R Hansen says: >I?ve checked into this, the votes for ?consistency? are substantial.
I see. I didn't know logic was a voting kind of thing, but new logics are always being invented I suppose.
Consistency means that you will never (properly) derive a statement and its negation. Suppose we investigate a new system in which the sign law is given as:
+ x + = + + x - = + - - x + = - - - x - = -
"The sign of the result is the same as the sign of the first factor".
You can easily find that the usual distributive law no longer holds. But so what? If we took the distributive law as a given (axiom) then we cannot use this new sign law, but if we are investigating a possible new system, we cannot assume all the results that follow from the starting points (not detailed here) of the old system. (But, are there slightly different but similar laws that do hold?)
In short, your example of "inconsistency" is too narrow, and amounts to a kind of circular reasoning. Of course one cannot introduce new laws or identities willy-nilly into a already established system without possibly introducing inconsistency, but the usual system of integers in arithmetic is not the only possible consistent system hat uses the signs +, -, x, /.
So the question becomes -- is there a scheme for introducing "/" with signs that is compatible with the above sign laws for "x"? Obviously order of operands will now be important, although we are not used to thinking about that too much in ordinary arithmetic.
That is mathematical thinking, and explanation such matters ("what is consistency?") should be part of an introduction to mathematical logic.