We investigate the polyhedral semantics of a recently devised intermediate logic, in which expressions are interpreted in terms of n-dimensional polyhedra. An intermediate logic is polyhedral complete if it is complete with respect to some class of polyhedra. The first major result of this paper is a necessary and sufficient condition for the polyhedral completeness of logic. This state, called the neural criterion, is expressed in Alexandrov’s conception of the neural of pose sets. It provides a pure combinatorial characterization of polyhetically complete logic.

    Using the Nerve Criterion, it is easy to show that there are many intermediate logics in a continuum that are not polyhedral perfect but have finite model properties. We also provide, with considerable combinatorial work, countable infinite class logic axiomatized by Jankov-Fine formulas for polytopically complete “star-like trees”. These “star-like logic” polyhedral completeness theorems are his second main result in this paper.

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