// reference · Replication
￩ Converse Conditional_
The converse conditional, also called replication or reverse implication, is written p ￩ q and means "p if q" or "p is implied by q". It is simply the conditional with the arrow pointing backwards: p ￩ q is equivalent to q ⇒ p. It is false only when q is true and p is false.
Truth table: p ￩ q
| p | q | p ￩ q★ |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | F |
| F | F | T |
Classification: Contingency · 4 rows
Definition
p ￩ q is true in every case except when the right operand q is true and the left operand p is false. In other words, q is the antecedent and p the consequent: p ￩ q ≡ q ⇒ p.
This operator shows up in logic texts as "replication" or "reverse implication", and in logic programming languages (Prolog) the rule p :- q reads exactly "p if q". In the calculator you write it with the arrow ￩.
How to read it
p ￩ q is read "p if q", "p whenever q", "p is implied by q", or "q is sufficient for p". It carries the same information as "if q, then p", but with the conclusion written first.
Its value is stylistic: it lets you state what you want to conclude first and the condition afterwards, as in "you pass if you study".
When it is true
The table has four rows. With p = T and q = T, p ￩ q = T. With p = T and q = F, p ￩ q = T. With p = F and q = T, p ￩ q = F (the only false case). With p = F and q = F, p ￩ q = T.
Three true rows and one false row: it is a contingency. Compare with p ⇒ q: the conditional's false row is (T, F) and the converse's is (F, T). They are mirror images.
Everyday example
"You get a refund if the product arrives damaged." Here p = "you get a refund" and q = "the product arrives damaged". The statement is violated only if the product arrives damaged and you get no refund. If it arrives intact, the guarantee says nothing and the statement remains true.
Properties and equivalences
Basic equivalence: p ￩ q ≡ q ⇒ p. Disjunctive form: p ￩ q ≡ p ∨ ¬q. Contrapositive: p ￩ q ≡ ¬p ⇒ ¬q.
Combined with the conditional it yields the biconditional: (p ⇒ q) ∧ (p ￩ q) ≡ p ⇔ q. Its negation is p ⇍ q ≡ ¬p ∧ q.
Common mistakes
Confusing it with the contrapositive. p ￩ q is not equivalent to p ⇒ q; it is its converse. Assuming both hold at once is the fallacy of affirming the consequent.
Swapping the operands in your head. Remember: the arrow points at the consequent. In p ￩ q the consequent is p (left) and the antecedent is q (right).
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 ⇒ q and p ￩ q? ▼
They are converses. p ⇒ q is "if p, then q" and is false at (T, F). p ￩ q is "p if q", equivalent to q ⇒ p, and is false at (F, T).
Why have an operator that only flips the arrow? ▼
To write the conclusion before the condition without reordering the formula, as Prolog does ("p :- q") or natural language ("you pass if you study"). Logically it adds no expressive power.
What is the negation of p ￩ q? ▼
p ⇍ q, which is equivalent to ¬p ∧ q: the consequent fails while the antecedent holds.