The Probabilistic Argument for a Nonclassical Logic of Quantum Mechanics

Patrick Suppes

Patrick Suppes

The aim of this paper is to state the single most powerful argument for use of a non-classical logic in quantum mechanics. In outline the argument is the following. The working logic of a science is the logic of the events and propositions to which probabilities are assigned. A probability should be assigned to every element of the algebra of events. In the case of quantum mechanics probabilities may be assigned to events but not, without restriction, to the conjunction of two events. The conclusion is that the working logic of quantum mechanics is not classical. The nature of the logic that is appropriate for quantum mechanics is examined.

S?awomir Bugajski

Stanley Gudder

Weidong Tang

Hardy-like proofs for some quantum mechanical properties, such as Hardy-like nonlocality and Hardy-like contextuality, provide us two probabilistic no-go theorems by logical contradictions. Inspired by this, one may conjure that there are some other nonclassical properties which can also be revealed by Hardy-like arguments. In this work, we present an unconventional kind of quantum pigeonhole effects using Hardy-like arguments and show how to systematically construct them. Besides, we introduce a kind of ``projected-stabilizer state, say projected-coloring graph states, associated with a special type of coloring graphs called projected-coloring graph states. We build a one-to-one connection between a given projected-coloring graph (state) and a Hardy-like quantum pigeonhole In this sense a Hardy-like quantum pigeonhole paradox may also be referred to as a ``quantum coloring paradox.

María Luisa Dalla Chiara, Roberto Giuntini