// reference · NOT OR
↓ NOR_
NOR is the negation of disjunction: p ↓ q is equivalent to ¬(p ∨ q) and reads "neither p nor q". It is true only when p and q are both false. Like NAND, it is a universal operator: it alone suffices to build all of propositional logic.
Truth table: p ↓ q
| p | q | p ↓ q★ |
|---|---|---|
| T | T | F |
| T | F | F |
| F | T | F |
| F | F | T |
Classification: Contingency · 4 rows
Definition
p ↓ q ("p NOR q") is true only when neither proposition is true. Its name contracts NOT OR. The symbol ↓ is known as the Peirce arrow; it is also written ⊽. The calculator accepts both.
How to read it
p ↓ q is read "neither p nor q", "not p and not q", or "it is not the case that p or q". The natural-language "neither... nor..." is exactly NOR.
Example: "She neither studied nor worked" formalizes as p ↓ q, which is true only if she did neither.
When it is true
The table has four rows. With p = T and q = T, p ↓ q = F. With p = T and q = F, p ↓ q = F. With p = F and q = T, p ↓ q = F. With p = F and q = F, p ↓ q = T.
One true row and three false rows: it is a contingency and its table is that of p ∨ q with every value flipped.
Everyday example
An automatic sprinkler that only turns on when it is not raining and the garden has not been watered by hand today. If either happens, the sprinkler stays off.
In electronics, the NOR gate was the foundation of the Apollo Guidance Computer: its entire logic was built from a single type of three-input NOR chip.
Properties and equivalences
Definition: p ↓ q ≡ ¬(p ∨ q). By De Morgan: p ↓ q ≡ ¬p ∧ ¬q.
It is commutative but neither associative nor idempotent: p ↓ p ≡ ¬p. Like NAND, NOR can build every other operator: ¬p ≡ p ↓ p; p ∨ q ≡ (p ↓ q) ↓ (p ↓ q); p ∧ q ≡ (p ↓ p) ↓ (q ↓ q).
Common mistakes
Mixing up NOR and NAND when applying De Morgan. p ↓ q ≡ ¬p ∧ ¬q (both false), whereas p ⊼ q ≡ ¬p ∨ ¬q (at least one false).
Chaining NOR without parentheses and expecting associativity. (p ↓ q) ↓ r and p ↓ (q ↓ r) produce different tables.
Equivalent expressions
Try it yourself
Edit the expression in the calculator and watch how every step of the table changes.
Open in the calculator →Related laws and rules
Frequently asked questions
What does NOR mean? ▼
NOT OR: the negation of disjunction. p ↓ q reads "neither p nor q" and is true only when both propositions are false.
Is NOR also a universal operator? ▼
Yes. NOR can build NOT (p ↓ p), OR ((p ↓ q) ↓ (p ↓ q)) and AND ((p ↓ p) ↓ (q ↓ q)), hence any Boolean function.
What is the difference between NOR and NAND? ▼
NOR is true only when both are false (one true row). NAND is false only when both are true (three true rows). They are the negations of OR and AND respectively.