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What is Negation in a System 2020

Dov Gabbay

The notion of negation is basic to any formal or informal logical system. When any such system is presented to us, it is presented either as a system without negation or as a system with some form of negation. In both cases we are supposed to know intuitively whether there is no negation in the system or whether the form of negation presented in the system is indeed as claimed. To be more specific, suppose Robinson Crusoe writes a logical system with Hilbert type axioms and rules, which includes a unary connective $*A$. He puts the document in a bottle and let it lose at sea. We find it and take a look. We ask: is the connective ``$*$'' a negation in the system? Yet the notion of what is negation in a formal system is not clear. When we see a unary connective $*A$, ($A$ a wff) together with<
>some other axioms for some additional connectives, how can we tell whether $*A$ is indeed a form of negation of $A$? Are there some axioms which the connective ``$*$'' must satisfy in order to qualify $*$ as a negation?

February 2021






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