# Inverse function in multi-valued logic through the Webb function

Let Webb function in multi-valued logic as $Webb(x, y) = W(x, y) = Inc(Max(x, y))$. There is a theorem about any function in any multi-valued logic can be represented through the Webb function.

Then basic logic operations (negation, conjunction and disjunction) can be represented by following way for ordinary binary logic:

~x = W(x, x)
x & y = ~(~x | ~y) = W(W(x, x), W(y, y))
x | y = Inc(W(x, y)) = W(W(x, y), W(x, y))


And for ternary logic:

Inc(x) = W(x, x)
Dec(x) = Inc(Inc(x))
~x = W(W(Dec(x), Inc(x)), Inc(W(Dec(x), x))
x & y = ~(~x | ~y)
x | y = Inc(Inc(W(x, y))


What is the generalized formula for basic logic operations in any multi-valued logic through the Webb function, especially negation?

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A reference to the source of the theorem you mention would be helpful. –  Levon Haykazyan Dec 9 '12 at 13:39