The expression "the reciprocal is true" is used when a person wants to demonstrate that feel the same about the other.
For example, when a friend says to a friend "our friendship is very important to me", and the friend thinks the same and wants to reciprocate the words, she says "the reciprocal is true".
The expression is based on the word reciprocal, which comes from reciprocity. What is reciprocal means what is mutual, which presents itself as a characteristic of the relationships between two or more parties, in order to establish and maintain a good relationship.
See too:reciprocal.
Reinforcement with the use of the word "true" confirms the reciprocal feeling, saying that it is true, that this is right. In other words, "the reciprocal is true" means that there is reciprocity.
The phrase is often used in moments of demonstration of friendship or of love. When "the reciprocal is true" is said in these contexts, it is expressed in response to someone who has shown their feelings towards that person or group of people.
It can also be used in affirming something corresponding to two things that are neither feelings nor related to people. An example is the usual expression "art imitates life", which at one point can be complemented in this way: "art imitates life, and the reverse is true". In other words, the reciprocal relationship, of exchange, also happens and life imitates art.
The negation of the sentence can be "the reciprocal is not true", or "the reciprocal is not always true", when there are disagreements between the two related parties. As in "Attitudes make up for words, but the reverse is not true" (Ulysses Franco).
The English translation would be the expression the opposite is true, or the reverse is true, as well as the word likewise, the latter in the sense of "in the same way".
See also the meaning of Vice versa.