scholarly journals A Comparison of Implications in Orthomodular Quantum Logic—Morphological Analysis of Quantum Logic

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
Mitsuhiko Fujio
Keyword(s):  
Modal Logic ◽  
Quantum Logic ◽  
Morphological Analysis ◽  
The Other ◽  
Kripke Semantics ◽  
Quantum Logics ◽  
Von Neumann ◽  
Morphological Operators ◽  
Complemented Lattices ◽  
Implication Problem

Morphological operators are generalized to lattices as adjunction pairs (Serra, 1984; Ronse, 1990; Heijmans and Ronse, 1990; Heijmans, 1994). In particular, morphology for set lattices is applied to analyze logics through Kripke semantics (Bloch, 2002; Fujio and Bloch, 2004; Fujio, 2006). For example, a pair of morphological operators as an adjunction gives rise to a temporalization of normal modal logic (Fujio and Bloch, 2004; Fujio, 2006). Also, constructions of models for intuitionistic logic or linear logics can be described in terms of morphological interior and/or closure operators (Fujio and Bloch, 2004). This shows that morphological analysis can be applied to various non-classical logics. On the other hand, quantum logics are algebraically formalized as orhomodular or modular ortho-complemented lattices (Birkhoff and von Neumann, 1936; Maeda, 1980; Chiara and Giuntini, 2002), and shown to allow Kripke semantics (Chiara and Giuntini, 2002). This suggests the possibility of morphological analysis for quantum logics. In this article, to show an efficiency of morphological analysis for quantum logic, we consider the implication problem in quantum logics (Chiara and Giuntini, 2002). We will give a comparison of the 5 polynomial implication connectives available in quantum logics.

scholarly journals Recursive enumerability and elementary frame definability in predicate modal logic

2019 ◽  
Vol 30 (2) ◽  
pp. 549-560 ◽  
Author(s):  
Mikhail Rybakov ◽  
Dmitry Shkatov
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
The Other ◽  
Kripke Semantics ◽  
First Order ◽  
Elementary Class ◽  
Recursive Enumerability ◽  
Kripke Frames ◽  
Recursively Enumerable ◽  
The Relationship

Abstract We investigate the relationship between recursive enumerability and elementary frame definability in first-order predicate modal logic. On one hand, it is well known that every first-order predicate modal logic complete with respect to an elementary class of Kripke frames, i.e. a class of frames definable by a classical first-order formula, is recursively enumerable. On the other, numerous examples are known of predicate modal logics, based on ‘natural’ propositional modal logics with essentially second-order Kripke semantics, that are either not recursively enumerable or Kripke incomplete. This raises the question of whether every Kripke complete, recursively enumerable predicate modal logic can be characterized by an elementary class of Kripke frames. We answer this question in the negative, by constructing a normal predicate modal logic which is Kripke complete, recursively enumerable, but not complete with respect to an elementary class of frames. We also present an example of a normal predicate modal logic that is recursively enumerable, Kripke complete, and not complete with respect to an elementary class of rooted frames, but is complete with respect to an elementary class of frames that are not rooted.

scholarly journals Well-definedness and observational equivalence for inductive–coinductive programs

2019 ◽  
Vol 29 (4) ◽  
pp. 419-468
Author(s):  
Henning Basold ◽  
Helle Hvid Hansen
Keyword(s):  
Fixed Point ◽  
Modal Logic ◽  
The Other ◽  
Observational Equivalence ◽  
Other Hand ◽  
Inductive Types ◽  
Point Types ◽  
Usual Topology

Abstract We define notions of well-definedness and observational equivalence for programs of mixed inductive and coinductive types. These notions are defined by means of tests formulas which combine structural congruence for inductive types and modal logic for coinductive types. Tests also correspond to certain evaluation contexts. We define a program to be well-defined if it is strongly normalizing under all tests, and two programs are observationally equivalent if they satisfy the same tests. We show that observational equivalence is sufficiently coarse to ensure that least and greatest fixed point types are initial algebras and final coalgebras, respectively. This yields inductive and coinductive proof principles for reasoning about program behaviour. On the other hand, we argue that observational equivalence does not identify too many terms, by showing that tests induce a topology that, on streams, coincides with usual topology induced by the prefix metric. As one would expect, observational equivalence is, in general, undecidable, but in order to develop some practically useful heuristics we provide coinductive techniques for establishing observational normalization and observational equivalence, along with up-to techniques for enhancing these methods.

Extremely undecidable sentences

1982 ◽  
Vol 47 (1) ◽  
pp. 191-196 ◽  
Author(s):  
George Boolos
Keyword(s):  
Modal Logic ◽  
The Other ◽  
Propositional Modal Logic ◽  
Image Position

Let ‘ϕ’, ‘χ’, and ‘ψ’ be variables ranging over functions from the sentence letters P0, P1, … Pn, … of (propositional) modal logic to sentences of P(eano) Arithmetic), and for each sentence A of modal logic, inductively define Aϕ by[and similarly for other nonmodal propositional connectives]; andwhere Bew(x) is the standard provability predicate for PA and ⌈F⌉ is the PA numeral for the Gödel number of the formula F of PA. Then for any ϕ, (−□⊥)ϕ = −Bew(⌈⊥⌉), which is the consistency assertion for PA; a sentence S is undecidable in PA iff both and , where ϕ(p0) = S. If ψ(p0) is the undecidable sentence constructed by Gödel, then ⊬PA (−□⊥→ −□p0 & − □ − p0)ψ and ⊢PA(P0 ↔ −□⊥)ψ. However, if ψ(p0) is the undecidable sentence constructed by Rosser, then the situation is the other way around: ⊬PA(P0 ↔ −□⊥)ψ and ⊢PA (−□⊥→ −□−p0 & −□−p0)ψ. We call a sentence S of PA extremely undecidable if for all modal sentences A containing no sentence letter other than p0, if for some ψ, ⊬PAAψ, then ⊬PAAϕ, where ϕ(p0) = S. (So, roughly speaking, a sentence is extremely undecidable if it can be proved to have only those modal-logically characterizable properties that every sentence can be proved to have.) Thus extremely undecidable sentences are undecidable, but neither the Godel nor the Rosser sentence is extremely undecidable. It will follow at once from the main theorem of this paper that there are infinitely many inequivalent extremely undecidable sentences.

Bell inequality for quNits with binary measurements

2003 ◽  
Vol 3 (2) ◽  
pp. 157-164
Author(s):  
H. Bechmann-Pasquinucci ◽  
N. Gisin
Keyword(s):  
Quantum Cryptography ◽  
Bell Inequality ◽  
The Other ◽  
Entangled State ◽  
Maximally Entangled State ◽  
Von Neumann ◽  
Chsh Inequality ◽  
Intermediate States ◽  
Von Neumann Measurements

We present a generalized Bell inequality for two entangled quNits. On one quNit the choice is between two standard von Neumann measurements, whereas for the other quNit there are N^2 different binary measurements. These binary measurements are related to the intermediate states known from eavesdropping in quantum cryptography. The maximum violation by \sqrt{N} is reached for the maximally entangled state. Moreover, for N=2 it coincides with the familiar CHSH-inequality.

scholarly journals A Conflict Tolerant Logic of Explicit Evidence

2021 ◽  
Vol 27 (1) ◽  
pp. 124-144
Author(s):  
Thomas Studer
Keyword(s):  
Modal Logic ◽  
Quantum Physics ◽  
Thought Experiment ◽  
The Other ◽  
Justification Logic ◽  
Epistemic Modal ◽  
Epistemic Reasoning ◽  
Basic Properties

Standard epistemic modal logic is unable to adequately deal with the FrauchigerRenner paradox in quantum physics. We introduce a novel justification logic CTJ, in which the paradox can be formalized without leading to an inconsistency. Still CTJ is strong enough to model traditional epistemic reasoning. Our logic tolerates two different pieces of evidence such that one piece justifies a proposition and the other piece justifies the negation of that proposition. However, our logic disallows one piece of evidence to justify both a proposition and its negation. We present syntax and semantics for CTJ and discuss its basic properties. Then we give an example of epistemic reasoning in CTJ that illustrates how the different principles of CTJ interact. We continue with the formalization of the Frauchiger–Renner thought experiment and discuss it in detail. Further, we add a trust axiom to CTJ and again discuss epistemic reasoning and the paradox in this extended setting.

Diversity and evolution of stem structure in Eleocharis (Cyperaceae)

Rodriguésia ◽  
2021 ◽  
Vol 72 ◽  
Author(s):  
Jaqueline Priscila Bispo de Almeida Cardoso Marcondes ◽  
Rafael Trevisan ◽  
Maria Cecilia de Chiara Moço ◽  
Cleusa Bona
Keyword(s):  
Cross Section ◽  
Morphological Analysis ◽  
Plant Size ◽  
Molecular Data ◽  
The Other ◽  
Air Gaps ◽  
Ecological Aspects ◽  
The Cross ◽  
Poorly Drained Soils ◽  
Stem Structure

Abstract Eleocharis (Cyperaceae) includes more than 300 species of perennial or annual herbs, frequently found on poorly drained soils. The species are morphologicaly similar, and the taxonomy is difficult because their vegetative and reproductive structures are very reduced. Previous study on the stem architecture in the subgenus Limnochloa showed that anatomical features help in the interpretation of the evolution, taxonomy and ecological aspects of the group. Our objectives were to add new characters from the stem (= culm) structure, to explore the characters in a greater number of Eleocharis species, representatives of the other subgenera, and add these data in a new phylogenetic analysis with molecular data. The study covered 68 species obtained from herbaria and fixed material. In addition to the stem architecture, the internal organization of the tissues, the cross-section format, the presence or absence of stretched cells in aerenchyma air gaps and the plant size were included in the morphological analysis. Our data confirm that spongy aerenchyma pattern is the ancestral condition while the mixed and septate patterns occurred independent and punctually. Only the cross-section format was variable among specimens while the other characters were uniform and relevant for taxonomic use.

Phylogenetic relationships of Mesoamerican species of the genus Sciurus (Rodentia: Sciuridae)

Zootaxa ◽  
2007 ◽  
Vol 1525 (1) ◽  
pp. 31-40 ◽  
Author(s):  
FEDERICO VILLALOBOS ◽  
FERNANDO CERVANTES-REZA
Keyword(s):  
Phylogenetic Relationships ◽  
Maximum Parsimony ◽  
Morphological Analysis ◽  
Taxonomic Status ◽  
The Other

The phylogenetic relationships of Sciurus species present in Mesoamerica are addressed using a morphological analysis under Maximum Parsimony. Our results recovered the existence of two clades: one comprising S. aureogaster, S. colliaei, S. variegatoides and Syntheosciurus brochus and the other clade composed by S. richmondi, S. granatensis, S. deppei, S. yucatanensis and Microsciurus alfari. The taxonomic status of the genera Microsciurus and Syntheosciurus is discussed as well the biogeographic implications of these findings.

Morphological variation in the genus Juliomys (Rodentia: Cricetidae: Sigmodontinae) and taxonomic status of Juliomys anoblepas (Winge 1887) from the Quaternary of Southeast Brazil

Zootaxa ◽  
2020 ◽  
Vol 4861 (3) ◽  
pp. 429-443
Author(s):  
CAROLINA PIRES ◽  
MARCELO WEKSLER ◽  
CIBELE R. BONVICINO
Keyword(s):  
Principal Component Analysis ◽  
Morphological Variation ◽  
Morphological Analysis ◽  
Principal Component ◽  
Taxonomic Status ◽  
Morphological Characters ◽  
The Other ◽  
Anterior Portion ◽  
Living Species ◽  
Qualitative Characters

The region of Lagoa Santa, Minas Gerais, Brazil, is one of the most important karstic areas of the Brazilian Quaternary due to the faunistic diversity of living and extinct forms. Among them, some taxa remain poorly studied, as is the case of Calomys anoblepas Winge 1887. Despite the recent allocation of the taxon within Juliomys, its description and morphological analysis are condensed, based on comparative few specimens and on few informative characters. In this study, we investigate characters proposed to distinguish species of Juliomys, and reevaluate the taxonomic status of the fossil Juliomys anoblepas. We analyzed 80 cranio-dental morphological characters in 233 specimens represented by the four species currently recognized: J. pictipes (Osgood 1933), J. rimofrons Oliveira & Bonvicino 2002, J. ossitenuis Costa, Pavan, Leite & Fagundes 2007, and J. ximenezi Christoff, Vieira, Oliveira, Gonçalves, Valiati & Tomasi 2016. We also performed principal component analysis on eight craniodental measurements available for the J. anoblepas hypodigm. The review of morphological systems and the evaluation of the characters used in the literature revealed that there are no diagnostic characters in the anterior portion of the skull and in the molar series of Juliomys, being difficult to differentiate the fossil from the other living species. Only six qualitative characters were variable and applicable to the hypodigm of J. anoblepas. Characters are polymorphic, invariable, or the fossil is not sufficiently complete to determinate its states. The taxon could not be morphometrically differentiated from J. pictipes and J. ossitenuis. Based on the results presented herein, we consider J. anoblepas as a nomen dubium and restrict its name to the taxon’s hypodigm. 

On Von Neumann Regular Rings

1974 ◽  
Vol 17 (2) ◽  
pp. 283-284 ◽  
Author(s):  
Kwangil Koh
Keyword(s):  
Regular Ring ◽  
The Other ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Research Problems ◽  
Von Neumann Regular ◽  
Regular Rings

Recently, in the Research Problems of Canadian Mathematical Bulletin, Vol. 14, No. 4, 1971, there appeared a problem which asks “Is a prime Von Neumann regular ring pimitive?” While we are not able to settle this question one way or the other, we prove that in a Von Neumann regular ring, there is a maximal annihilator right ideal if and only if there is a minimal right ideal.

A new species of Ferula (Apiaceae) from Malta

Phytotaxa ◽  
2018 ◽  
Vol 382 (1) ◽  
pp. 74 ◽  
Author(s):  
SALVATORE BRULLO ◽  
CRISTIAN BRULLO ◽  
SALVATORE CAMBRIA ◽  
GIANPIETRO GIUSSO DEL GALDO ◽  
CRISTINA SALMERI ◽  
...  
Keyword(s):  
New Species ◽  
Close Relationships ◽  
Morphological Analysis ◽  
The Other ◽  
Taxonomic Group ◽  
Herbarium Material ◽  
Reproductive Structures ◽  
Leaf Segments ◽  
A New Species ◽  
Shape And Size

Ferula melitensis, a new species from the Malta Archipelago, is described and illustrated. It belongs to the taxonomic group of F. communis, showing close relationships mainly with F. tunetana, F. arrigonii, F. communis s.str. and F. glauca, from which it differs in several features regarding the stem, leaves (mainly for shape and size of terminal segments), flowers and mericarps; other significant differences regard the ecology and phenology. The investigations, carried out on living plants, herbarium material and literature data, regard the morphological analysis of vegetative and reproductive structures, the anatomy of terminal leaf segments and mericarps, as well as the shape and size of seedlings. Previous phylogenetic approaches, based on nuclear and plastidial genome, have allowed to clarify better the relationships among F. melitensis and the other species of this group.

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