Consider a sudoku puzzle for which there is a unique solution. In solving the puzzle, one enters in pencil what, a priori, each of the $81$ small squares could be, given the (at least $17$) clues that start the puzzle.
Suppose that, to progress, one has to guess, for a small square $s$, which of two potential numbers the solution requires, in order to solve the sudoku inductively. Suppose one happens to pick the wrong number and proceeds to eliminate from each of the remaining small squares the potential numbers according to the wrong choice, following the rules of the game.
Eventually, at least for some sudokus, one would find a contradiction in the sense that a row, column, or box would contain two numbers that are the same. Call the second such number (or small square) to be written down (or filled in tentatively) $c$.
Now consider the metric $\delta$ defined on a $9\times 9$ array as the distance between entries of the array in units of the length between two adjacent, non-diagonal entries.
One now has, I suppose, the distance from $s$ to $c$ given by $\delta$ superimposed on the sudoku puzzle; that is, a distance from the initial false assumption entry $s$ to the contradictory entry $c$.
Has such a thing been studied before?
It reminds me of nonclassical logics that are paraconsistent; that is, logics for which the principle of explosion and the law of noncontradiction do not hold.
This leads me to the point of this post. (The previous question is a primer, acting as motivation but still requiring a response.)
So the main question is . . .
Are there metric spaces defined on nonclassical logics which measure some notion of the distances from a given statement to contradictions?
Please help :)
Indeed the "physical distance between $s$ and $c$" metric is uninteresting. Here is a simple example where they are at opposite corners of the $9 \times 9$ grid.
$$\begin{array}{ccc|ccc|ccc} s & 9 & 8 & 7 & 6 & 5 & 4 & 3 & a \\ \cdot & \cdot &\cdot &\cdot &\cdot &\cdot &\cdot & \cdot & 5\\ \cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot & 6\\ \hline \cdot & \cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot & 7\\ \cdot & \cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot & 8\\ \cdot & \cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot & 9\\ \hline \cdot & \cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot & 3\\ \cdot & \cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot & 4\\ \cdot & \cdot &\cdot &\cdot &\cdot &\cdot &\cdot & 1 & c \end{array}$$
$s$ can be filled with $1$ or $2$, which makes $a$ to be $2$ or $1$ respectively, which makes $c$ to be $1$ or $2$ respectively. So if the player chooses $s=1$ they would only find out the error when $c=1$ causes a contradiction.