>>12202708The main confusion in the thread is between the "you-can-change-the-order" aspects of algebraic commutativity, and relational symmetry.
The word commutativity refers to binary operations which form expressions (which do not necessarily have truth-value taken by themselves), and the expression is the same thing in either order (this is a distinct point). + is commutative for real numbers x and y since for all x and y, x + y = y + x. Now, I just made a truth-claim-definition, hinging on the equality sign, but " 2 + 3 " for example is an expression which can't be judged true or false. It's simply an expression (where the word "expression" itself carries the right open, un-resolved philosophical content.)
Relations are the stuff where judgment/truth value can be inserted. W P A, for example, where W denotes George Washington, A denotes John Adams, and the relation P means immediate successor to the office of President of the United States. Similarly LEQ, GEQ, and opeartors which invoke the possibility of truth-judgment.
