The logic of proofs, semantically

Annals of Pure and Applied Logic 132 (1):1-25 (2005)
Abstract
A new semantics is presented for the logic of proofs (LP), [1, 2], based on the intuition that it is a logic of explicit knowledge. This semantics is used to give new proofs of several basic results concerning LP. In particular, the realization of S4 into LP is established in a way that carefully examines and explicates the role of the + operator. Finally connections are made with the conventional approach, via soundness and completeness results.
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Melvin Fitting (2012). Prefixed Tableaus and Nested Sequents. Annals of Pure and Applied Logic 163 (3):291 - 313.

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