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Duality in propositional logic between conjunction and disjunction, $K$ and $A$ means that for any "identity", such as $KpNp = 0$ (ignoring the detail of how to define this notion in propositional logic), if we replace all instances of $K$ by $A$, all instances of $A$ by $K$, all instances of 1 by 0, and all instances of 0 by 1, the resulting equation will also consists of an "identity", $ApNp = 1$. Suppose that instead of conjunction "$K$" and disjunction "$A$", we consider any pair of "dual" operations $\{Y, Z\}$ of the 16 logical operations such that they qualify as isomorphic via the negation operation $N$, where $Y$ does not equal $Z$. By "isomorphic" I mean that the sub-systems $(Y, \{0, 1\})$, $(Z, \{0, 1\})$ are isomorphic in the usual way via the negation operation $N$, for example $K$ and $A$ qualify as "isomorphic" in the sense I've used it here.

If we have an identity involving operations $A_1, \dots, A_x$, and replace each instance of each operation by its "dual" $A'_1$, ..., $A'_x$, replace each instance of 1 by 0, and each instance of 0 by 1, is the resulting equation also an identity? If so, how does one prove this? How does one show that the equation obtained via the "duality" transformation here is also an identity?

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    Every time one uses non-standard notation, it is good practice to explain it.2012-02-11

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The resulting equation is also an identity. This is because any of the $16$ operations can be put in canonical disjunctive normal form using only $\land$, $\lor$, and $\lnot$. Then the replacement procedure described in the post becomes the standard one, and we are dealing with ordinary duality.

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    Huh? Negation "preserves", if you must use that term, conjunction when going from ({0, 1}, K) to ({0, 1}, A), and "preserves" disjunction when going from ({0, 1}, A) to ({0, 1}, K)... it consists of a bijective map H: H(x +$_1$ y)=(H(x) +$_2$ H(y)), where "+$_1$" and "+$_2$" indicate distinct members of {K, A}, since the De Morgan laws hold. I don't know what you mean by "isomorphisms" in this post.2012-02-11