I'm writing an undergrad philosophy paper. My take on the issue is that the conceptual problem I'm addressing is only a problem because the word 'is' and 'relation' are too slippery. By more precisely defining the relationship in question, I suspect that the problem can be solved.
I'm just learning predicate logic now, and otherwise math is something I'm looking forward to knowing more about, but don't know much about now (so please go easy on the notation).
Do I understand these relationships correctly (by example):
antitransitivity: links in a chain - link a is connected (related) to b, b to c, but not a to c.
intransitivity: strands in a rope - where each fiber might run all the way from end to end, but not necessarily. So if a fiber is adjacent to a fiber at some point in its length they have a relationship for the purpose of this example. It may be that fiber a connects with fibers b and c, but not necessarily, although it must connect with fiber b.
transitivity: 'is an ancestor of': Adam begat Seth, Seth begat Enoch, Enoch begat Methuselah. so, Adam is an ancestor of x would apply to Seth, Enoch, and Methuselah and that would be a transitive relationship.
reflexivity 'is' of identity. Bruce Wayne 'is' Batman, and Batman 'is' Bruce Wayne. or "x shares my hair color"
irreflexivity 'is not' of identity. a 'is not' b. or "x does not share my hair colour"
symmetry links in a chain might work here too. A link a is connected to b in the same way b is connected to a? or 'in contact with' e.g. a hand is in contact with a rail, and a rail is in contact with a hand.
asymmetry "entails" rain entails wetness, but wetness does not entail rain.
trichotomous this one's a stretch. Two twins are born. Either a came out before b, or b before a, or they came out at the same time.
Hopefully I understood the concepts above, but the following three I don't really get.
anti-symmetry
quasi-reflexivity
co-reflexivity
Thank you for your help.
I hadn’t seen the terms coreflexive relation and quasi-reflexive relation before, but the definitions are fairly straightforward, especially that of coreflexivity.
A relation $R$ on a set $A$ is coreflexive if it is a subset of the identity relation on $A$; this is equivalent to saying that for all $x,y\in A$, if $\langle x,y\rangle\in R$, then $x=y$. In particular, the identity relation itself and the empty relation are both coreflexive. A silly but less trivial example on the set of people: is male and is the same person as. Being male, I stand in this relation to myself, but me sister Patty does not stand in this relation to anyone. Clearly one can stand in this relation only to oneself, but roughly half the population, being female, does not stand in this relation to anyone.
A relation $R$ on a set $A$ is quasi-reflexive if every element of $A$ that is related to at least one element of $A$ is related to itself. However, there might be elements of $A$ that are not related to any element of $A$, including themselves. An example on the set of people is has at least one living parent in common with: if you have a living parent in common with anyone, you have one in common with yourself. I have no living parent, however, so I do not stand in this relation to myself.