ReflexiveWhile having same etymology as Reflection, the term Reflexive means to apply or to refer to oneself.
In elementary algebra (mathematics), being considered an implicit binary relation, elements linked to oneself are called reflexive; by extension, a binary relation where all elements are reflexive is itself called reflexive (which does not mean the relation applies to itself). Reflexivity is then the property of relations that are reflexive in that extended meaning. Yet another unrelated meaning of reflexive is relative to reflex, which paradoxically means absence of conscious thought (reflection). This glossary entry thus shows how the meaning (and spelling) of words drift; earlier entries were even more confused (and wrong) because of discrepancy in drift between french and english languages!
To come back to our main topic, which is Reflection, the question of whether the adjective "irreflexive" is reflexive or not is yet another well-known variation of Epimenides' paradox, that shows that a meaningful discourse must restrict reflection severely. The solution to the paradox is to distrust reflexive definitions, and instead strive towards well-founded definitions. and when trying to give a well defined meaning to reflective objects, replace reflection by recursive equations and induction principles. It seems that everytime a reflective object is to be used statically, it can be expressed this way, because using such an object means firstly implementing its abstract idea in a "real" substratum world in which it has such clear meaning. This does not mean that reflection in general can be reduced to mere abstraction, just in the same way as logical thought does not reduce to deduction.