## Opaque (1998)

### BibTeX

@MISC{Krause98opaque,

author = {Décio Krause and Steven French},

title = {Opaque},

year = {1998}

}

### OpenURL

### Abstract

predicates, veiled sets and their logic

### Citations

538 | The Emperor’s New Mind - Penrose - 1989 |

110 |
Introduction to Mathematical Logic
- Church
- 1956
(Show Context)
Citation Context ...is section are performed in the quasi-set theory Q. We will not provide here all the details but only the main definitions and results, which are similar to those of classical logic (see for instance =-=[4]-=-). The proofs can be adapted without difficulty from the most general case of Schödinger logics presented in [7]. We call L the language of Lop. Let D be a quasi-set such that D = m ∪ M where m is a p... |

64 | Contributions to the Founding of the Theory of Transfinite Numbers - Cantor - 1955 |

48 |
Quantum Mechanics: An Empiricist View
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- 1991
(Show Context)
Citation Context ...n other words, there exist D1 and D2 such that D1 ∩ D2 = ∅ , D1 ∪ D2 = D and D1 = {x ∈ D : P(x)} ̸= ∅. 1 We have chosen this formulation as neutral between the claims of realist and anti-realist; see =-=[38]-=-, for example.OPAQUE PREDICATES, VEILED SETS AND THEIR LOGIC 3 The set D1 is the extension of P and if a ∈ D1, we say that ‘a has the property P’; otherwise, that is, if a ∈ D2, we say that ‘a does n... |

41 |
An Interpretive Introduction to Quantum Field Theory
- Teller
- 1995
(Show Context)
Citation Context ...ect to all their quantum numbers, so there is no way of distinguishing them. Even so, physicists reason as if there are six ‘entities’ (it is difficult to use the word ‘individuals’ in this case —see =-=[36]-=-) in that level. Then consider the predicate ‘to be one of the electrons of the level 2p of a sodium atom’. How may we ascribe to this predicate a well-defined extension? This is not possible without ... |

19 | a quasi-set theory’, Notre Dame - Krause, ‘On - 1992 |

17 | Science and humanism - Schrödinger - 1952 |

12 |
A formal framework for quantum nonindividuality, Relatório interno, Departamento de mateemática, Setor de Ciências exatas, Universidade Federal do
- KRAUS, FRENCH
- 1993
(Show Context)
Citation Context ...vertheless, our approach here is different and has not to our knowledge been explored in the literature. Quantum particles may also be regarded as ‘non-individuals’ in a certain sense [31] (see [14], =-=[24]-=-) and now the issues shifts from the applicability of the predicates to the determinateness of the objects. It is this which marks the difference between vagueness and what we shall call opacity. With... |

10 |
Quasiset theories for microobjects: a comparision
- Chiara, L, et al.
- 1998
(Show Context)
Citation Context ...y, what this intuitively means that, for every x, x belongs or does not belong to the quasi-set. So, the present case is distinct from that involving fuzzy sets or quasets (concerning the latter, see =-=[13]-=-.OPAQUE PREDICATES, VEILED SETS AND THEIR LOGIC 11 by an indistinguishable one, all happens (to us, who are behind the veil) as if nothing had occurred at all. 13 Such quasi-sets could be called veil... |

10 |
Axioms for collections of indistinguishable objects’, Logique et Analyse 153–154
- Krause
- 1996
(Show Context)
Citation Context ...ematical structure in which the language of Lop is to be interpreted. Thus, instead of using a set theory like Zermelo-Fraenkel in the metamathematics, we use the quasi-set theory Q (for details, see =-=[23]-=-). 3 Roughly speaking, the theory Q is a mathematical device for treating collections of indistinguishable objects. The theory allows the presence of a certain kind of Urelemente (the so called m-atom... |

9 | Principles of mathematical logic - Hilbert, Ackermann - 1958 |

9 |
Vague identity and quantum indeterminacy, Analysis 54
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- 1994
(Show Context)
Citation Context ...ned individual, such that the application of the predicate regarding its location cannot be determined. As we have indicated elsewhere, this is certainly one way of considering the quantum situation (=-=[26]-=-, [16], [18]). Nevertheless, our approach here is different and has not to our knowledge been explored in the literature. Quantum particles may also be regarded as ‘non-individuals’ in a certain sense... |

9 | The problem of vague predicates - Parikh - 1983 |

8 | The Model-Theoretic Approach to the philosophy - Costa, Newton, et al. - 1990 |

8 | Language-mastery and the sorites paradox - Wright - 1976 |

7 |
An intensional Schrödinger logic”, Notre Dame
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- 1997
(Show Context)
Citation Context ...efinitions and results, which are similar to those of classical logic (see for instance [4]). The proofs can be adapted without difficulty from the most general case of Schödinger logics presented in =-=[7]-=-. We call L the language of Lop. Let D be a quasi-set such that D = m ∪ M where m is a pure qset whose elements are indistinguishable from one another and M is a non-empty set (in Q). By a frame for L... |

7 | Quantum Physics and the Identity of Indiscernibles’, The British Journal for the Philosophy of Science 39 - French, Redhead - 1988 |

7 |
Vague identity and quantum nonindividuality
- French, Krause
- 1995
(Show Context)
Citation Context ...dividual, such that the application of the predicate regarding its location cannot be determined. As we have indicated elsewhere, this is certainly one way of considering the quantum situation ([26], =-=[16]-=-, [18]). Nevertheless, our approach here is different and has not to our knowledge been explored in the literature. Quantum particles may also be regarded as ‘non-individuals’ in a certain sense [31] ... |

5 | Toraldo di - Chiara, L - 1992 |

5 |
Individuality and physics’, The Listener 70
- Post
- 1963
(Show Context)
Citation Context ... [16], [18]). Nevertheless, our approach here is different and has not to our knowledge been explored in the literature. Quantum particles may also be regarded as ‘non-individuals’ in a certain sense =-=[31]-=- (see [14], [24]) and now the issues shifts from the applicability of the predicates to the determinateness of the objects. It is this which marks the difference between vagueness and what we shall ca... |

5 | Science Theory and - Schrodinger - 1957 |

3 | Language and Philosophy - Black - 1949 |

3 |
Identity and individuality in classical and quantum physics
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- 1989
(Show Context)
Citation Context ...]). Nevertheless, our approach here is different and has not to our knowledge been explored in the literature. Quantum particles may also be regarded as ‘non-individuals’ in a certain sense [31] (see =-=[14]-=-, [24]) and now the issues shifts from the applicability of the predicates to the determinateness of the objects. It is this which marks the difference between vagueness and what we shall call opacity... |

3 |
Quantum vagueness
- French, Krause
(Show Context)
Citation Context ...al, such that the application of the predicate regarding its location cannot be determined. As we have indicated elsewhere, this is certainly one way of considering the quantum situation ([26], [16], =-=[18]-=-). Nevertheless, our approach here is different and has not to our knowledge been explored in the literature. Quantum particles may also be regarded as ‘non-individuals’ in a certain sense [31] (see [... |

3 |
The theory of quasi-sets and ZFC are equiconsistent
- Krause
- 1995
(Show Context)
Citation Context ...ing the indistinguishability relation instead of equality) which has quasi-cardinal 1. In Q, we can prove that such a qset exists. 7 For further information on this particular topic, see for instance =-=[22, 8, 23]-=-. 8 Another axiom says that every quasi-set has an unique quasi-cardinal which is a cardinal defined in the ‘classical’ part of Q. So, the quasi-cardinals are ‘sets’ in Q, obeying the standard axioms ... |

3 |
Quasi-set theory for bosons and fermions
- Krause, Sant’Anna, et al.
- 1999
(Show Context)
Citation Context ...ry is compatible with the existence of distinct and abosolutely indistinguishable m-atoms. This is important, for example, for obtaining a description of quantum statistics in the framework of Q (see =-=[25]-=- for details on this point). In other words, it is consistent with Q to maintain that x has α elements, which may be regarded as absolutely indistinguishable objects. Since the elements of x may share... |

3 |
Some remarks on vague predicates', Theoria - Segunda Epoca 10
- Terricabras, Trillas
- 1989
(Show Context)
Citation Context ...ic and semantic elements. It is precisely such a framework that we sketch below. From a more mathematical point of view, vague predicates are considered to differ from the usual ‘Fregean’ predicates (=-=[37]-=-) in the following sense. A unary predicate letter P of (say) a first-order language is Fregean if it provides a bipartition on a domain D of objects to which the language makes reference. In other wo... |

2 | Experiment and theory in physics - Born - 1943 |

2 |
Schrödinger logics’, Studia Logica 53
- Costa, A, et al.
- 1994
(Show Context)
Citation Context ... The reasons to use such a logic will be explained in the last section. In what follows we present the main features of the logic Lop, which is a slight modified version of the system Sω presented in =-=[6]-=-. Let us begin by defining the concept of type: we call Π the set of types, recursively defined as being the smallest set such that: (a) e1, e2 ∈ Π, and (b) if τ1, . . . , τn ∈ Π, then 〈τ1, . . . , τn... |

2 |
Set-theoretical models for quantum systems’, forthcoming in Dalla
- Costa, A, et al.
- 1995
(Show Context)
Citation Context ...ing the indistinguishability relation instead of equality) which has quasi-cardinal 1. In Q, we can prove that such a qset exists. 7 For further information on this particular topic, see for instance =-=[22, 8, 23]-=-. 8 Another axiom says that every quasi-set has an unique quasi-cardinal which is a cardinal defined in the ‘classical’ part of Q. So, the quasi-cardinals are ‘sets’ in Q, obeying the standard axioms ... |

2 | Some foundational problems in mathematics suggested by physics, Synthese 62 - Chiara, L - 1985 |

2 | Toraldo di Francia, G., ‘Identity questions from quantum theory - L - 1995 |

2 | The logic of quanta’, forthcoming - French, Krause - 1996 |

2 |
Vagueness and alternative logic” Erkenntnis 19
- Putnam
- 1983
(Show Context)
Citation Context ...ather the instructions we already have determine that the line is not to be drawn” [ibid.]. However, if ‘the line is not to be drawn’, then clearly classical logic is not to be used. Following Putnam =-=[32]-=-, Mott has recently suggested that vague predicates can be accommodated within intuitionistic logic if they are taken to be partial, in precisely the above sense; that is, in certain situations, their... |

1 | An approach to intensional semantics’, Synthese 73 - Chiara, L - 1987 |

1 | Models and analogies in science, Un. of Notre Dame - Hesse - 1970 |

1 |
On the intuitionistic solution of the Sorites Paradox’, Pacific Phil. Quarterly 75
- Mott
- 1994
(Show Context)
Citation Context ...icates can be accommodated within intuitionistic logic if they are taken to be partial, in precisely the above sense; that is, in certain situations, their application simply cannot be decided at all =-=[27]-=-. Thus, the application of a colour predicate can be decided simply by using our sense organs, in those situations where it can be decided at all and it is precisely because of the existence of the co... |

1 | Foundations: essays in philosophy - Ramsey - 1978 |