Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-11T00:29:47.257Z Has data issue: false hasContentIssue false
  • English
  • Français

The metatheory of the classical propositional calculus is not axiomatizable

Published online by Cambridge University Press:  12 March 2014

Ian Mason
Ian Mason*
Affiliation:
Department of Philosophy, Stanford University, Stanford, California 94305
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we investigate the first order metatheory of the classical propositional logic. In the first section we prove that the first order metatheory of the classical propositional logic is undecidable. Thus as a mathematical object even the simplest of logics is, from a logical standpoint, quite complex. In fact it is of the same complexity as true first order number theory.

This result answers negatively a question of J. F. A. K. van Benthem (see [van Benthem and Doets 1983]) as to whether the interpolation theorem in some sense completes the metatheory of the calculus. Let us begin by motivating the question that we answer. In [van Benthem and Doets 1983] it is claimed that a folklore prejudice has it that interpolation was the final elementary property of first order logic to be discovered. Even though other properties of the propositional calculus have been discovered since Craig's orginal paper [Craig 1957] (see for example [Reznikoff 1965]) there is a lot of evidence for the fundamental nature of the property. In abstract model theory for example one finds that very few logics have the interpolation property. There are two well-known open problems in this area. These are

1. Is there a logic satisfying the full compactness theorem as well as the interpolation theorem that is not equivalent to first order logic even for finite models?

2. Is there a logic stronger than L(Q), the logic with the quantifierthere exist uncountably many, that is countably compact and has the interpolation property?

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 2 , June 1985 , pp. 451 - 457
Copyright
Copyright © Association for Symbolic Logic 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Barwise, J. and Feferman, S., Model theoretic languages, Springer-Verlag, Berlin (to appear).Google Scholar
Craig, W., Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory, this Journal, vol. 22 (1957), pp. 269–285.Google Scholar
Feferman, S. and Vaught, R., The first order properties of products of algebraic systems, Fundamenta Mathematicae, vol. 47 (1959), pp. 57–103.CrossRefGoogle Scholar
Friedman, H., One hundred and two problems in mathematical logic, this Journal, vol. 40 (1975), pp. 113–129.Google Scholar
Makowski, J. A., Shelah, S. and Stavi, J., ⊿-logics and generalized quantifiers, Annals of Mathematical Logic, vol. 10 (1976), pp. 155–192.CrossRefGoogle Scholar
Mason, I., Theundecidabilityof the metatheories of propositional calculi, Abstracts of Papers Presented to the American Mathematical Society, vol. 5 (1984), p. 130 (abstract 84T-03-74).Google Scholar
Mundici, D., An algebraic result about soft model theoretical equivalence relations with an application to H. Friedman's fourth problem, this Journal, vol. 46 (1981), pp. 523–530.Google Scholar
Rabin, M., A simple method for undecidability proofs and some applications, Logic, methodology and philosophy of science. II (Bar-Hillel, Y., editor), North-Holland, Amsterdam, 1965, pp. 55–68.Google Scholar
Reznikoff, I., Tout ensemble de formules de la logique classique est équivalent à un ensemble indépendant, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciénces, vol. 260 (1965), pp. 2385–2388.Google Scholar
Skolem, T., Untersuchungen über die Axiome des Klassenkalküls and über Produktations- und Summationsprobleme, welche gewisse Klassen von Aussagen betreffen, Skrifter utgit av Videnskapssel-skapet i Kristiania, I: Matematisk-Naturvidenskabelig Klasse, 1919, no. 3.Google Scholar
van Benthem, J. F. A. K. and Doets, H., Higher-order logic, Handbook of philosophical logic (Gabbay, D. and Guethner, F., editors), vol. 1, Reidel, Dordrecht, 1983, pp. 275–329.CrossRefGoogle Scholar