Deduction-Detachment Theorem and Gentzen-Style Deductive Systems

In Janusz Czelakowski (ed.), Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science. Springer Verlag (2018)

Abstract
Logical implication is an attempt to catch the essence of causeeffect relationships of the real world in the context of formal deductive systems. The Deduction-Detachment Theorem being, in its turn, a statement about essential logical properties of classical implication, was therefore of great interest for logicians. Although a statement about a Hilbert-style deductive system, DDT can be formulated by means of Gentzen-style rules, and as such seems to be a statement about the metatheoretical properties of Hilbert-style deductive systems. As is often the case with metatheoretical properties, DDT leaves the question about its meaning and scope a bit vague or at least requires a higher order abstraction layer to formalize them. Closure relations, discussed in this paper, present a convenient context to give a precise formal statement of the DDT and its connection with Hilbert- and pertinent Gentzen-style deductive systems.
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DOI 10.1007/978-3-319-74772-9_3
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