Canonicity and Bi-Approximation in Non-Classical Logics

Non-classical logics, or variants of non-classical logics, have rapidly been developed together with the progress of computer science since the 20th century. Typically, we have found that many variants of non-classical logics are represented as ordered algebraic structures, more precisely as lattice expansions. From this point of view, we can think about the study of ordered algebraic structures, like lattice expansions or more generally poset expansions, as a universal approach to non-classical logics. Towards a general study of non-classical logics, in this dissertation, we discuss canonicity and bi-approximation in non-classical logics, especially in lattice expansions and poset expansions. Canonicity provides us with a connection between logical calculi and space-based semantics, e.g. relational semantics, possible world semantics or topological semantics. Note that these results are traditionally considered over bounded distributive lattice-based logics, because they are based on Stone representation. Today, thanks to the recent generalisation of canonical extensions, we can talk about the canonicity over poset expansions. During our investigation of canonicity over poset expansions, we will find the notion of bi-approximation, and apply it to non-classical logics, especially with resource sensitive logics.