The axiomatization of Horst Wessel’s strict logical consequence relation

Andrzej Pietruszczak

DOI: http://dx.doi.org/10.12775/LLP.2004.007

Abstract

In his book from 1984 Horst Wessel presents the system of strict logical consequence Fs (see also (Wessel, 1979)). The author maintained that this system axiomatized the relation |=s of strict logical consequence between formulas of Classical Propositional Calculi (CPC). Let |= be the classical consequence relation in CPC. The relation |=s is defined as follows: \phi |=s \psi iff \phi |= \psi, every variable from \psi occurs in \phi and neither \phi is a contradiction nor \psi is a tautology. Clearly, if \phi |=s \psi, then neither \phi is a tautology nor \psi is a contradiction.

Intuitions connected with the relation |=s were presented in (Wessel, 1984). The analysis of the relation |=s is also carried out in (Pietruszczak, 2004). In the present paper we will show that the system Fs is not a complete axiomatization of the relation |=s. Moreover, we will present the system VF s that is an «extension to completeness» of the Fs.

Keywords

Horst Wessel’s strict logical consequence relation

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References

Asser, G., 1959, Einführung in die matematische Logik, Teil I, Leipzig.

Pietruszczak, A., 1997, “Aksjomatyzacja relacji ścisłego wynikania logicznego Horsta Wessla”, pp. 281–297 in: Byt, Logos, Matematyka. FLFL 1995, J. Perzanowski and A. Pietruszczak (eds.), The NCU Press, Toruń.

Pietruszczak, A., 1998, “Zur Axiomatisierungder strikten logischen FolgebeziehungHorst Wessels”, pp. 215–228 in: Terminigebrauch und Folgebeziehung, U. Scheffler and K. Wuttich (eds.), Logische Philosophie, Bd. 1, Logos Verlag Berlin.

Pietruszczak, A., 2004, “The consequence relation preserving logical information”, this volume, pp. 89–120.

Wessel, H., 1979, “Ein System der strikten logischen Folgebeziehung”, in: “Begriffsschrift”. Jenaer Frege-Konferenz, Wissenschaftliche Beiträge der Friedrich-Schiller-Universität Jena.

Wessel, H., 1984, Logik, Deutsche Verlag der Wissenschaften Berlin.

Wessel, H., 1999, Logik, Logische Philosophie, Bd. 2, Logos Verlag Berlin.

Financed by MNiSW on the basis of agreement no. 706/P-DUN/2018 (dated 10/05/18). Project 1: “Preparation of articles in English for eight editions of the journal Logic and Logical Philosophy over the period 2018–19; Vol. 27, No. 1–4 (2018), Vol. 28, No. 1–4 (2019)”; amount from the DUN grant: 64800 zł. Project 4: “Digitalisation of eight editions of the journal Logic and Logical Philosophy over the period 2018-19; Vol. 27, No. 1–4 (2018), Vol. 28, No. 1–4 (2019)”; amount from the DUN grant: 18600 zł.

ISSN: 1425-3305 (print version)
ISSN: 2300-9802 (electronic version)