Relations are Not Sets of Ordered Pairs Ingvar Johansson,
Institute for Formal Ontology and Medical Information Science,
Saarbrcken 2004-10-09
Slide 2
The Importance of Relations The question of relations is one of
the most important that arise in philosophy, as most other issues
turn on it: monism and pluralism; the question whether anything is
wholly true except the whole of truth, or wholly real except the
whole of reality; idealism and realism, in some of their forms;
perhaps the very existence of philosophy as a subject distinct from
science and possessing a method of its own. (Bertrand Russell,
Logical Atomism, 1924.)
Slide 3
General Message Features of representations must not be
conflated with what is represented.
Slide 4
Specific Message An internal relation can be represented by,
but not be identical with an ordered pair or a set of ordered
pairs.
Slide 5
Interesting Consequence There are no anti-symmetrical internal
relations in the language-independent world.
Slide 6
Identity and Representation Circles are not identical with an
equation. Relations are not identical with sets of ordered pairs,
In both cases, it is merely a matter of representation. (x - a) 2 +
(y - b) 2 = r 2 y x b a
Slide 7
Relations in Set Theory relation, a two-or-more-place property
(e.g. loves or between), or the extension of such a property. In
set theory, a relation is any set of ordered pairs (or triples,
etc., but these are reducible to pairs). For simplicity, the formal
exposition here uses the language of set theory, although an
intensional (property-theoretic) view is later assumed. (The
Cambridge Dictionary of Philosophy, 1999.)
Slide 8
Quotation Because relations are sets of ordered pairs, we can
combine them using set of operations of union, intersection, and
complement. These are called Boolean operations.
Slide 9
Relation Terminology Relation relations Relatum relata Binary,
ternary, , n-ary Symbolism: Rab (=aRb), Rabc, Rabcd, etc. Ordered
pair: Set of ordered pairs: : such that or ,,...
Slide 10
Kinds of Ontological Relations Relations of Existential
dependence; relata are existentially dependent on each other, e.g.,
perceived color and perceived spatial extension. Intentional
relations; can exist with only one relatum, e.g., love. External
relations; independent of the qualities of the relata related, e.g.
spatial distance. Internal relations (comparative, grounded);
dependent on the qualities of the relata related, e.g. larger than,
lighter than, rounder than.
Slide 11
Internal Relations Between Quality Instances are Grounded in
Universals The thing T 1 is larger than the thing T 2 because T 1
has an area-instance, a, that is larger than the area-instance of T
2, b. Every area-instance that instantiates the same universal as a
is larger than every area-instance that instantiates the same
universal as b. T 2 : b T 1 : a
Slide 12
Internal Relations have Specific Features Possibly, a exists
but not b, or vice versa (compare existential dependence).
Necessarily: if both a and b exist, then the relation of being
larger than is instantiated (compare external relations). b a
Slide 13
Internal Relations can be Perceived a is larger than b b is
larger than c a is larger than c a and d are equally large a is
more similar to b than to c a d b c
Slide 14
Internal Relations require a Determinable a is larger than b a
and d are equally large a is more similar to b than to c a, b, c,
and d are different determinate instances of the same determinable.
a d b c
Slide 15
Internal Relations require a Determinable Statements describing
internal relations: a is smaller than b, a is heavier than b, a is
more electrically charged than b, a has greater intensity than b,
etc. These internal relations have different relata of the same
kind. Resemblance (both exact and non-exact) is always resemblance
in a certain respect. A determinate-determinable distinction (W.E.
Johnson) has to be made explicit.
Slide 16
Exact Resemblance When two things are equally large there is a
relation of exact resemblance between the two quality instances in
question. But there is only one universal. When two particulars are
qualitatively identical there is necessarily a relation of exact
resemblance between them.
Slide 17
Non-Exact Resemblance When two things have different areas
there is a relation of non-exact resemblance between the two area
instances in question. There are then two universals. Between these
universals there is an internal relation, a determinate kind of
resemblance. When the universals are instantiated there is also an
internal relation between the corresponding instances.
Slide 18
Internal Relations can be Mind-Independent We discover that a
is larger than b, that a and d are equally large, and that a is
more similar to b than to c. True of both binary and ternary
relations. a d b c
Slide 19
Internal Relations are Epiphenomena a and d are equally large.
a is larger than b. There are epiphenomenal natural facts.
Epiphenomena add to being. a d b c
Slide 20
Language Lumps Together What is true of the ordinary language
terms, are true of larger than, too. But not of 2.13 times as large
as. blue green y red
Slide 21
Internal Relations in Set Theory a is larger than b = def :
such that x is larger than y ? No, circular definition. ,,,, ? No,
cant distinguish between extensionally equivalent relations. a d b
c
Slide 22
Internal Relations in Set Theory a is larger than b = def ? No,
cant distinguish between extensionally equivalent relations. But,
of course, can be used to represent the fact that a is larger than
b. a d b c
Slide 23
Properties of Internal Relations larger than (L) is asymmetric:
xy (Lxy Lyx) Lab & Lba equally large (E) is symmetric: xy (Exy
Eyx) Ead & Eda a d b c
Slide 24
Anti-Symmetrical Relations larger than or equal to is
anti-symmetric: xy (x y) & (y x) x = y. (a d) & (d a) &
(a = d). In the individual case there is only symmetry. (a b) &
(b a) In the individual case there is only asymmetry. No case is
anti-symmetric. a d b c
Slide 25
Anti-Symmetrical Relations and their Relata Anti-symmetry: (a d
& d a) (a = d). If a and d are numbers, then = means numerical
identity. If a and d are quality instances, then = means
qualitative identity. If a and d are universals, then = means
numerical identity.
Slide 26
Representations and Represented: Possible Mistake There are no
anti-symmetrical internal relational universals. Predicates ( ) can
be anti-symmetric. Sets of ordered pairs can be anti-symmetric.
Representations of internal relations can be anti-symmetric but
internal relations cannot.
Slide 27
Representations and Represented: Logical Constants and the
World The statement The spot is red or green contains a
disjunction. The language-independent world contains no
corresponding disjunctive fact. The spot is red or green
Slide 28
Cant Disjunctions be Truthmakers even for Categorical
Assertions? The spot is red or green. The spot is red. Isnt in both
cases a disjunction a truthmaker? v v v v v v. Answer: In the
latter case, we talk as if there is in the world only one property.
Language lumps together.
Slide 29
Representations and Represented: Scales and Quantities 0 1 2 3
4 5 6 7 8 9 10 meter 1 m represents the standard meter. 7 m
represents all things that are seven times as long as the standard
meter. 0 m represents nothing at all, but it is part of the
scale.
Slide 30
The End There are in the language-independent world no
disjunctive facts. There are in the language-independent world no
zero quantities. There are in the language-independent world no
anti-symmetrical relations. In language, there are disjunctions,
zero quantities, as well as anti-symmetrical relations.