Non-classical logics (many-valued, modal, substructural...)
Osnova sekce
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In the next few topics we will deal with non-classical logics from different perspective. Their motivation ranges from reactions to the (perceived) problems of classical logics (as in Brouwer), desire to add a new expressive power (model logics), to use logical machinary to analyse computations (substructural logics, etc.). And there is always some motivation to touch on philosophical issues.
Before we start, we need to get familiar with the notion of algebraic sementics which plays an important role in many non-classical logics.
Read for instance
https://plato.stanford.edu/entries/logic-algebraic-propositional/#IntuPropLogi
in particular Section 5.
For many purposes, it suffices to start with some knowledge of lattices
https://en.wikipedia.org/wiki/Lattice_(order)#Bounded_lattice
Boolean algebras
https://en.wikipedia.org/wiki/Boolean_algebra_(structure)
and Heyting algebras
https://en.wikipedia.org/wiki/Heyting_algebra
Boolean algebras are algebras for the classical logic, while Heyting algebras are algebras for intuitionistic logic (in the sense of algebraic semantics introduced above).
For more details, see the lecture notes for the Boolean algebras.