Abstract
It is shown that the implicational fragment of Anderson and Belnap's R, i.e. Church's weak implicational calculus, is not uniquely characterized by MP (modus ponens), US (uniform substitution), and WDT (Church's weak deduction theorem). It is also shown that no unique logic is characterized by these, but that the addition of further rules results in the implicational fragment of R. A similar result for E is mentioned.
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References
A. Church, The weak theory of implication, in A. Menne, A. Wilhelmy and H. Angstl, Kontrolliertes Denken, München, Karl Alber, 1951, pp. 22–37.
A. R. Anderson and N. D. Belnap, Jr., Entailment, vol. I, Princeton University Press, 1975.
D. Batens and J. P. Van Bendegem, Relevant derivability and classical derivability in Fitch-style and axiomatic formulations of relevant logics, Logique et Analyse 109 (1985), pp. 22–31.
M. R. Diaz, Topics in the Logic of Relevance, München, Philosophia Verlag, 1981.
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I am grateful to the referees; their comments enabled me to correct a mistake and to clarify several passages.
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Batens, D. Relevant implication and the weak deduction theorem. Stud Logica 46, 239–245 (1987). https://doi.org/10.1007/BF00372548
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DOI: https://doi.org/10.1007/BF00372548