Deduction Theorem and Peirce Law in General Algebraic Logic: Constructive Proofs in General Sentential Logic and Universal Algebra

In this book we study the issues of Deduction Theorem and Peirce Law within the context of finitary universal Horn theories andequivalence between them. As basic general results, we first obtain a model-theoretic chracterization of sushtheories having Deduction Theorem with Peirce Law and then prove that equivalence between them preserves both Deduction Theorem and Peirce law. Next, we argue that our Deduction Theorem scema for enlargable multiple-conclusionsequent calculi with structural rules found earlier respects Peirce Law.As a consequence, we provide a natural and quite useful semanticsof such calculi. Finally, we explore the issues involved within the context of the Weak Contraposition extensions of so-called contraposable propositional calculi of the mentioned kind.After all, we successfully apply our generic elaborationto both certain sentential logics and varieties of algebras, providing constructive and quite transparent proofsof Deduction Theorem with Peirce Law for the formers as well asimplicativity (or, at least, restricted equational definabilityof principal congruences) for the latters.