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On a New Concept of Truth in Empirical Theories

Jiří Souček

Published Feb 1, 2016• 0 citations
OpenAlex

Abstract

We introduce a new concept of truth in empirical theories based on the experience in the study of quantum mechanics. In this concept a statement can be true, false or undecidable.

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In the proposed non-Aristotelian finitary logic (NAFL), truths for formal propositions can exist only with respect to axiomatic theories, essentially as temporary axiomatic declarations in the human mind. An undecidable proposition P in a consistent NAFL theory T is true/false with respect to T if and only if it has been axiomatically declared as true/false by virtue of its provability/refutability in an interpretation T* of T. In the absence of any such axiomatic declarations, P is in a superposed state of 'neither true nor false' and the consistency of T requires the existence of a non-classical model for T in which P&¬P is the case. Here T* is an axiomatic NAFL theory that, like T, resides in the human mind and acts as the 'truth-maker' for (a model of) T. Quantum superposition is justified by identifying 'axiomatic declarations' for the truth/falsity of P (by virtue of its provability/refutability in T*) with 'measurement' in the real world. NAFL also explains and de-mystifies the phenomenon of entanglement. NAFL severely restricts classical infinitary reasoning, but possibly provides sufficient machinery for a consistent axiomatization of quantum mechanics.
In the present paper, the decision problem of the Schrödinger equation (asking whether or not a given Hamiltonian operator has the nonempty solution set) is represented as a logical statement. As it is shown in the paper, the law of excluded middle would be applicable to the introduced statement if and only if quantum fundamentalism (asserting that everything in the universe is ultimately describable in quantum-mechanical terms) held. But, since the decision problem of the Schrödinger equation is in general undecidable, such a statement is allowed to be other than true or false, explicitly, it may fail to have truth values at all. This makes possible to abandon the law of excluded middle together with quantum fundamentalism in the proposed intuitionistic interpretation of quantum mechanics.
The problem is posed of establishing a possible relationship between a new type of Multi-verse representation, Gödel undecidability theorems and the logic of classical, quantum mechanics and quantum gravity. For this purpose example cases of multi-verses are first discussed in the context of non-relativistic classical, quantum mechanics and quantum gravity. As a result, it is confirmed that thanks to Gödel theorems non-relativistic classical and quantum mechanics, as well as quantum gravity theory are incomplete. As a consequence, they necessarily admit undecidable logical propositions and therefore obey a three-way boolean logical, i.e., a propositional logic with the three different logical truth values true, false and undecidable.