Description
Volume II of "Lectures On Mathematical Logic" focuses on the observation that in everyday arguments, statements are linguistically transformed and connected in formal ways, regardless of their content. This understanding leads to the development of Gentzen's calculi of derivations, which are presented for positive, minimal, intuitionist, and classical logic. Each of these calculi produces a lattice-like ordered structure by identifying interdeducible formulas. The generation of filters in these structures leads to corresponding modus ponens calculi, which are semantically complete and express the algorithms for generating semantical consequences. The operators that transform derivations between different calculi are also studied, as well as operators that eliminate defined predicate and function symbols.
In this volume, logic starts from the observation that in everyday arguments, as brought forward say by a lawyer, statements are transformed linguistically, connecting them in formal ways irrespective of their contents. Understanding such arguments as deductive situations, or sequents in the technical terminology, the transformations between them can be expressed as logical rules. This leads to Gentzen's calculi of derivations, presented first for positive logic and then, depending on the requirements made on the behaviour of negation, for minimal, intuitionist and classical logic. Identifying interdeducible formulas, each of these calculi gives rise to a lattice-like ordered structure. Describing the generation of filters in these structures leads to corresponding modus ponens calculi, and these turn out to be semantically complete because they express the algorithms generating semantical consequences, as obtained in Volume One of these lectures. The operators transforming derivations from one type of calculus into the other are also studied with respect to changes of the lengths of derivations, and operators eliminating defined predicate and function symbols are described expli
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