RELATIONS
A relation is a two-place predicate R that associates two objects, a and b. This is 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 no more or less logical than any other, but it would be a valid point in a detective novel. It can only be rejected by accusing Peter of lying.
If Peter confessed to meeting Paul at the bar, then we must assume that Paul met Peter. Paul cannot deny the obvious.
2. Reflexivity
A reflexive relation relates a being to itself. It is noted “aRa”.
“— Being contemporary of —” is a reflexive relationship: a is his 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. For example, the principle “charity begins at home” for example forces the reflexivity of the relationship “a makes charity to b”. Nevertheless, the love of others can be used to encourage self-care:
If you love all humanity, then you should try to love yourself as well!
A counselor’s competence can be challenged by inviting them to make a reflexive use of their talents:
Doctor, heal thyself!
Such replies correspond to the ad hominem variety, setting up practices against words.
3. Transitivity
A relation is transitive if, when it connects a to b and b to c, then it also connects a to c.
In other words, “aRb and bRc” implies that “aRc”.
However if a loves b, and if b loves c, then a does not necessarily love c. Thus a relationship of love is not transitive.
The relation “being 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.
These 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.
These inferences are part of the unnoticed evidence exploited by everyday reasoning and argument. They are sometimes considered to be “quasi-logical”, see quasi-logic; but being sound and valid does not preclude being an argument.
4. Conversion
S. Conversion