| Summary: | In intuitionistic propositional logic, one of the so-called De Morgan's laws is not valid. This thesis studies the non intuitionistically valid one, namely, $ neg( phi wedge psi)= neg phi vee neg psi$, (denoted by (DML)), with examples and applications in topology, algebra, analysis, logic and topos theory. In particular, we recall the Gleason cover of a topos which is a universal construction of a De Morgan topos covering the given one. This construction is then used in connection with the Hahn-Banach theorem in any topos of sheaves on a locale, and in order to obtain the real closure of an ordered field in any topos of sheaves on a Boolean space. We also show that an algebraic analogue of (DML) may be related to the Zariski spectrum of a ring. Finally, we examine (DML) in the contexts of model theory and locale theory.
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