# Natural Deduction for the Sheffer Stroke and Peirce’s Arrow (and any Other Truth-Functional Connective)

## Abstract

Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke (nand) and Peirce’s arrow (nor). The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic versions of the connectives in question.

This is a preview of subscription content, access via your institution.

## Notes

1. The general elimination rules coincide with the elimination rules of Parigot’s [12] free deduction; he showed how to obtain the standard elimination rules from them by systematic simplification. A version of the simplified system of classical multiple-conclusion natural deduction was studied by Parigot [13]. The general elimination rules for ∧ and → were studied by von Plato [14].

2. The use of the special symbol ⊥ is not strictly necessary, as it could be replaced by explicitly contradictory premises (e.g., C|D, C, and D; or more simply C|C and C) when it appears as a premise, and by an arbitrary formula when it appears as the conclusion. This has the advantage that the resulting rules mention no symbols other than |, i.e., are pure. Indeed, this was in part Hazen and Pelletier’s goal and their rules used this approach. This alternative approach can however not be used if no such “explicit contradiction” can be expressed. See also Section 7 below.

## References

1. Baaz, M., & Fermüller, C.G. (1996). Intuitionistic counterparts of finitely-valued logics. In Proceedings of 26rd international symposium on multiple-valued logic (pp. 136–141). Los Alamitos: IEEE Press. doi:10.1109/ISMVL.1996.508349.

2. Baaz, M., Fermüller, C.G., & Zach, R. (1993). Systematic construction of natural deduction systems for many-valued logics. In Proceedings of 23rd international symposium on multiple-valued logic (208–213). Los Alamitos: IEEE Press. doi:10.1109/ISMVL.1993.289558.

3. Baaz, M., Fermüller, C.G., & Zach, R. (1994). Elimination of cuts in first-order finite-valued logics. Journal of Information Processing and Cybernetics EIK, 29(6), 333–355. http://ucalgary.ca/rzach/papers/mvlcutel.html.

4. Fitch, F. (1952). Symbolic Logic: An introduction. New York: Ronald.

5. Gentzen, G. (1934). Untersuchungen über das logische Schließen I–II. Math Z, 39, 176–210,405–431.

6. Harrop, R. (1960). Concerning formulas of the types ABC, A → (E x)B(x). Journal of Symbolic Logic, 25(1), 27–32. http://www.jstor.org/stable/2964334.

7. Hazen, A.P., & Pelletier, F.J. (2014). Gentzen and Jaśkowski natural deduction: fundamentally similar but importantly different. Studia Logica, 102(6), 1103–1142. doi:10.1007/s11225-014-9564-1.

8. van Heijenoort, J. (Ed.) (1967). From Frege to Gödel. A source book in mathematical logic (1879–1931). Cambridge: Harvard University Press.

9. Jaśkowski, S. (1934). On the rules of suppositions in formal logic. No. 1 in Studia Logica, Seminarjum Filozoficzne. Wydz. Matematyczno-Przyrodniczy UW, Warsaw, reprinted in (McCall 1967, 232–258).

10. Lemmon, E.J. (1965). Beginning logic. London: Nelson.

11. McCall, S. (Ed.) (1967). Polish logic 1920–1939. London: Oxford University Press.

12. Parigot, M. (1992a). Free deduction: an analysis of “computations” in classical logic. In A. Voronkov (Ed.), Logic programming, Lecture Notes in Computer Science (Vol. 592 pp. 361–380). Berlin: Springer. doi:10.1007/3-540-55460-2_27.

13. Parigot, M. (1992b). λ μ-Calculus: an algorithmic interpretation of classical natural deduction. In Proceedings LPAR’92 logic programming and automated reasoning, LNAI 624 (pp. 190–201). Berlin: Springer. doi:10.1007/BFb0013061.

14. von Plato, J. (2001). Natural deduction with general elimination rules. Archive for Mathematical Logic, 40(7), 541–567. doi:10.1007/s001530100091.

15. Prawitz, D. (1965). Natural deduction. Stockholm studies in philosophy 3. Stockholm: Almqvist & Wiksell.

16. Price, R. (1961). The stroke function and natural deduction. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 7, 117–123.

17. Read, S. (1999). Sheffer’s stroke: a study in proof-theoretic harmony. Danish Yearbook of Philosophy, 34, 7–23.

18. Schönfinkel, M. (1924). Über die Bausteine der mathematischen logik. Mathematische Annalen, 92, 305–316. English translation in (van Heijenoort 1967, pp. 355–366).

19. Suppes, P. (1957). Introduction to logic. New York: Van Nostrand Reinhold.

20. Takeuti, G. (1987). Proof theory, 2nd edn. Studies in logic 81. Amsterdam: North-Holland.

21. Troelstra, A.S., & Schwichtenberg, H. (2000). Basic proof theory, 2nd edn. Cambridge: Cambridge University Press.

22. Zach, R. (1993). Proof theory of finite-valued logics. Diplomarbeit, Technische Universität Wien, Vienna. http://ucalgary.ca/rzach/papers/ptmvl.html.

## Acknowledgments

I am grateful to Allen Hazen and Jeff Pelletier for helpful comments as well as the question which originally prompted this paper.

## Author information

Authors

### Corresponding author

Correspondence to Richard Zach.

## Rights and permissions

Reprints and Permissions

Zach, R. Natural Deduction for the Sheffer Stroke and Peirce’s Arrow (and any Other Truth-Functional Connective). J Philos Logic 45, 183–197 (2016). https://doi.org/10.1007/s10992-015-9370-x

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1007/s10992-015-9370-x

### Keywords

• Natural deduction
• Sequent calculus
• Sheffer stroke