In mathematics, the relation of reciprocity corresponds to the relation of symmetry: a relation R is symmetric in the domain in which it is defined, if for all the pairs of elements a and b both aRb and bRa hold. The relation “being the friend of” is symmetrical:

Peter is the friend of Paul = Paul is the friend of Peter = Peter and Paul are friends.

1. Returning and anticipating the good

In natural language, the reciprocal relation is defined on the basis of a set of actions which binds two persons. That is to say that if a does something positive to/for b, then b must reciprocate, doing the same thing to/for a. This is the principle of returning favors. The individuals a and b are equal in this relationship. The principle of reciprocity acts as a constraint:

If you treat me to dinner, I must treat you to dinner.

As a form of natural morality, the imperative of reciprocity is expressed by the principle:

Do to others as you would have them do to you. (Luke Gospel, 6:31)
Do not do to others what you would not have them do to you.

This principle is applied in the argument:

I’m polite to you, so be polite to me.

The speaker defines him/herself and defines his or her partner as members of the same category, who must be treated in the same way, S. Rule of Justice.

1. Returning and anticipating the wrong:
Reciprocity as retaliation

Eye for eye, tooth for tooth
If your disappointed lover disfigured you with vitriol, the court grants you the right to treat him likewise.

The law of retaliation is a primitive rule of justice that if A has wronged B in some way, then B can legitimately do the same wrong to A. In contemporary times, we might consider nuclear deterrence, based on the certainty of reciprocal destruction, as a concrete application of this principle. This theory corresponds to a particular case of the, You too!” argument.

Reciprocity as a legal principle allows different states to assert their equal international dignity, and possibly to justify a retaliatory measure:

If country A requires a visa for the nationals of country B, it is right that country B also requires that nationals of country A acquire a visa.