What is the difference between “not both” and “neither”?
Quick answer
“Not both A and B” means A and B are not true at the same time; one of them may still be true. “Neither A nor B” means A is false and B is false. For example, if A is “I chose tea” and B is “I chose coffee,” choosing tea alone satisfies “not both,” but it does not satisfy “neither.” [S1]
The question, the details, the next step.
I keep mixing up “not both” and “neither.” What’s the simplest way to tell them apart?
Ask whether the statement rules out the two things happening together, or rules out each thing separately.
“Not both A and B” rules out only the case where A and B are both true. A alone or B alone is still allowed. “Neither A nor B” rules out A and also rules out B, so both must be false. In logic, “not both” is the negation of a conjunction; “neither” says both individual conditions are false. [S1]
For a familiar example, let A mean “I chose tea” and B mean “I chose coffee.” “I did not choose both” allows tea only, coffee only, or neither. “I chose neither” allows just one of those possibilities: no tea and no coffee.
Can you show every possible case? I want to see exactly where they differ.
Here are all four combinations. “True” means the item was chosen; “False” means it wasn’t.
- A true, B true: “not both” is false; “neither” is false.
- A true, B false: “not both” is true; “neither” is false.
- A false, B true: “not both” is true; “neither” is false.
- A false, B false: “not both” is true; “neither” is true.
The middle two cases are the key difference: exactly one item is true. Those cases make “not both” true, but “neither” false. In short, “not both” excludes the both-true case, whereas “neither” is true only in the both-false case. [S1]
How do I use this when a sentence is worded less clearly, like “we don’t serve both breakfast and lunch”?
First name the two conditions plainly: A = “breakfast is served” and B = “lunch is served.” In the logical reading, “we don’t serve both breakfast and lunch” denies that A and B are both true. It still allows breakfast only, lunch only, or neither.
That differs from “we serve neither breakfast nor lunch,” which says breakfast is not served and lunch is not served. If the wording matters for a real decision, don’t assume the intended meaning from a compressed phrase alone; ask the speaker whether one of the two services might still be available. The logic shows what each reading means, but it cannot establish what a particular speaker intended.
Can I use a quick rule to check my answer on a homework question?
Yes. Translate the two options into A and B, then test the one-item-only cases. If either “A only” or “B only” makes the statement true, the statement can be “not both.” “Neither” is true only when both A and B are false. [S1]
A copyable check:
- A = [first condition]
- B = [second condition]
- “Not both”: Is it false when A and B are both true? If yes, check whether either single-true case is allowed.
- “Neither”: Are A and B each false?
For example, with A = “I chose tea” and B = “I chose coffee,” choosing coffee only makes “not both” true, but makes “neither” false. If a problem uses different wording or defines its terms specially, follow those definitions; this comparison assumes the ordinary logical meanings of “not both” and “neither.” [S1]
A practical checklist
Label the two conditions A and B.
For “not both,” cross out only the case where A and B are both true.
For “neither,” keep only the case where A and B are both false.
Test the A-only and B-only cases; they distinguish the statements.
If interpreting someone’s wording, ask what they intended rather than guessing.
A few more useful answers
Is “not both” the same as “either one or the other”?
Not necessarily. “Not both” allows neither item as well as either one alone. It simply rules out both being true together. [S1]
Can “neither A nor B” be true if one item is true?
No. “Neither” requires A to be false and B to be false. [S1]
Does this explain what a speaker meant by an unclear sentence?
No. It explains the logical readings, not a particular speaker’s intention. Ask the speaker if the practical meaning is important.
Sources
References used for this guide.
- [S1] OpenStax Contemporary Mathematics: Compound Statements openstax.org · Checked