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A comparative study of various model-theoretic treatments of negation: A history of formal negation. (English) Zbl 0972.03028
Gabbay, Dov M. (ed.) et al., What is negation? Dordrecht: Kluwer Academic Publishers. Appl. Log. Ser. 13, 23-51 (1999).
Summary: There are different treatments of the semantics of negation in nonclassical logics, the most familiar of which is probably the Kripke definition of negation for intuitionistic logic. In two recent papers I have explored generalizations of Goldblatt’s semantics for negation in orthologic (which uses the perp relation), and related these to the Kripke definition of negation for intuitonistic logic, and to the definition of De Morgan negation in relevance logic (which standardly uses the ‘star operator’ $$*$$). The present paper in part surveys this earlier work, but the main focus is to connect the alternative four-valued semantics of De Morgan negation (using ‘overdetermined’ and ‘underdetermined’ truth values) to the star semantics (and hence indirectly to the perp semantics). This connection is based on semi-published work of mine from the 1960’s.
For the entire collection see [Zbl 0957.00012].

##### MSC:
 03B60 Other nonclassical logic 03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) 03B20 Subsystems of classical logic (including intuitionistic logic) 03G12 Quantum logic