Décio Krause

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.

Quasi-set theory has been proposed as a means of handling collections of indiscernible objects. Although the most direct application of the theory is quantum physics, it can be seen per se as a non-classical logic (a non-reflexive logic). In this paper we revise and correct some aspects of quasi-set theory as presented in [12], so as to avoid some misunderstandings and possible misinterpretations about the results achieved by the theory. Some further ideas with regard to quantum field theory are also advanced in this paper

A investigação de um domínio amplo da realidade, como a evolução dosseres vivos, pode dar origem a diferentes “teorias”, cada uma consoantecom uma particular perspectiva que se considere. Para que se proceda auma análise detalhada dos pressupostos e conceitos que baseiam umadeterminada visão, o método axiomático parece ser a melhor das opções.Neste artigo, são discutidas algumas teorias da biologia evolutiva de umponto de vista axiomático, mostrando-se de que forma se pode apresentarum “predicado de Suppes” para a teoria sintética da evolução, comodetalhado em outro local . Estaformulação é utilizada, dentre outras coisas, para se discutir a noção deaptidão darwiniana, bem como para explorar as relações entre a teoriadarwiniana da seleção natural e a genética de populações.The investigation of a wide field of knowledge, as evolution of living beings,may originate different “theories”, each one acting as a particularperspective we have about the domain. In order to proceed a philosophicalanalysis of the underlying assumptions of a particular view, it seems thatthe use of the axiomatic method is still the better way. Here, we discuss some theories of evolutionary biology from an axiomatic point of view, bymeans of a Suppes predicate for the synthetic theory of evolution, alreadydetailed in MAGALHÃES and KRAUSE . This formulation is hereused, among other things, to deal with the concept of Darwinian fitness, soas to explore the relationships between Darwinian theory and populationgenetics

Sortal predicates have been associated with a counting process, which acts as a criterion of identity for the individuals they correctly apply to. We discuss in what sense certain types of predicates suggested by quantum physics deserve the title of ‘sortal’ as well, although they do not characterize either a process of counting or a criterion of identity for the entities that fall under them. We call such predicates ‘quantum-sortal predicates’ and, instead of a process of counting, to them is associated a ‘criterion of cardinality’. After their general characterization, it is discussed how these predicates can be formally described.

We discuss some logico-mathematical systems which deviate from classical logic and mathematics with respect to the concept of identity. In the first part of the paper we present very general formulations of the principle of identity and show how they can be ‘relativized’ to objects and to properties. Then, as an application, we study the particular cases of physics and logic . In the last part of the paper, we discuss the alphabar logics, that is, those logical systems which violate a formulation of one of the most fundamental versions of the principle of identity; in these logics, there are formulas which are not deducible from themselves

This paper intends to introduce the three issues of Principia which will appear in a sequel honoring Newton da Costa’s 80th birthday. Instead of presenting the papers one by one, as it is common in presentations such as this one, we have left the papers speak by themselves, and instead we have preferred to present to the Brazilian readers, specialty to our students, some aspects of Newton da Costa’s conception of science and of the scientific activity, grounded on the concept of quasi-truth, which he contributed to develop in a rigorous way. Da Costa is known as one of the founding fathers of paraconsistent logic, but his contributions go also to the foundations of physics, theoretical computation, model theory, algebraic logic, lattice theory, applications of non-classical logics to law and technology, etc. But perhaps his main contribution was to provide a basis for the birth of a school of logic in our country, serving as teacher and inspiring new researchers for generations. It is a pleasure to have had so enthusiastic acceptation from the editors of Principia to organize these volumes. I would like to thank the contributors and the editors of Principia, specially Prof. Cezar Mortari for his help in organizing the issue.

Inspired in Quine's well known slogans “To be is to be the value of a variable” and "No entity without identity", we provide a way of enabling that non-individual entities (as characterized in the text) can also be values of variables of an adequate "regimented" language, once we consider a possible meaning of the background theory Quine reports to ground his view. In doing that, we show that there may exist also entities without identity, and emphasize the importance of paying attention to the metalanguage of scientific theories, for they may be also fundamental in determining the theory's ontological commitment.

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 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, 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.

Our aim in this paper is to take quite seriously Heinz Post’s claim that the non-individuality and the indiscernibility of quantum objects should be introduced right at the start, and not made a posteriori by introducing symmetry conditions. Using a different mathematical framework, namely, quasi-set theory, we avoid working within a label-tensor-product-vector-space-formalism, to use Redhead and Teller’s words, and get a more intuitive way of dealing with the formalism of quantum mechanics, although the underlying logic should be modified. We build a vector space with inner product, the Q-space, using the non-classical part of quasi-set theory, to deal with indistinguishable elements. Vectors in Q-space refer only to occupation numbers and permutation operators act as the identity operator on them, reflecting in the formalism the fact of unobservability of permutations. Thus, this paper can be regarded as a tentative to follow and enlarge Heinsenberg’s suggestion that new phenomena require the formation of a new “closed” (that is, axiomatic) theory, coping also with the physical theory’s underlying logic and mathematics

The ‘ontic’ form of structural realism , roughly speaking, admits a complete elimination of the objects in the discourse of scientific theories, leaving us with structures only. As put by the defenders of such a claim, the idea is that all there is are structures and, if the relevant structures are to be set-theoretical constructs , as it has also been claimed, then the relations which appear in such structures should be taken to be ‘relations without the relata’. As far as we know, there is not a definition of structure in standard mathematics which fits their intuitions, and even category theory seems not to correspond adequately the OSR claims. Since OSR is also linked with the semantic approach to theories, the structures to be dealt with are taken to be set-theoretical constructs. But these are ‘relational’ structures where the involved relations are built from basic objects , and so no complete elimination of the relata is possible, although it would be adequate for characterizing OSR. In this paper we present a definition of a kind of relation that does not depend on the particular objects being related in the sense that the ‘relation’ continues to hold even if the relata are exchanged by suitable objects. Although there is not a ‘complete’ elimination of relata, our definition might be viewed as an alternative way of finding adequate mathematical ‘set-theoretical’ frameworks for describing at least some of the intuitions regarding OSR within a ‘set-theoretical’ framework

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 that identity lacks sense for elementary particles of modern physics, and the way which physicists deal with this concept; normally, they understand identity as meaning indistinguishability . 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

PORTUGUESE: Neste artigo, apresentaremos uma visão particular do desenvolvimento de teorias científicas que denominamos (inspirados em Ortega y Gasset) "perspectivismo". Discutiremos como, através desse enfoque, é possível compatibilizar diversas descrições aparentemente distintas e incompatíveis de uma suposta realidade que se investiga. Fazemos isso distinguindo entre a "realidade" (R) e a "descrição empírica da realidade" (Re). Aceitando que podemos ter diversas descrições empíricas de uma mesma realidade, discutimos o caso particular em que esse esquema é utilizado nos debates atuais acerca da ontologia da física quântica, em especial da mecânica quântica não relativista. Como se sabe, essa teoria é compatível com distintas ontologias (ou metafísicas), em particular, pode ser vista como comprometida com uma ontologia de indivíduos, mas também com uma ontologia de não indivíduos. Ambas as alternativas são plausíveis e merecem ser desenvolvidas. Propomos que o perspectivismo aqui apresentado é neutro com relação à escolha de uma metafísica (ontologia), bem como aos debates entre realistas e antirrealistas em filosofia da ciência. Concluímos que passar do pluralismo aceito pelo perspectivista ao realismo ou antirrealismo envolve comprometimento com teses mais robustas acerca da relação entre realidade e descrição da realidade.ENGLISH: In this article we present a particular view of the development of scientific theories which (following Ortega y Gasset), we term "perspectivism". Making use of this view, we discuss how we can accommodate distinct and apparently incompatible descriptions of a supposed reality under investigation. We distinguish between a "reality" (R) and its "empirical description", (Re). Acknowledging that we can have diverse empirical descriptions of the same reality, we discuss the particular case in which the view is applied to current debates on the ontology of quantum physics, especially of non-relativistic quantum mechanics. As it is well known, this theory is compatible with treating quantum objects either as individuals or as non-individuals. Both ontologies are possible and deserve to be developed. We propose that the perspectivism developed here is neutral not only with respect to the choice of an ontology, but also concerning the debates among realists and anti-realists in the philosophy of science. We conclude that to go beyond the pluralism accepted by perspectivism, and to adopt realism or anti-realism, involves commitment to strong theses about the relation between reality and its descriptions. PORTUGUESE: Neste artigo, apresentaremos uma visão particular do desenvolvimento de teorias científicas que denominamos (inspirados em Ortega y Gasset) "perspectivismo". Discutiremos como, através desse enfoque, é possível compatibilizar diversas descrições aparentemente distintas e incompatíveis de uma suposta realidade que se investiga. Fazemos isso distinguindo entre a "realidade" (R) e a "descrição empírica da realidade" (Re). Aceitando que podemos ter diversas descrições empíricas de uma mesma realidade, discutimos o caso particular em que esse esquema é utilizado nos debates atuais acerca da ontologia da física quântica, em especial da mecânica quântica não relativista. Como se sabe, essa teoria é compatível com distintas ontologias (ou metafísicas), em particular, pode ser vista como comprometida com uma ontologia de indivíduos, mas também com uma ontologia de não indivíduos. Ambas as alternativas são plausíveis e merecem ser desenvolvidas. Propomos que o perspectivismo aqui apresentado é neutro com relação à escolha de uma metafísica (ontologia), bem como aos debates entre realistas e antirrealistas em filosofia da ciência. Concluímos que passar do pluralismo aceito pelo perspectivista ao realismo ou antirrealismo envolve comprometimento com teses mais robustas acerca da relação entre realidade e descrição da realidade.