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But some logics may require the addition of some new bridge principles (mixing rules and axioms) in order to ensure the preservation of completeness.
But some logics were designed for studying aspects of rational behaviour, and in recent years it has become increasingly common to link these logics to suitable games.
For instance the model-theoretic consequence relation for some logics of time presupposes some facts about the physical structure of time.
Some logics may be formulated using axioms (rather than axiom schemas) together with the rule of condensed detachment without loss of theorems.
But both classes obtained from the definitions have been considered in the algebraic studies of many specific logics (for some logics one, for other the other) the natural class to study.
It is also known that for some logics with an algebraic semantics (relative to some set of equations), the natural class of algebras that corresponds to the logic is not an algebraic semantics (for any set of equations) of it.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com