I would like to express the well-known arithmetic relations of less-than ($<$) and greater-than ($>$) using some [semblance of] first order logic (FLO) or at least some form of abstract notation that specifically isn't "arithmetically aware" -- that is, I would like to define analogues of the mentioned relations for abstract symbols, but where the following holds:
For every $x$ and $y$ in a set of symbols S, if [the tuple] $<x, y>$ (the brackets are not related to the less-than and greater-than arithmetic symbols) is in the set $L$ then the tuple $<y, x>$ must be in the set $G$.
What I am trying to express is the equivalent of the relationship in arithmetics where given two [natural] numbers, if the first number is less than the second number then the second number must be greater than the first.
Using what I know of some variant(s) of first order logic notation, if less-than is defined as the set $L$, a binary relation, and greater-than is defined as the set $G$, also a binary relation, the implication could be expressed like so:
$$ \forall x,y \in S ((x < y) \implies (y > x)) $$
As with all binary relations, to clarify, $(x < y)$ expresses $xLy$ and $<x,y> \in L$, while $(y > x)$ expresses $xGy$ and $<y,x> \in G$.
Please forgive my possibly poor annotation and mistakes in syntax, if any, this is as much learning FLO for me as anything else.
My question is, would the aforementioned statement be a good representation of the desired relationship between the $<$ and $>$ relations, for e.g. purposes of "boostrapping" what we actually learn from the equivalent constructs in arithmetic?
Seen from a more humourly side, if Vulcans were to use our FLO symbolism and teach arithmetics to their young through FLO precepts, would the statement be any good?
My ultimate purpose is to experiment with a computer program that can be taught basics of arithmetics through being fed a logical program, with no prior understanding of the former. It's a lofty goal, I know, but I'd like to start somewhere.