Institutions with Proofs
The already familiar semantic consequence relation E ⊨ E′ between sets of sentences constitutes the semantic way to establish truth because it involves the models and the satisfaction relation between models and sentences. The syntactic approach to truth consists of establishing consequence relations, called proofs, between sets of sentences involving only syntactic entities. Therefore this approach is beyond models and satisfaction relation between models and sentences. The syntactic approach to truth, called ‘proof theory’, is in many ways complementary to model theory. However the relationship between model theory and proof theory is crucial for any logical system. For example the correctness of a proof theory can be established only in the presence of a model theory.
KeywordsProof System Proof Theory Partial Algebra Proof Rule Substitutivity Rule
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