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A compact \([0,1]\)-valued first-order Łukasiewicz logic with identity on Hilbert space. (English) Zbl 1220.03010
Author’s 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 \(\langle v,w\rangle \geq 0\) for all \(v,w \in X\). The scalar product \(\langle v,w\rangle \in [0,1]\) 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 Łukasiewicz 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.”

MSC:
03B50 Many-valued logic
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