Logic and Structure

Dalen, Dirk van

doi:10.1007/978-3-662-02962-6

Cited by 80 publications

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“…If the type domain Θ contains first-and second-order types only and at least one second-order type, then we shall say that the signature Σ g and the language LpΣ g q have the second order (see [2,Appendix], [4,4]). In this case the notations Σ g 2 and LpΣ g 2 q will be used.…”

Section: Parenthesismentioning

confidence: 99%

“…The semantics for the language LpΣ g 2 q differs both from the standard semantics (see [2,Appendix], [3, §16]) and from the Henkin semantics (see [2,Appendix], [3, §21], [4,4], and [5,6,7]), which restricts the range of values of the evaluation γpx τ q for a variable x τ of a second-order type τ by some subset of the set Ppτ pAqq of the terminal τ pAq.…”

Section: Introductionmentioning

confidence: 99%

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A Sequence of Models of Generalized Second-order Dedekind Theory of Real Numbers with Increasing Powers

Zakharov

Rodionov

2020

ARJOM

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The paper is devoted to construction of some closed inductive sequence of models of the generalized second-order Dedekind theory of real numbers with exponentially increasing powers. These models are not isomorphic whereas all models of the standard second-order Dedekind theory are. The main idea in passing to generalized models is to consider instead of superstructures with the single common set-theoretical equality and the single common set-theoretical belonging superstructures with several generalized equalities and several generalized belongings for first and second orders. The basic tools for the presented construction are the infraproduct of collection of mathematical systems different from the factorized Loś ultraproduct and the corresponding generalized infrafiltration theorem. As its auxiliary corollary we obtain the generalized compactness theorem for the generalized second-order language.

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Section: Parenthesismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Sequence of Models of Generalized Second-order Dedekind Theory of Real Numbers with Increasing Powers

Zakharov

Rodionov

2020

ARJOM

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“…where A ⊔ B stands for the logical disjunction of A and B, the double-bracket notation denotes a valuation [2,3], that is, a mapping from the set of atomic propositions, symbolized by P, to the Boolean domain B 2 (the set of truth values, true and false, renamed to 1 and 0, correspondingly), i.e., v :…”

Section: Introductionmentioning

confidence: 99%

“…Now, recall that in accordance with the Kochen-Specker theorem [4], it is impossible to assign truth values to all experimental propositions associated with a quantum system in a consistent manner. In detail, this theorem (usually called the KS theorem) asserts that there always exists a set S of compatible experimental propositions about the quantum system characterized by N -dimensional Hilbert space H, such that all propositions in S cannot have truth values satisfying the rules (1) and (2).…”

Section: Introductionmentioning

confidence: 99%

Algebraic structures identified with bivalent and non-bivalent semantics of experimental quantum propositions

Bolotin

2019

Quantum Stud.: Math. Found.

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The failure of distributivity in quantum logic is motivated by the principle of quantum superposition. However, this principle can be encoded differently, i.e., in different logico-algebraic objects. As a result, the logic of experimental quantum propositions might have various semantics. E.g., it might have either a total semantics, or a partial semantics (in which the valuation relationi.e., a mapping from the set of atomic propositions to the set of two objects, 1 and 0 -is not total), or a many-valued semantics (in which the gap between 1 and 0 is completed with truth degrees). Consequently, closed linear subspaces of the Hilbert space representing experimental quantum propositions may be organized differently. For instance, they could be organized in the structure of a Hilbert lattice (or its generalizations) identified with the bivalent semantics of quantum logic or in a structure identified with a non-bivalent semantics. On the other hand, one can only verify -at the same time -propositions represented by the closed linear subspaces corresponding to mutually commuting projection operators. This implies that to decide which semantics is proper -bivalent or non-bivalent -is not possible experimentally. Nevertheless, the latter allows simplification of certain no-go theorems in the foundation of quantum mechanics. In the present paper, the Kochen-Specker theorem asserting the impossibility to interpret, within the orthodox quantum formalism, projection operators as definite {0, 1}-valued (pre-existent) properties, is taken as an example. The paper demonstrates that within the algebraic structure identified with supervaluationism (the form of a partial, non-bivalent semantics), the statement of this theorem gets deduced trivially.

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“…Formal inferencing, in contrast to rational reconstruction, is already treated in the literature extensively. For sociological examples, see for instance~Péli et al1994;Péli 1997;Péli and Masuch 1997;Kamps Pólos 1999!. …”

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confidence: 99%

5. A Logical Toolkit for Theory (Re) Construction

Bruggeman

Vermeulen

2002

Sociological Methodology

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The social sciences have achieved highly sophisticated methods for data collection and analysis, leading to increased control and tractability of scientific results. Meanwhile, methods for systematizing these results, as well as new ideas and hypotheses, into sociological theories have seen little progress, leaving most sociological arguments ambiguous and difficult to handle, and impairing cumulative theory development. Sociological theory, containing many valuable ideas and insights, deserves better than this. As a way out of the doldrums, this paper presents a systematic approach to computer-supported logical formalization, that is widely applicable to sociological theory and other declarative discourse. By increasing rigor and precision of sociological arguments, they become better accessible to critical investigation, thereby raising scientific debate to a new level. The merits of this approach areWe are grateful to Jaap Kamps and Gábor Péli for their comments on an earlier version, and to an anonymous reviewer for comments on a later version. Basic ideas for the heuristics presented were developed by László Pólos, Gábor Péli, Breanndán Ó Nualláin, Michael Masuch, Jaap Kamps, and the authors of this paper, who all worked at CCSOM, now called the Applied Logic Laboratory, in Amsterdam. CCSOM was supported from 1990 until 1995 by the Netherlands Organizations for Scientific Research through a PIONIER project awarded to Michael Masuch #PGS50-334!. Bruggeman did part of this work at the Rijksuniversiteit Groningen and at the Universiteit Twente. He can be contacted at the Department of Sociology and Anthropology, Universiteit van Amsterdam, Oudezijds Achterburgwal 185, 1012 DK Amsterdam, Netherlands, bruggeman@pscw.uva.nl.Ivar Vermeulen is at ALL, UvA, Nieuwe Achtergracht 166, 1018 VW Amsterdam, ivar@ccsom.uva.nl.*University of Amsterdam 183 demonstrated by applying it to an actual fragment from the sociological literature.The very first lesson that we have a right to demand that logic shall teach us is how to make our ideas clear; and a most important one it is, depreciated only by minds who stand in need of it.To know what we think, to be masters of our own meaning, will make a solid foundation for great and weighty thought. It is most easily learned by those whose ideas are meagre and restricted; and far happier they than such as wallow helplessly in a rich mud of conceptions.-C. S. Peirce, How to Make Our Ideas Clear

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Logic and Structure

Cited by 80 publications

References 0 publications

A Sequence of Models of Generalized Second-order Dedekind Theory of Real Numbers with Increasing Powers

A Sequence of Models of Generalized Second-order Dedekind Theory of Real Numbers with Increasing Powers

Algebraic structures identified with bivalent and non-bivalent semantics of experimental quantum propositions

5. A Logical Toolkit for Theory (Re) Construction

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