An institution-independent proof of Craig Interpolation Property
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- We formulate a general institution-independent (i.e. independent of
the details of the actual logic or institution) version of the Craig Interpolation
Theorem and prove it in dependence of Birkhoff-style axiomatizability properties
of the actual logic.
We formalise Birkhoff-style axiomatizability within the general abstract
model theoretic framework of institution theory by the novel concept of Birkhoff
Our proof destills a set of conditions behind the Craig Interpolation Property,
which are easily satisfied in the applications. This together with the generality
of our approach leads to a wide range of applications for our result, including
conventional and non-conventional logics (many of them from algebraic specification
theory), such as general algebra, classical model theory, partial algebra,
rewriting logic, membership algebra, etc. all of them in various versions
and with various types of sentences (including infinitary ones).
In dependence of axiomatizability properties many other applications are
expected for various institutions or logics.