Non-Reflexive Logics and its Applications to the Foundations of Quantum Mechanics
Newton da Costa has studied the foundations of quantum mechanics, and of quantum theories in general, since a long time ago. Motivated by Erwin Schrödinger, who sustained that the notion of identity would not be appliable to quantum objects, he developed a first-order system where the expression a = b is not a formula for some objects denoted by a and b in order to grasp Schrödinger’s intuitions from a logical point of view. These logics, where the standard notion of identity is questioned, where later termed non-reflexive. Then, reasoning as a legitimate logician, da Costa guessed that a suitable semantics for his system would not be formulated within a standard set theory, where the notion of identity holds for all objects, proposing that a theory of quasi-sets should be developed to cope with such a semantics. The present author took this idea seriously, and extended da Costa’s system to higher-order logics (simple theory of types) and developed quasi-set theory, providing a semantics (build in such a mathematical basis) for both the first and the higher-order systems. Quasi-set theory was then applied in several aspects of the quantum realm, and is still today a widespread field of investigation here and abroad. In this talk we revise da Costa’s original account and present some of his motivations for developing a non-reflexive interpretation of quantum mechanics, qualifying this term and providing the main related ideas.