“Neither A nor B” is NOT A AND NOT B
“Neither A nor B” means both fail: NOT A AND NOT B, which is NOR(A, B). The common slip, NOT A OR NOT B, is “not both”, and the two differ on the rows where exactly one of A and B holds.
The table compares NOT A AND NOT B with NOT A OR NOT B. The right formula is TRUE on one row only, A = FALSE, B = FALSE. The slip is TRUE on three rows; on A = FALSE, B = TRUE and on A = TRUE, B = FALSE it admits a case where one of the two holds, which “neither” forbids. The equivalence check reports A = FALSE, B = TRUE as the counterexample.
By De Morgan, “neither A nor B” is NOT (A OR B), and pushing the negation inside flips OR to AND. The slip keeps OR. It is the mirror image of the De Morgan slip on AND, and it has the same cause: the operator must change when the negation moves, and people forget.
In rules the phrase appears as exclusions: “no refund if the item is neither damaged nor late”, “the account is eligible if it has neither a block nor an overdue balance”. The NOR gate page shows the same function in hardware, and the pitfall on “not over AND” shows the other half of De Morgan.
right := NOT A AND NOT B · outputwrong := NOT A OR NOT B · output Truth table
They differ on this row: #1, #2
| # | A | B | right | wrong |
|---|---|---|---|---|
| 0 | FALSE | FALSE | TRUE | TRUE |
| 1 | FALSE | TRUE | FALSE | TRUE |
| 2 | TRUE | FALSE | FALSE | TRUE |
| 3 | TRUE | TRUE | FALSE | FALSE |
Plain-language reading
Beginnerright is true exactly when A is false and B is false.
In every other case, right is false.
About the operators:
- AND is true only when all of its parts are true.
- NOT reverses a value: true becomes false and false becomes true.
Intermediateright is true exactly when A is false and B is false.
In every other case, right is false.
Advancedright is true exactly when A is false and B is false [conjunction ∧].
In every other case, right is false.
Formal notation: ¬A ∧ ¬B
The common slip
Translating “neither A nor B” as NOT A OR NOT B. That formula is satisfied by a case where only one of them fails, which is not “neither”. Only NOT A AND NOT B has the single TRUE row the sentence describes.
Questions
Is “neither nor” the same as NOR?
Yes. NOR(A, B) is NOT (A OR B), which by De Morgan is NOT A AND NOT B; the three spellings have the same table, with one TRUE row.
How do I say “not both”?
NOT (A AND B), which is NOT A OR NOT B, the NAND gate: TRUE unless both hold. It has three TRUE rows, not one.
Related
Tools
- Logical equivalence checker: two expressions, exact verdict, counterexample
- Truth table generator: any Boolean expression, up to 12 variables, with intermediate columns
Practice
See also
- De Morgan's laws: NOT (A AND B) = NOT A OR NOT B, with proof and examples
- NOR gate: truth table for 2, 3 and 4 inputs, universal gate
- NAND gate: truth table for 2, 3 and 4 inputs, universal gate
- NOT (A AND B) = NOT A OR NOT B, not NOT A AND NOT B
- NOR elimination: NOR(A, B) = NOT (A OR B)
- Lesson 6: De Morgan’s laws (intermediate, 12 min)
Pitfalls
Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build . Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.