The Uncertainty Relation for Quantum propositions

TL;DR

This paper explores an uncertainty principle in fuzzy logic propositions with probabilistic truth values, drawing parallels to quantum mechanics, and introduces quantum coherent propositions and cat state propositions.

Contribution

It establishes a logical uncertainty relation analogous to quantum physics and defines quantum coherent and cat state propositions within fuzzy logic.

Findings

Uncertainty relation applies to fuzzy propositions with probabilistic truth values.

Quantum coherent propositions minimize the logical uncertainty relation.

Only one type of compound quantum-coherent propositions exists: the cat state propositions.

Abstract

Logical propositions with the fuzzy modality "Probably" are shown to obey an uncertainty principle very similar to that of Quantum Optics. In the case of such propositions, the partial truth values are in fact probabilities. The corresponding assertions in the metalanguage, have complex assertion degrees which can be interpreted as probability amplitudes. In the logical case, the uncertainty relation is about the assertion degree, which plays the role of the phase, and the total number of atomic propositions, which plays the role of the number of modes. In analogy with coherent states in quantum physics, we define "quantum coherent propositions" those which minimize the above logical uncertainty relation. Finally, we show that there is only one kind of compound quantum-coherent propositions: the "cat state" propositions.