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Negative Predication and Distinctness

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Abstract

It is argued that the intuitionistic conception of negation as implication of absurdity is inadequate for the proof-theoretic semantic analysis of negative predication and distinctness. Instead, it is suggested to construe negative predication proof-theoretically as subatomic derivation failure, and to define distinctness—understood as a qualified notion—by appeal to negative predication. This proposal is elaborated in terms of intuitionistic bipredicational subatomic natural deduction systems. It is shown that derivations in these systems normalize and that normal derivations have the subexpression (incl. subformula) property. A proof-theoretic semantics for negative predication and distinctness is defined, and an intuitionistic conception of truth based on canonical derivations is proposed. An application to the doctrine of the Trinity, to which negative predication and distinctness are central, illustrates the systems and their proof-theoretic semantics.

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Notes

  1. I would like to thank Peter Schroeder-Heister for pointing this out to me.

  2. In this section we shall, for exegetical reasons, stick to the three classical languages of Western academic Christian theology. We shall use the New International Version (NIV) translation of the Bible and the translation of the dogmatic source texts offered in [7] (=DH). A prominent passage in the New Testament is Matthew 28:19 where the exhortation is made to baptize                [in the name of the Father and of the Son and of the Holy Spirit].

  3. For the Roman Catholic tradition see, in particular, the Apostolic Creed (2nd century; DH 30), the Quicumque (6th century; also known as the “Athanasian” Creed; DH 75-76), the Niceno-Constantinopolitan Creed (325, 381; DH 125, 150), the Eleventh Council of Toledo (675; DH 525-532), the Fourth Council of the Lateran (1215; DH 800, 803-806), the Council of Florence (1442; DH 1330-1333). The Lutheran tradition acknowledges all of the above creeds (BSLK [9]: 37-64). The Orthodox tradition uses the Apostolic Creed and the Niceno-Constantinopolitan Creed without the Filioque.

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Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft (DFG), grant WI 3456/4-2.

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  1. Institut für Philosophie, Goethe-Universität Frankfurt am Main, Norbert-Wollheim-Platz 1, 60629, Frankfurt am Main, Germany

    Bartosz Więckowski

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Więckowski, B. Negative Predication and Distinctness. Log. Univers. 17, 103–138 (2023). https://doi.org/10.1007/s11787-022-00321-9

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