Décio Krause

Is there vagueness in the world? This is the central question that we are concerned with. Focusing on identity statements around which much of the recent debate has centred, we argue that `vague identity' arises in quantum mechanics in one of two ways. First, quantum particles may be described as individuals, with `entangled' states understood in terms of non-supervenient relations. In this case, the vagueness is ontic but exists at the level of these relations which act as a kind of `veil'. Secondly, the particles can be regarded as non-individuals, where this is understood as a lack of self-identity and given formal expression in terms of quasi-set theory. Here we have ontic vagueness at perhaps the most basic metaphysical level. Our conclusion is that there is genuine vagueness `in the world' but how it is understood depends on the metaphysical package adopted.

Paraconsistent logics are logics that can be used to base inconsistent but non-trivial systems. In paraconsistent set theories, we can quan- tify over sets that in standard set theories, if consistent, would lead to contradictions, such as the Russell set, R = fx : x =2 xg. Quasi-set theories are mathematical systems built for dealing with collections of indiscernible elements. The basic motivation for the development of quasi-set theories came from quantum physics, where indiscernible entities need to be considered. Usually, the way of dealing with indiscernible objects within clas- sical logic and mathematics is by restricting them to certain structures, in a way so that the relations and functions of the structure are not sufficient to individuate the objects; in other words, such structures are not rigid. In quantum physics, this idea appears when symmetry conditions are introduced, say by choosing symmetric and anti-symmetric functions in the relevant Hilbert spaces. But in standard mathematics, such as that built in Zermelo-Fraenkel set theory, any structure can be extended to a rigid structure. That means that, although we can deal with certain objects as they were indiscernible, we realize that from out- side of these structures these objects are no more indiscernible, for they can be individualized in the extended rigid structures: ZF is a theory of individuals, distinguishable objects. In quasi-set theory, it seems that there are structures that cannot be extended to rigid ones, so it seems that they provide a natural mathematical framework for expressing quantum facts without certain symmetry suppositions. There may be situations, however, in which we may need to deal with inconsistent bits of infor- mation in a quantum context, even if these informations are concerned with ways of speech. Furthermore, some authors think that superposi- tions may be understood in terms of paraconsistent logics, and even the notion of complementarity was already treated by such a means. This is, apparently, a nice motivation to try to merge these two frameworks. In this work, we develop the technical details, by basing our quasi-set theory in the paraconsistent system C1. We also elaborate a new hierarchy of paraconsistent calculi, the paraconsistent calculi with indiscernibility. For the finalities of this work, some philosophical questions are outlined, but this topic is left to a future work.

This paper is the sequel of a previous one where we have introduced a paraconsistent logic termed paraclassical logic to deal with 'complementary propositions'. Here, we enlarge upon the discussion by considering certain 'meaning principles', which sanction either some restrictions of 'classical' procedures or the utilization of certain 'classical' incompatible schemes in the domain of the physical theories. Here, the term 'classical' refers to classical physics. Some general comments on the logical basis of a scientific theory are also put in between the text, motivated by the discussion of complementarity.

In this paper we try to justify our way of looking for an alternative approach to quantum mechanics, which is based on a non-classical logic. We consider two specific questions related to quantum theory, namely, entanglement and the indiscernibility of quanta. We characterize individuals, and then explain in what sense entanglement is a concept which can be applied to individuals in a restricted sense only. Then, we turn to indiscernibility and, after realizing that this concept is of a fundamental importance, we mention the ‘traditional’ theory of identity of standard logic and mathematics, which underly the basic formalism of quantum theory. Then we propose to call the Problem of Identity the question whether identity of objects can be justified, and under what conditions. As in the Hume’s celebrated Problem of Induction, we conclude that the attribution of transtemporal identity to an object has no logic justification, and must be considered as a metaphysical hypothesis. Numerical identity is also put aside for similar reasons. Then we guess that identity is just an useful concept, but which in certain fields, mainly in the quantum realm, could be substituted by a weaker concept of indiscernibility. This assumption motivates us to look for an interpretation of quantum mechanics based on a non-classical logic, termed non-reflexive, and the corresponding mechanics is called non-reflexive quantum mechanics.

Discutimos algumas quest~oes relacionadas a uma possvel teoriade quase-objetos, isto e, objetos para os quais as condic~oes usuais de identidaden~ao se aplicariam. Desde que a principal motivac~ao para uma tal teoriae a mec^anica qu^antica, a discuss~ao e realizada tendo-se essa intepretac~ao pretendidaem mente.We discuss some questions related to a possible theory of quasiobjects,that is, objects for which the standard identity conditions should nothold. Since the main motivation for such a theory is quantum mechanics,the discussion is posed with this intended interpretation in mind.