan’s Law

The negation of a disjunctive sentence is the conjunction of the negated sentences and the negation of a conjunctive sentence is the disjunction of the negated sentences.