Definition: Negation of a String

Let $L$ be a logical calculus with a syntax allowing to express a negation of a string $\phi$. Without loss of generality, assume that the underlying alphabet contains a special negation letter "$\neg$" such that $\neg \phi$ denotes the negation of the string $\phi$. The negation of the string $\phi$ has the following semantics: For any interpretation $I,$ the valuations $[[\phi]]_I$ and $[[\neg \phi]]_I$ are related to each other as follows: * if $[[\phi]]_I=1$, then $[[\neg \phi]]_I=0$, * if $[[\neg \phi]]_I=1$, then $[[\phi]]_I=0$.

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