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Condensed detachment as a rule of inference

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Abstract

Condensed detachment is usually regarded as a notation, and “defined” by example. In this paper it is regarded as a rule of inference, and rigorously defined with the help of the Unification Theorem of J. A. Robinson. Historically, however, the invention of condensed detachment by C. A. Meredith preceded Robinson's studies of unification. It is argued that Meredith's ideas deserve recognition in the history of unification, and the possibility that Meredith was influenced, through Łukasiewicz, by ideas of Tarski going back at least to 1939, and possibly to 1930 or earlier, is discussed. It is proved that a term is derivable by substitution and ordinary detachment from given axioms if and only if it is a substitution instance of a term which is derivable from these axioms by condensed detachment, and it is shown how this theorem enables the ideas of Łukasiewicz and Tarski mentioned above to be formalized and extended. Finally, it is shown how condensed detachment may be subsumed within the resolution principle of J. A. Robinson, and several computer studies of particular Hilbert-type propositional calculi using programs based on condensed detachment or on resolution are briefly discussed.

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  1. University of Auckland, Auckland, New Zealand

    J. A. Kalman

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Kalman, J.A. Condensed detachment as a rule of inference. Stud Logica 42, 443–451 (1983). https://doi.org/10.1007/BF01371632

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