Journal of Logic and Computation

Journal of Logic and Computation Advance Access published online on March 13, 2009
Journal of Logic and Computation, doi:10.1093/logcom/exp017

This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions Google Scholar Articles by Mundici, D. Search for Related Content Social Bookmarking          What's this?

© The Author, 2009. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Papers

A Compact [0,1]-valued First-order ukasiewicz Logic with Identity on Hilbert Space

Daniele Mundici

Department of Mathematics, ‘Ulisse Dini’, University of Florence, Viale Morgagni 67/A, I-50134 Florence, Italy.
E-mail: mundici{at}math.unifi.it

Received 28 August 2008.

By an MV-set, we understand a pair (E,X) where X is a set of unit vectors in a Hilbert space E such that the linear span of X is dense in E, and v,w 0 for all v,w X. The scalar product v,w [0,1] is the identity degree of v and w. Building on MV-sets and continuous functions and relations defined on them, we construct a compact [0,1]-valued first-order ukasiewicz logic, whose set of unsatisfiable formulas is recursively enumerable. In the particular case when X is an orthonormal basis of E we recover classical Skolem first-order logic with identity, constants, functions and relations. Our main tools are the Kolmogorov dilation theorem for positive semidefinite kernels, and the Tarski–Seidenberg decision method for elementary algebra and geometry.

Keywords: First-order ukasiewicz logic; many-valued logic; compact logic; ukasiewicz calculus; skolemization; Skolem normal form; Hilbert space; Kolmogorov dilation; positive definite kernel; reproducing kernel; (auto)correlation matrix; positive semidefinite matrix; Tarski-Seidenberg decision method

References

1. Baaz M, Metcalfe G. Herbrand theorem, skolemization and proof systems for ukasiewicz logic. Journal of Logic and Computation. [Epub ahead of print, doi:10.1093/logcom/exn059, November 17, 2008].
2. Belluce LP, Chang CC. A weak completeness theorem for infinite valued first-order logic. Journal of Symbolic Logic (1963) 28:43–50.[CrossRef]
3. Berg C, Christensen JPR, Ressel P. Harmonic analysis on semigroups. In: Graduate Texts in Mathematics (1984) 100. Springer.
4. Chang CC, Keisler HJ. Model Theory. In: Studies in Logic and the Foundations of Mathematics (1973) 73. Amsterdam: North-Holland.
5. Cignoli R, D’Ottaviano IML, Mundici D. Algebraic foundations of many-valued reasoning. In: Trends in Logic (2000) 7. Kluwer.
6. Hájek P. Metamathematics of fuzzy logic. In: Trends in Logic (1998) 4. Kluwer.
7. Hájek P. Function symbols in fuzzy predicate logic. In: Proceedings East-West Fuzzy Colloquium 2000 (2000) IPM. 2–8.
8. Hájek P. Arithmetical complexity of fuzzy predicate logics—a survey. In: Soft Computing (2005) 9:935–941.[CrossRef][Web of Science]
9. Hirsch F, Lacombe G. Elements of functional analysis. In: Graduate Texts in Mathematics (1999) 192. Springer.
10. Horn RA, Johnson CR. Matrix Analysis (1985) Cambridge University Press. (Reprinted, 1990).
11. ukasiewicz J, Tarski A. Investigations into the sentential calculus. In: Logic, Semantics, Metamathematics—Tarski A, ed. (1956) Clarendon Press. 38–59. (Reprinted, Hackett, 1983).
12. Mishra B. Algorithmic Algebra. In: Texts and Monographs in Computer Science (1993) Springer.
13. Mostowski A. Axiomatizability of some many valued predicate calculi. Fundamenta Mathematicae (1957) 44:165–190.

CiteULike    Connotea    Del.icio.us    What's this?

This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions Google Scholar Articles by Mundici, D. Search for Related Content Social Bookmarking          What's this?