Décio Krause

In this paper we consider the notions of structure and models within the semantic approach to theories. To highlight the role of the mathematics used to build the structures which will be taken as the models of theories, we review the notion of mathematical structure and of the models of scientific theories. Then, we analyse a case-study and argue that if a certain metaphysical view of quantum objects is adopted, one seeing them as non-individuals, then there would be strong reasons to ask for a different mathematical framework for describing the structures that would be the models of the corresponding theory. In departing from the standard frameworks (those worked on within standard mathematics), we hope to bring to the scene, within the scope of the semantic approach, the importance of paying attention to some fundamental concepts usually only superficially touched by philosophers of science (if touched).

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.

Some of the forerunners of quantum theory regarded the basic entities of such theories as 'non-individuals'. One of the problems is to treat collections of such 'things', for they do not obey the axioms of standard set theories like Zermelo- Fraenkel. In this paper, collections of objects to which the standard concept of identity does not apply are termed 'quasi-sets'. The motivation for such a theory, linked to what we call 'the Manin problem', is presented, so as its specific axioms. At the end, it is shown how quantum statistics can be obtained within quasi-set thbeory.

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.

Recently, in the debate about the ontology of quantum mechanics some authors have defended the view that quantum particles are individuals in a primitive sense, so that individuality should be preferred over non-individuality (the alternative option). Primitive individuality involves two main claims: (1) every item is identical with itself and (2) it is distinct from every other item. Non-relativistic quantum mechanics is said to provide positive evidence for that position, since in every situation comprising multiple particles there is a well-defined number of them to begin with, and so they must be distinct from each other. We argue that the link between a well-defined number of items and the relation of identity that is being claimed to hold is not imposed by quantum mechanics, but rather by a metaphysical view. Formal evidence is advanced in favor of the thesis that counting may be performed for items without identity (non-individuals), so that quantum mechanics needs not be viewed as endorsing an ontology of individuals

In this paper I consider some logical and mathematical aspects of the discussion of the identity and individuality of quantum entities. I shall point out that for some aspects of the discussion, the logical basis cannot be put aside; on the contrary, it leads us to unavoidable conclusions which may have consequences in how we articulate certain concepts related to quantum theory. Behind the discussion, there is a general argument which suggests the possibility of a metaphysics of non-individuals, based on a reasonable interpretation of quantum basic entities. I close the paper with a suggestion that consists in emphasizing that quanta should be referred to by the cardinalities of the collections to which they belong, for which an adequate mathematical framework seems to be possible.

We show that using quasi-set theory, or the theory of collections of indistinguishable objects, we can define an algebra that has most of the standard properties of an orthocomplete orthomodular lattice, which is the lattice of all closed subspaces of a Hilbert space. We call the mathematical structure so obtained $\mathfrak{I}$-lattice. After discussing (in a preliminary form) some aspects of such a structure, we indicate the next problem of axiomatizing the corresponding logic, that is, a logic which has $\mathfrak{I}$-lattices as its Lindembaum algebra, which we postpone to a future work. Thus we conclude that the initial intuitions by Birkhoff and von Neumann that the ``logic of quantum mechanics" would be not classical logic (a Boolean algebra), is consonant with the idea of considering indistinguishability right from the start, that is, as a primitive concept. In the first sections, we present the main motivations and a ``classical'' situation which mirrors that one we focus on the last part of the paper. This paper is our first analysis of the algebraic structure of indiscernibility.

Despite the discrepancies between quantum objects and `classical' ones, mainly with regard to the fact that the latter may be thought of as `individuals', contrary to the former, we still regard the quanta as `things' in our ordinary discourse as well as in the logico-mathematical basis of quantum theories. This paper considers some possibilities for accomodating the logico-mathematical framework of the theories which deal with such a strange ontology where the inhabitants are things devoid of identity and both having and not having certain properties.``All right'', said the Cat; and this time it vanished quite slowly, beginning with the end of the tail, and ending with the grin, which remained some time after the rest of it had gone.

Schrödinger logics are logical systems in which the principle of identity is not true in general. The intuitive motivation for these logics is both Erwin Schrödinger's thesis (which has been advanced by other authors) that identity lacks sense for elementary particles of modern physics, and the way which physicists deal with this concept; normally, they understandidentity as meaningindistinguishability (agreemment with respect to attributes). Observing that these concepts are equivalent in classical logic and mathematics, which underly the usual physical theories, we present a higher-order logical system in which these concepts are systematically separated. A classical semantics for the system is presented and some philosophical related questions are mentioned. One of the main characteristics of our system is that Leibniz' Principle of the Identity of Indiscernibles cannot be derived. This fact is in accordance with some authors who maintain that quantum mechanics violates this principle. Furthermore, our system may be viewed as a way of making sense some of Schrödinger's logical intuitions about the nature of elementary particles.

In this paper we discuss some questions proposed by Prof. Newton da Costa on the foundations of quasi-set theory. His main doubts concern the possibility of a reasonable semantical understanding of the theory, mainly due to the fact that identity and difference do not apply to some entities of the theory’s intended domain of discourse. According to him, the quantifiers employed in the theory, when understood in the usual way, rely on the assumption that identity applies to all entities in the domain of discourse. Inspired by his provocation, we suggest that, using some ideas presented by da Costa himself in his seminars at UFSC and by one of us in some papers, these difficulties can be overcome both on a formal level and on an informal level, showing how quantification over items for which identity does not make sense can be understood without presupposing a semantics based on a ‘classical’ set theory.