This Element is an introduction to recent work proofs and models in
philosophical logic, with a focus on the semantic paradoxes the
sorites paradox. It introduces and motivates different proof
systems and different kinds of models for a range of logics,
including classical logic, intuitionistic logic, a range of
three-valued and four-valued logics, and substructural logics. It
also compares and contrasts the different approaches to
substructural treatments of the paradox, showing how the structural
rules of contraction, cut and identity feature in paradoxical
derivations. It then introduces model theoretic treatments of the
paradoxes, including a simple fixed-point model construction which
generates three-valued models for theories of truth, which can
provide models for a range of different non-classical logics. The
Element closes with a discussion of the relationship between proofs
and models, arguing that both have their place in the philosophers'
and logicians' toolkits.
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