Jon,
That is an extensional definition of a relation:
Following the pattern of the functional case, let the notation
“L ⊆ X × Y” bring to mind a mathematical object specified by
three pieces of data, the set X, the set Y, and a particular
subset of their cartesian product X × Y}. As before we have
two choices, either let L = (X, Y, graph(L)) or let “L” denote
graph(L) and choose another name for the triple.
Nominalists prefer extensional definitions. But Peirce would
usually state intensional definitions (rules) for the functions
or relations he was considering.
Alonzo Church (1941) stated the intensional definition:
A function is a rule of correspondence by which when anything is
given (as argument) another thing (the value of the function for
that argument) may be obtained. That is, a function is an operation
which may be applied on one thing (the argument) to yield another
thing (the value of the function).
For further discussion of the distinction between intensions
extensions, see pp. 1 to 3 of Church's book:
http://www.jfsowa.com/logic/alonzo.htm
By the way, Church was not a nominalist. See the transcript of his
talk "On the ontological status of women and abstract entities":
http://www.jfsowa.com/ontology/church.htm
John
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