  In this paper we provide a formalism, Sudoku logic, in which a solution is
logically deducible if for every cell of the grid we can provably exclude all
but a single option. We prove that the deductive system of Sudoku logic is
sound and complete, and as a consequence of that we prove that a Sudoku puzzle
has a unique solution if and only if it has a deducible solution. Using the
classification of fundamental Sudoku transformations by Adler and Adler we then
formalize the notion of symmetry in Sudoku and provide a formal proof of
Gurth's Symmetrical Placement Theorem. In the concluding section we present a
Sudoku formula that captures the idea of a Sudoku grid having a unique
solution. It turns out that this formula is an axiom of Sudoku logic, making it
possible for us to offer to the Sudoku community a resolution of the Uniqueness
Controversy: if we accept Sudoku logic as presented in this paper, there is no
controversy! Uniqueness is an axiom and, as any other axiom, may freely be used
in any Sudoku deduction.
