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


  Abstract

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 isin X. The scalar product <v,w> isin [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 Lukasiewicz 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 Lukasiewicz logic; many-valued logic; compact logic; Lukasiewicz calculus; skolemization; Skolem normal form; Hilbert space; Kolmogorov dilation; positive definite kernel; reproducing kernel; (auto)correlation matrix; positive semidefinite matrix; Tarski-Seidenberg decision method


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