Relation:
Connection, tie between things; something that conditions several things.
This is a fundamental abstract idea, but everyone knows instances of it,
like loves, hates, is greater than, is the same as, knows, believes, is an
element of and many, many more, of many kinds, that may relate all kinds of
things one can think of in any kind of connection one can think of to any kind
of things one can think of.
1. Relations in logic
2. Relations analysed settheoretically
3. Relations as structures
It is not easy to analyse relations, and one of the odd things about human
beings is that it took over 2000 years for logicians to analyse relations
explicitly in logic. The merit for first seeing that there is a need for a
logic of relations belongs to the 19th Century English mathematician Augustus de
Morgan, who noted that up to then logic had no place for intuitively perfectly
valid arguments like "if this is the head of a horse, this is the head of an
animal".
After some initial work of his, the logic of relations was rather quickly
set up, independently, by Frege and Peirce, who also independently (Peirce
together with a pupil of his, Mitchell) developed the logic of
quantifiers for
it.
1. Relations in logic
In logic, a relation is expressed by a
predicate with two or more subjects and may
be written with "R" for the relation, and "x" and "y" for the subjects, as
"R(x,y)", "xRy", "Rxy" etc. The notation is a
matter of choice, and the three given ones are common.
Now this gives, within formal logic, the means for quite sophisticated
analyses and terminology of relations, since one can front statements about
relations with quantifiers ("for every x there is some y such that x loves y"
etc.), and introduce new terminology for properties of such expressions. One
example is that a relation R is symmetric iff (x)(y)(Rxy iff Ryx);
another is that a relation R is reflexive iff (x)(Rxx), and quite few
more.
There are many more of these properties of relations, but one thing to note
is that one can and does introduce these in fact as properties of the
linguistic expressions one uses for the relations, which one defines in terms
of what subjects make statements that involve the relation true.
2. Relations analysed settheoretically
A further step is to analyze relations settheoretically, by taking "R" as referring to a set
of pairs:
{(a,b): (a,b)eR} = {(x,y): R(x,y)} whence (a,b)eR iff R(a,b). Here
"{(a,b): (a,b)eR}" = "the set of pairs (a,b) such that (a,b) is an element of
R" and " "{(x,y): R(x,y)}" = "the set of pairs (x,y) such that x has R to y".
Now, one can use "R" or "{R}" as a name for either of last sets. This will
be a set of pairs. Thus, if one speaks of love, the corresponding sets will be
the set of pairs in which the first element is a lover and the second a
beloved.
There are at least two problems with this otherwise neat analysis, that was first thought of by
Frege
and Peirce, it seems, who also saw the problems that follow, if not in the same terms.
One: In some
imaginable simple Universe of Discourse, such as
{Adam, Eve} with Adam and Eve being the only
humans who happen to love and tease each other, whence on the given analysis,
supposing there being only Adam and Eve, the relations of teasing and loving
have just the same elements (namely (Adam, Eve) and (Eve,Adam)).
And the problem is that  in this simple universe  loving and teasing are
true of just the same pairs of elements; loving and teasing are construed as
sets; sets are identical if they are true of just the same elements; and so 
in this simple universe  loving=teasing, which is not intuitively correct.
Two: It seems to follow that
relations "are" sets of pairs,
which may not be quite what one intuitively expects that a relation "is".
Loving someone, one may say with some intuitive justification, just does not
reduce to being a member of a set of pairs in which one is a first member and
one's beloved a second member  there is, at least, more to it than that,
notably the relation of being in love.
Let's first dispose of the first problem, by noting it also holds for
oneplace
predicates i.e.
properties, and noting that for any predicate that is
coextensive with another predicate in a certain domain, although intuitively
the predicates have different meanings, one can
at least alleviate the problem by adding individuals to the domains or adding
predicates to the predicates that do distinguish them, or both.
This does not dissolve the underlying problem of the interpretation of
predicates in settheory: They are identified with the
set of things the predicate is true of, and not with a new sort of
entity, that is unlike the things the predicate is true of, such as a really
existing property or relation. But supposing there are only things and sets of
things, this identification or confusion makes sense, especially since it
leads to lots of formulas and definitions that also make sense, and preserve
the great majority of one's intuitions about relations.
Next, we consider relations as sets of pairs (or
tuples). As just mentioned, one reason for this reductive analysis of a
relation to a set of the pairs (or tuples) of things that the relation is true
of is that it requires no more entities than are already supposed in set
theory: things (of any kind) and
sets of things, that may also
be sequences i.e. ordered sets.
And part of the problem is that even if one insists that a relation or
property intuitively just is not the same kind
of thing as the things it truly attributes, then it remains difficult to say
what kind of thing the meaning of a relationterm or propertyterm would be.
3. Relations as structures
The most plausible answer seems to run like this: Relations and properties
are names of structures  kinds of things
that remain the same while one or more of their parts are changed, somewhat
like a room, which remains the same while furniture in it is removed or
replaced, or like a medicinestrip, with say 10 containers for pills that may
be pushed out of them, and be put back, or replaced by other items that fit,
or indeed again like one's body that remains much the same while some of its
parts are renewed.
One reason this answer is plausible is that it does give an interpretation
to relation terms and property terms that gives them their own kind of
referent; another reason is that structures are
analysed much like predicates  indeed, if
P(a,b) is a statement with a twoplace predicate P true of the
pair (a,b),
then the structure involved may be written as: structure(P,a,b).
This in turn may be analysed using
Cartesian Products: A structure is an entity
that is like a Cartesian Product of which the first member is a constant, that
often represents a complicated entity into which the things that satisfy it
may enter as parts or a type of action in which some of the states of one
thing condition some of the states of another thing.
