Erkenntnis

, Volume 78, Issue 2, pp 399–419 | Cite as

Recovering Quantum Logic Within an Extended Classical Framework

Original Paper

Abstract

We present a procedure which allows us to recover classical and nonclassical logical structures as concrete logics associated with physical theories expressed by means of classical languages. This procedure consists in choosing, for a given theory \({{\mathcal{T}}}\) and classical language \({{\fancyscript{L}}}\) expressing \({{\mathcal{T}}, }\) an observative sublanguage L of \({{\fancyscript{L}}}\) with a notion of truth as correspondence, introducing in L a derived and theory-dependent notion of C-truth (true with certainty), defining a physical preorder\(\prec\) induced by C-truth, and finally selecting a set of sentences ϕV that are verifiable (or testable) according to \({{\mathcal{T}}, }\) on which a weak complementation is induced by \({{\mathcal{T}}. }\) The triple \((\phi_{V},\prec,^{\perp})\) is then the desired concrete logic. By applying this procedure we recover a classical logic and a standard quantum logic as concrete logics associated with classical and quantum mechanics, respectively. The latter result is obtained in a purely formal way, but it can be provided with a physical meaning by adopting a recent interpretation of quantum mechanics that reinterprets quantum probabilities as conditional on detection rather than absolute. Hence quantum logic can be considered as a mathematical structure formalizing the properties of the notion of verification in quantum physics. This conclusion supports the general idea that some nonclassical logics can coexist without conflicting with classical logic (global pluralism) because they formalize metalinguistic notions that do not coincide with the notion of truth as correspondence but are not alternative to it either.

Notes

Acknowledgments

The authors are greatly indebted with Prof. Carlo Dalla Pozza and Dr. Marco Persano for reading the manuscript and providing useful remarks and suggestions.

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Dipartimento di Fisicadell’Università del Salento and Sezione INFNLecceItaly
  2. 2.Center Leo Apostel (CLEA)Vrije Universiteit Brussel (VUB)BrusselsBelgium

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