// reference · NIF
⇏ Negated Conditional_
The negated conditional, written p ⇏ q, asserts that "p does not imply q": p is true and yet q is false. It is the one situation where the conditional p ⇒ q fails, so p ⇏ q ≡ ¬(p ⇒ q) ≡ p ∧ ¬q. Understanding it is understanding what it takes to refute an implication.
Truth table: p ⇏ q
| p | q | p ⇏ q★ |
|---|---|---|
| T | T | F |
| T | F | T |
| F | T | F |
| F | F | F |
Classification: Contingency · 4 rows
Definition
p ⇏ q is true when the antecedent p is true and the consequent q is false, and false in the other three cases. It is exactly the table of p ⇒ q with every value flipped.
Most textbooks do not list it as a standalone operator; they write ¬(p ⇒ q). The calculator offers it as its own symbol ⇏ so the step shows up in a single column.
How to read it
p ⇏ q is read "p does not imply q", "p but not q", or "p without q". Each reading describes a counterexample to the implication "if p, then q".
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 = T (the only true case). With p = F and q = T, p ⇏ q = F. With p = F and q = F, p ⇏ q = F.
One true row and three false rows: it is a contingency with the same distribution as conjunction — no coincidence, since p ⇏ q ≡ p ∧ ¬q.
Everyday example
Someone claims: "if you study, you pass". To refute it you need a concrete case of someone who studied and did not pass: p ∧ ¬q. That is exactly p ⇏ q. A student who did not study (p false) is no counterexample, whether they passed or not.
Properties and equivalences
Definition: p ⇏ q ≡ ¬(p ⇒ q). Conjunctive form: p ⇏ q ≡ p ∧ ¬q. Also p ⇏ q ≡ ¬(¬p ∨ q) by De Morgan.
It is not commutative: p ⇏ q and q ⇏ p are true in different rows ((T, F) and (F, T) respectively). The latter coincides with p ⇍ q, the negated converse conditional.
Common mistakes
Thinking that negating "if p then q" gives "if p then not q" (p ⇒ ¬q). It does not: the negation of a conditional is a conjunction, p ∧ ¬q, not another conditional.
Believing that q being false is enough to refute p ⇒ q. If p is also false, the conditional remains vacuously true; a counterexample requires p to be true.
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
Is the negation of "if p then q" the same as "if p then not q"? ▼
No. The negation of p ⇒ q is p ∧ ¬q: p happens and q does not. "If p then not q" (p ⇒ ¬q) has a different table and is true, for example, whenever p is false.
How many true rows does p ⇏ q have? ▼
Only one: p = T, q = F. It is the only possible counterexample to the conditional p ⇒ q.
How do I write p ⇏ q with basic operators? ▼
As p ∧ ¬q, or equivalently ¬(¬p ∨ q).