Why are are trichotomous strict orders called total orders, when by definition they are not total? Consider an arbitrary set equipped with a strict order $(X,<)$ such that we have:
$$\forall a,b\in X\left(a\neq b\implies (a<b)\lor (a>b)\right)$$
I frequently see this relation being called a "strict total order" yet the term itself is an oxymoron. For in order for a relation to be total it must be reflexive which would contradict it being a strict order.
Also while on the topic, I would be grateful if someone could explain to me why in order theory there is a focus on partial orders rather then strict orders. 
Now I understand any work on partial orders can be transplanted over to give analogous theorems on strict orders, in the same way elementary theorems in linear algebra regarding the rows of a matrix can analogously be translated into theorems on the columns of a matrix. Thus making the distinction between whether we choose strict or partial orders somewhat trivial. However the concept of a strict order seem a bit more intuitive at least when viewing finite posets, for instance the nodes/vertices in a hasse diagram could be viewed as the directed acyclic graphs formed by the covering relations of a strict order.
 A: The point is that a strict order and a non-strict order are essentially the same thing.  Given a strict order $<$, you can get a non-strict order by defining $x\leq y$ to mean "$x<y$ or $x=y$".  Conversely, given a non-strict order $\leq$, you can get a strict order by defining $x<y$ to mean "$x\leq y$ and $x\neq y$".
So "strict total order" just means "the strict version of a total order", where we're axiomatizing the relation $<$ rather than the relation $\leq$.  It really is essentially the same thing, since you can convert between a strict total order and an ordinary (non-strict) total order as described above.  Reflexivity really has nothing to do with totality at all: rather, demanding reflexivity just means you are axiomatizing $\leq$ instead of $<$ (regardless of whether your order is total).
Since strict orders and non-strict orders are pretty much interchangeable, it usually doesn't matter which one you take as your basic definition or object of study.  It's just a matter of convention that non-strict orders are what is usually axiomatized.  Each version has a few circumstances where it is more useful (for instance, strict orders tend to be more convenient when talking about well-foundedness, and non-strict orders are more natural if you are also interested in preorders).
