Abstract
Formal calculi of deduction have proved useful in logic and in the foundations of mathematics, as well as in metamathematics. Examples of some of these uses are:
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(a)
The use of formal calculi in attempts to give a secure foundation for mathematics, as in the original work of Frege.
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(b)
To generate syntactically an already given semantical consequence relation, e.g. in some branches of technical modal logic.
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(c)
Formal calculi can serve as heuristic devices for finding metamathematical properties of the consequence relation, as was the case, e.g., in the early development of infinitary logic via the use of cut-free Gentzen sequent calculi.
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(d)
Formal calculi have served as the objects of mathematical study, as in traditional work on Hilbert’s consistency programme.
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(e)
Certain versions of formal calculi have been used in attempts to formulate philosophical insights into the nature of reasoning.
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Sundholm, G. (1983). Systems of Deduction. In: Gabbay, D., Guenthner, F. (eds) Handbook of Philosophical Logic. Synthese Library, vol 164. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7066-3_2
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