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8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation

[ways relations can be categorised and formalised]

12 ideas
If a relation is symmetrical and transitive, it has to be reflexive [Russell]
     Full Idea: It is obvious that a relation which is symmetrical and transitive must be reflexive throughout its domain.
     From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II)
     A reaction: Compare Idea 13543! The relation will return to its originator via its neighbours, rather than being directly reflexive?
'Asymmetry' is incompatible with its converse; a is husband of b, so b can't be husband of a [Russell]
     Full Idea: The relation of 'asymmetry' is incompatible with the converse. …The relation 'husband' is asymmetrical, so that if a is the husband of b, b cannot be the husband of a.
     From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], V)
     A reaction: This is to be contrasted with 'non-symmetrical', where there just happens to be no symmetry.
'Reflexiveness' holds between a term and itself, and cannot be inferred from symmetry and transitiveness [Russell]
     Full Idea: The property of a relation which insures that it holds between a term and itself is called by Peano 'reflexiveness', and he has shown, contrary to what was previously believed, that this property cannot be inferred from symmetry and transitiveness.
     From: Bertrand Russell (The Principles of Mathematics [1903], §209)
     A reaction: So we might say 'this is a sentence' has a reflexive relation, and 'this is a wasp' does not. While there are plenty of examples of mental properties with this property, I'm not sure that it makes much sense of a physical object. Indexicality...
Nothing is genuinely related to itself [Armstrong]
     Full Idea: I believe that nothing is genuinely related to itself.
     From: David M. Armstrong (What is a Law of Nature? [1983], 10.7)
A relation is 'Euclidean' if aRb and aRc imply bRc [Cresswell]
     Full Idea: A relation is 'Euclidean' if aRb and aRc imply bRc.
     From: Max J. Cresswell (Modal Logic [2001], 7.1.2)
     A reaction: If a thing has a relation to two separate things, then those two things will also have that relation between them. If I am in the same family as Jim and as Jill, then Jim and Jill are in the same family.
A relation is not reflexive, just because it is transitive and symmetrical [Bostock]
     Full Idea: It is easy to fall into the error of supposing that a relation which is both transitive and symmetrical must also be reflexive.
     From: David Bostock (Intermediate Logic [1997], 4.7)
     A reaction: Compare Idea 14430! Transivity will take you there, and symmetricality will get you back, but that doesn't entitle you to take the shortcut?
Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock]
     Full Idea: A relation is 'one-many' if for anything on the right there is at most one on the left (∀xyz(Rxz∧Ryz→x=y), and is 'many-one' if for anything on the left there is at most one on the right (∀xyz(Rzx∧Rzy→x=y).
     From: David Bostock (Intermediate Logic [1997], 8.1)
Reflexive relations are syntactically polyadic but ontologically monadic [Molnar]
     Full Idea: Reflexive relations are, and non-reflexive relations may be, monadic in the ontological sense although they are syntactically polyadic.
     From: George Molnar (Powers [1998], 1.4.5)
     A reaction: I find this a very helpful distinction, as I have never quite understood reflexive relations as 'relations', even in the most obvious cases, such as self-love or self-slaughter.
A reflexive relation entails that the relation can't be asymmetric [David]
     Full Idea: An asymmetric relation must be irreflexive: any case of aRa will yield a reductio of the assumption that R is asymmetric.
     From: Marian David (Truth-making and Correspondence [2009], 4)
'Multigrade' relations are those lacking a fixed number of relata [MacBride]
     Full Idea: A 'unigrade' relation R has a definite degree or adicity: R is binary, or ternary....or n-ary (for some unique n). By contrast a relation is 'multigrade' if it fails to be unigrade. Causation appears to be multigrade.
     From: Fraser MacBride (Relations [2016], 1)
     A reaction: He also cites entailment, which may have any number of premises.
A relation is a set consisting entirely of ordered pairs [Potter]
     Full Idea: A set is called a 'relation' if every element of it is an ordered pair.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 04.7)
     A reaction: This is the modern extensional view of relations. For 'to the left of', you just list all the things that are to the left, with the things they are to the left of. But just listing the ordered pairs won't necessarily reveal how they are related.
Being taller is an external relation, but properties and substances have internal relations [Macdonald,C]
     Full Idea: The relation of being taller than is an external relation, since it relates two independent material substances, but the relation of instantiation or exemplification is internal, in that it relates a substance with a property.
     From: Cynthia Macdonald (Varieties of Things [2005], Ch.6)
     A reaction: An interesting revival of internal relations. To be plausible it would need clear notions of 'property' and 'substance'. We are getting a long way from physics, and I sense Ockham stropping his Razor. How do you individuate a 'relation'?