According to the semantic view, a theory is characterized by a class of mod- els. In this paper, we examine critically some of the assumptions that underlie this approach. First, we recall that models are models of something. Thus we cannot leave completely aside the axiomatization of the theories under consider- ation, nor can we ignore the metamathematics used to elaborate these models, for changes in the metamathematics often impose restrictions on the resulting models. Second, based on a parallel between (...) van Fraassen’s modal interpre- tation of quantum mechanics and Skolem’s relativism regarding set-theoretic concepts, we introduce a distinction between relative and absolute concepts in the context of the models of a scientific theory. And we discuss the significance of that distinction. Finally, by focusing on contemporary particle physics, we raise the question: since there is no general accepted unification of the parts of the standard model (namely, QED and QCD), we have no theory, in the usual sense of the term. This poses a difficulty: if there is no theory, how can we speak of its models? What are the latter models of? We conclude by noting that it is unclear that the semantic view can be applied to contemporary physical theories. (shrink)
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. (shrink)
The physics and metaphysics of identity and individuality Content Type Journal Article DOI 10.1007/s11016-010-9463-7 Authors Don Howard, Department of Philosophy and Graduate Program in History and Philosophy of Science, University of Notre Dame, Notre Dame, IN 46556, USA Bas C. van Fraassen, Philosophy Department, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, USA Otávio Bueno, Department of Philosophy, University of Miami, Coral Gables, FL 33124, USA Elena Castellani, Department of Philosophy, University of Florence, Via Bolognese 52, 50139 (...) Florence, Italy Laura Crosilla, Department of Pure Mathematics, School of Mathematics, University of Leeds, Leeds, LS2 9JT UK Steven French, Department of Philosophy, University of Leeds, Leeds, UK Décio Krause, Department of Philosophy, Federal University of Santa Catarina, 88040-900 Campus Trindade, Florianópolis, SC Brazil Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796. (shrink)
In this paper we discuss two approaches to the axiomatization of scientific theories in the context of the so called semantic approach, according to which (roughly) a theory can be seen as a class of models. The two approaches are associated respectively to Suppes’ and to da Costa and Chuaqui’s works. We argue that theories can be developed both in a way more akin to the usual mathematical practice (Suppes), in an informal set theoretical environment, writing the set theoretical predicate (...) in the language of set theory itself or, more rigorously (da Costa and Chuaqui), by employing formal languages that help us in writing the postulates to define a class of structures. Both approaches are called internal , for we work within a mathematical framework, here taken to be first-order ZFC. We contrast these approaches with an external one, here discussed briefly. We argue that each one has its strong and weak points, whose discussion is relevant for the philosophical foundations of science. (shrink)
According to the semantic view, a theory is characterized by a class of models. In this paper, we examine critically some of the assumptions that underlie this approach. First, we recall that models are models of something. Thus we cannot leave completely aside the axiomatization of the theories under consideration, nor can we ignore the metamathematics used to elaborate these models, for changes in the metamathematics often impose restrictions on the resulting models. Second, based on a parallel between van Fraassen’s (...) modal interpretation of quantum mechanics and Skolem’s relativism regarding set-theoretic concepts, we introduce a distinction between relative and absolute concepts in the context of the models of a scientific theory. And we discuss the significance of that distinction. Finally, by focusing on contemporary particle physics, we raise the question: since there is no general accepted unification of the parts of the standard model (namely, QED and QCD), we have no theory, in the usual sense of the term. This poses a difficulty: if there is no theory, how can we speak of its models? What are the latter models of? We conclude by noting that it is unclear that the semantic view can be applied to contemporary physical theories. (shrink)
In this first paper of a series of works on the foundations of science, we examine the significance of logical and mathematical frameworks used in foundational studies. In particular, we emphasize the distinction between the order of a language and the order of a structure to prevent confusing models of scientific theories (as set-theoretical structures) with first-order structures (called here order-1 structures), and which are studied in standard (first-order) model theory. All of us are, of course, bound to make abuses (...) of language even in putatively precise contexts. This is not a problem—in fact, it is part of scientific and philosophical practice. But it is important to be sensitive to the dierent uses that structure, model, and language have. In this paper, we examine these topics in the context of classical logic; only in the last section we touch upon briefly on non-classical ones. (shrink)
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. (shrink)
In this paper, we examine the concept of particle as it appears in quantum field theories (QFT), focusing on a puzzling situation regarding this concept. Although quantum ‘particles’ arise from fields, which form the basic ontology of QFT, and thus a certain concept of ‘particle’ is al- ways available, the properties ascribed to such ‘particles’ are not completely in agreement with the mathematical and logical description of such fields, which should be taken as individuals.
http://dx.doi.org/10.5007/1808-1711.2009v13n3p251 Neste artigo discutimos algumas questões propostas por Newton da Costa relacionadas aos fundamentos da teoria de quase-conjuntos. Seus questionamentos aqui considerados tratam da possibilidade de uma compreenão semântica da teoria, principalmente devido ao fato de que identidade e diferença podem não ser aplicáveis para algumas das entidades no domínio pretendido da teoria. De acordo com ele, o modo usual de se compreender os quantificadores utilizados na teoria depende da hipótese de que a identidade deve valer para todas as entidades (...) no domínio de discurso. Inspirados pelas suas questões, sugerimos que essas dificuldades podem ser superadas tanto em um nível formal quanto em um nível informal, mostrando como a quantificação sobre itens para os quais a identidade não faz sentido pode ser entendida sem pressupor uma semântica baseada em uma teoria de ‘clássica’ de conjuntos. (shrink)
Este artigo pretende introduzir os três volumes de Principia que aparecerão em sequência homenageando os 80 anos do professor Newton da Costa. Ao invés de apresentar os artigos um a um, como se faz usualmente em uma introdução como esta, preferimos deixar os artigos falarem por si, e oforoecer aos leitores brasileiros, especialmente nossos estudantes, alguns aspectos da concepção de ciência e da atividade científica de Newton da Costa, fundamentadas no conceito de quase-verdade, que ele contribuiu para desenvolver de modo (...) rigoroso. Da Costa e conhecido como urn dos fundadores da lógica paraconsistente, mas suas contribuições alcançam também os fundamentos da física, da ciência da computação,a teoria dos modelos, a lógica algébrica, a teoria dos reticulados, as aplicações de lógicas não-clássicas à ciência do direito e à tecnologia, etc. No entanto, talvez sua maior contruição tenha sido proporcionar a base para a criação de uma escola de lógica em nosso país (Brasil), à qual serviu como professor e inspirador par gerações. É com satisfação que vimos uma imediata aceitaçãoo pelos editors de Principia para a organização desses volumes. Gostaria de agradecer a todos os que contribuiram com artigos e aos editors da revista, em especial ao professor Cezar Mortari pela ajuda na organização desta homenagem. (shrink)
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. 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. (shrink)
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. (shrink)
Quasi-set theory is a theory for dealing with collections of indistinguishable objects. In this paper we discuss some logical and philosophical questions involved with such a theory. The analysis of these questions enable us to provide the first grounds of a possible new view of physical reality, founded on an ontology of non-individuals, to which quasi-set theory may constitute the logical basis.
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. (shrink)
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. (shrink)
In this paper we argue that physical theories, including the most recent ones, even if only implicitly, talk of `objects' (or `things') of some sort (really, of several sorts), and question the logico-mathematical apparatus we still use to formulate them, taking into account what such theories presuppose about these entities. I shall point out that despite the discourse (or at least some discourses) goes in the direction of assuming that these quantum objects would be `new entities' of some kind, distinct (...) from the traditional physical objects of classical physics, the logico-mathematical framework we use is still the old one, grounded on classical logic and set theory, which are committed to atavistic concepts based on individuals and distinguishable things, in complete disagreement with our present day conception of quanta. So, the use of such apparatus would impede us to be in complete agreement with the ontological commitment the theories of \textit{quanta} seem to propose. Thus, I move in the direction of joining those who try to question the `logic of quantum mechanics' from a different point of view, looking for a formal rationale for a new ontology. As a consequence of this move, we can revisit Einstein's ideas on physical reality and see that, from the perspective of considering a new kind of object, here termed `non-individuals', it is possible to sustain that they still obey some of Einstein's conditions for `physical realities', so that it will be possible to talk of a `principle of separability' in a sense which is not in complete disagreement with quantum mechanics. So, Einstein's departure from quantum mechanics might be softened at least concerning a form of his realism (locality still remains a challenge of course), for we guess that the incompatibility between quantum mechanics (field theories included) and some form of `separability' makes sense only if the objects of discourse are thought as `classical' objects, typical of classical ontology. (shrink)
The concept of indiscernibility in a structure is analysed with the aim of emphasizing that in asserting that two objects are indiscernible, it is useful to consider these objects as members of (the domain of) a structure. A case for this usefulness is presented by examining the consequences of this view to the philosophical discussion on identity and indiscernibility in quantum theory.
The concept of indiscernibility in a structure is analysed with the aim of emphasizing that in asserting that two objects are indiscernible, it is useful to consider these objects as members of (the domain of) a structure. A case for this usefulness is presented by examining the consequences of this view to the philosophical discussion on identity and indiscernibility in quantum theory.
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. (shrink)
In this paper we make some general remarks on the use of non-classical logics, in particular paraconsistent logic, in the foundational analysis of physical theories. As a case-study, we present a reconstruction of P.\ -D.\ F\'evrier's 'logic of complementarity' as a strict three-valued logic and also a paraconsistent version of it. At the end, we sketch our own approach to complementarity, which is based on a paraconsistent logic termed 'paraclassical logic'.
It has been suggested that quantum particles are genuinelyvague objects (Lowe 1994a). The present work explores thissuggestion in terms of the various metaphysical packages that areavailable for describing such particles. The formal frameworksunderpinning such packages are outlined and issues of identityand reference are considered from this overall perspective. Indoing so we hope to illuminate the diverse ways in whichvagueness can arise in the quantum context.
The 'ontic' form of structural realism (OSR), roughly speaking, aims at a complete elimination of objects of 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 do not correspond adequately to the OSR claims. Since OSR is also linked to the semantic approach to theories, the structures to be dealt with are (at least in principle) to be taken as set theoretical constructs. But these are 'relational' structures where the involved relations are built from basic objects (in short, the rank of the relation is greater than the rank of the relata), and so no elimination of the relata is possible, although it would be interesting for characterizing OSR. In this paper we present a definition of a relation which 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 other suitable ones. Although there is not a 'complete' elimination of the relata, there is an elimination of 'particular' relata, and so 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. (shrink)
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. (shrink)
H. Post's conception of quantal particles as non-individuals is set in a formal logico-mathematical framework. By means of this approach certain metaphysical implications of quantum mechanics can be further explored.
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. (shrink)
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 understand identity as meaning indistinguishability (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. (shrink)
In this paper we present an overview of Professor Newton C. A. da Costa’s work in logic, emphasizing the main results obtained by him in the several areas of his research activity. The text furnish a detailed bibliographic reference of his works, which are listed in the last section.
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-<span class='Hi'>Fraenkel</span>. In this paper, collections of objects to which the standard concept of identity (Leibinizian 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. (shrink)
