A relation is a two-place predicate R associating two objects, a and b, denoted by “aRb”. Relations are characterized by three general properties, symmetry, transitivity, and reflexivity.
— Symmetry, or Reciprocity: The same relationship holds between “a and b” and “b and a”.
— Reflexivity: The relationship connects an object to itself.
— Transitivity: The relationship connecting a to b and b to c also connects a to c.
1. Symmetry, or reciprocity
A relation is symmetric or reciprocal if it relates both a to b and b to a. In other words, both “aRb” and “bRa” hold. If a loves b, b does not necessarily love a: a love relationship is not symmetrical. “Meeting” is a symmetric relationship. The following argument is neither more nor less logical than any other, but it would make a valid point in any detective novel; it can only be rejected by accusing Peter of lying:
If Peter confessed to having met Paul at the bar, we must assume that Paul met Peter. Paul cannot deny the obvious.
2. Reflexivity
A reflexive relation relates a being to itself, noted “aRa”. “— being contemporary of —” is a reflexive relationship: a is its own strict contemporary. For the average person, the causal relationship is not reflexive; only God is causa sui, his own cause.
The reflexive relation can be used ad hominem. The principle “charity begins at home” for example forces the reflexivity of the relationship “a makes charity to b”; all the same, the love of others can be used to encourage self-care:
You who love the whole of humanity, you should try to love yourself as well!
The competence of an adviser can be challenged by inciting him to make a reflexive application of his talents:
Physician, heal thyself!
Such replies correspond to the ad hominem variety setting up practices against words.
3. Transitivity
A relation is transitive if, when it relates a to b and b to c, it also connects a to c; in other words, “aRb and bRc” imply that “aRc”.
If a loves b, and if b loves c, then a does not necessarily love c; a relationship of love is thus not transitive. The relation “— be the father of —” is not transitive, but “— being an ancestor of —” is transitive. If a is an ancestor of b and if b is an ancestor of c, then a is an ancestor of c.
Inferences based on the transitivity of a predicate apply whenever at least three objects are positioned on a graduated scale:
If a is bigger, older, richer … than b
and b larger, older, richer … than c,
Then a is bigger, older, richer … than c.
Inferences based on these properties are part of the unnoticed evidences exploited by everyday reasoning and argument. They are sometimes considered to be “quasi-logical”, S. Quasi-logic; but being sound and valid does not preclude being an argument.
4. Conversion
S. Conversion