Bitopology and Four-valued Logic

Non-involutive twist-structures

“…This algebraic framework allows us to rigorously formulate a very natural and expected connection between bilattice-based logics on the one hand and the topological setting of d-frames and bitopological spaces on the other. We show in particular how many well-known structures can be seen as special cases of non-involutive bilattices, namely pre-bilattices, bilattices with an involutive negation, and the nd-frames of [10]. If we further introduce Nelson-type implications into the language, we can show how N4-lattices, Nelson algebras and implicative bilattices nicely fit into the picture as well.…”

“…On the other hand, a clear parallelism also seems to exist between bilattices and other formalisms motivated by the attempt to deal with inconsistency in computer science, notably the theory of d-frames and bitopological spaces [11]. This latter connection, however, had never been clearly stated in formal terms until the recent paper [10] introduced a mathematical framework that may be a possible way of bridging this gap. The present paper is an attempt at connecting, further exploring and developing both the above-mentioned links, introducing a uniform logical and algebraic framework which encompasses paraconsistent Nelson systems, bilattice-based logics and (the finitary aspects of) d-frame theory.…”

“…If one wants to have a negation connective in the language, then the only available candidate in the literature until recently was "strong negation" (essentially the same in bilattices and Nelson lattices), which requires the two domains to be isomorphic lattices. The situation changed with the recent [10], which introduced a novel and very natural way of defining a weaker negation operator that allows for the positive and the negative domain to be truly independent of each other, and gives rise to interesting structures that are moreover supported by a clear topological interpretation. To continue the discussion of the halting problem from this perspective, negation would allow us to formalise evidence for the statement "it is not true that the machine will stop".…”

“…The present work expands and exploits the main ideas of [10] introducing algebraic structures, called non-involutive bilattices, that have a pre-bilattice reduct and a negation operator that is no longer required to be involutive nor to satisfy all the De Morgan identities. This algebraic framework allows us to rigorously formulate a very natural and expected connection between bilattice-based logics on the one hand and the topological setting of d-frames and bitopological spaces on the other.…”

Bilattice logic properly displayed

“…Yet more recently, bilattices with a negation not necessarily satisfying the involution law (¬¬a = a) have been introduced with motivations of domain theory and topological duality (see [26]), and the study of the corresponding logics has been started [31]. These logics are weaker than the one considered in the present paper, and so adapting our display calculus formalism to them might prove a more challenging task (in particular, the translations introduced in Section 5 may need to be redefined, as they rely on the maps p and n being lattice isomorphisms, which is no longer true in the non-involutive case).…”