// reference · NOT
¬ Negation_
Negation is the only logical operator that acts on a single proposition: it takes a truth value and flips it. When p is true, ¬p is false; when p is false, ¬p is true. It is the building block from which every other connective can be defined.
Truth table: ¬p
| p | ¬p★ |
|---|---|
| T | F |
| F | T |
Classification: Contingency · 2 rows
Definition
The negation of a proposition p, written ¬p, is the proposition that is true exactly when p is false. Formally it is a unary truth function: it takes one value (T or F) and returns the opposite.
Textbooks and calculators use several symbols for the same thing: ¬p (the standard notation), ~p (very common in older logic texts), !p (borrowed from programming languages), and p' (Boolean algebra, where the prime is written after the variable). Our calculator accepts all four and treats them identically.
How to read it
¬p is read "not p", "it is not the case that p", or "p is false". If p stands for "it is raining", then ¬p is "it is not raining".
When negation applies to a compound expression you must use parentheses: ¬(p ∧ q) negates the whole conjunction, whereas ¬p ∧ q negates only p. The two have completely different truth tables, so always check the scope of a negation.
When it is true
The truth table of ¬p has only two rows because there is a single variable. If p = T, then ¬p = F. If p = F, then ¬p = T. There are no other cases: negation always produces the opposite of its input.
Since it has one true row and one false row, ¬p is a contingency — neither a tautology nor a contradiction. Its value depends entirely on p.
Everyday example
Let p be "the exam is on Monday". Its negation ¬p is "the exam is not on Monday". Notice that ¬p does not say when the exam is; it only rules out Monday. This is a frequent mistake: negating a statement does not assert a specific opposite, it just denies the original.
Another example: if p is "everyone passed", then ¬p is "not everyone passed", i.e. "at least one person failed". It is not the same as "nobody passed".
Properties and equivalences
Double negation: ¬¬p ≡ p. Negating twice returns the original value, which lets you simplify expressions by cancelling pairs of negations.
Law of excluded middle: p ∨ ¬p is always true (a tautology). Law of non-contradiction: p ∧ ¬p is always false (a contradiction). These two laws underpin proof by contradiction.
Negation can also be expressed with other operators: ¬p ≡ p ⊼ p (NAND of p with itself) and ¬p ≡ p ↓ p (NOR of p with itself). This is why NAND and NOR are called universal gates in digital electronics.
Common mistakes
Getting the scope wrong: ¬p ∨ q is not the same as ¬(p ∨ q). The first negates only p; the second negates the entire disjunction. In the calculator, use parentheses whenever a negation should cover more than one variable.
Forgetting De Morgan's laws: ¬(p ∧ q) is not ¬p ∧ ¬q but ¬p ∨ ¬q. When you push a negation inside, conjunction becomes disjunction and vice versa.
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 is the difference between ¬p, ~p and !p? ▼
None in meaning: all three denote the negation of p. ¬ is the standard symbol in mathematical logic, ~ is common in textbooks, and ! comes from programming. The calculator accepts all of them.
What happens if I negate twice? ▼
You get the original proposition back: ¬¬p ≡ p. This law of double negation holds in classical logic and lets you simplify expressions by removing pairs of negations.
How do I negate a compound expression like p ∧ q? ▼
Put the negation in front and wrap the expression in parentheses: ¬(p ∧ q). By De Morgan's laws this is equivalent to ¬p ∨ ¬q.
Is negation a tautology or a contradiction? ▼
Neither. ¬p is a contingency because it is true in one row (when p is false) and false in the other.