The deduction theorem for quantum logic—some negative results

1990 ◽  
Vol 55 (2) ◽  
pp. 615-625 ◽  
Author(s):  
Jacek Malinowski
Keyword(s):  
Quantum Logic ◽  
Deduction Theorem ◽  
Orthomodular Lattices ◽  
Consequence Operation ◽  
Negative Results

AbstractWe prove that no logic (i.e. consequence operation) determined by any class of orthomodular lattices admits the deduction theorem (Theorem 2.7). We extend those results to some broader class of logics determined by ortholattices (Corollary 2.6).

scholarly journals The computing power of Turing machine based on quantum logic

10.29007/k8cb ◽  
2018 ◽  
Author(s):  
Yun Shang ◽  
Xian Lu ◽  
Ruqian Lu
Keyword(s):  
Quantum Logic ◽  
Turing Machine ◽  
Turing Machines ◽  
Computational Power ◽  
Orthomodular Lattices ◽  
Locally Finite ◽  
Computing Power ◽  
Truth Value ◽  
Quantum Language

Turing machines based on quantum logic can solve undecidableproblems. In this paper we will give recursion-theoreticalcharacterization of the computational power of this kind of quantumTuring machines. In detail, for the unsharp case, it is proved that&#931<sup>0</sup><sub>1</sub>&#8746&#928<sup>0</sup><sub>1</sub>&#8838L<sup>T</sup><sub>d</sub>(&#949,&#931)(L<sup>T</sup><sub>w</sub>(&#949,&#931))&#8838&#928<sup>0</sup><sub>2</sub>when the truth value lattice is locally finite and the operation &#8743is computable, whereL<sup>T</sup><sub>d</sub>(&#949,&#931)(L<sup>T</sup><sub>w</sub>(&#949,&#931))denotes theclass of quantum language accepted by these Turing machine indepth-first model (respectively, width-first model);for the sharp case, we can obtain similar results for usual orthomodular lattices.

scholarly journals Quantum categories for quantum logic

2019 ◽  
Vol 25 (1) ◽  
pp. 70-87
Author(s):  
Владимир Леонидович Васюков
Keyword(s):  
Quantum Logic ◽  
Orthomodular Lattice ◽  
Heyting Algebra ◽  
Theoretic Approach ◽  
Orthomodular Lattices ◽  
C Algebras ◽  
Categorical Structure ◽  
Abstract Case ◽  
Complete Preorder ◽  
Categorical Semantics

The paper is the contribution to quantum toposophy focusing on the abstract orthomodular structures (following Dunn-Moss-Wang terminology). Early quantum toposophical approach to "abstract quantum logic" was proposed based on the topos of functors $\mathsf{[E,Sets]}$ where $\mathsf{E}$ is a so-called orthomodular preorder category – a modification of categorically rewritten orthomodular lattice (taking into account that like any lattice it will be a finite co-complete preorder category). In the paper another kind of categorical semantics of quantum logic is discussed which is based on the modification of the topos construction itself – so called $quantos$ – which would be evaluated as a non-classical modification of topos with some extra structure allowing to take into consideration the peculiarity of negation in orthomodular quantum logic. The algebra of subobjects of quantos is not the Heyting algebra but an orthomodular lattice. Quantoses might be apprehended as an abstract reflection of Landsman's proposal of "Bohrification", i.e., the mathematical interpretation of Bohr's classical concepts by commutative $C^*$-algebras, which in turn are studied in their quantum habitat of noncommutative $C^*$-algebras – more fundamental structures than commutative $C^*$-algebras. The Bohrification suggests that topos-theoretic approach also should be modified. Since topos by its nature is an intuitionistic construction then Bohrification in abstract case should be transformed in an application of categorical structure based on an orthomodular lattice which is more general construction than Heyting algebra – orthomodular lattices are non-distributive while Heyting algebras are distributive ones. Toposes thus should be studied in their quantum habitat of "orthomodular" categories i.e. of quntoses. Also an interpretation of some well-known systems of orthomodular quantum logic in quantos of functors $\mathsf{[E,QSets]}$ is constructed where $\mathsf{QSets}$ is a quantos (not a topos) of quantum sets. The completeness of those systems in respect to the semantics proposed is proved.

scholarly journals Many-valued logics—implications and semantic consequences

2013 ◽  
Vol 5 (2) ◽  
pp. 145-166
Author(s):  
Katalin Pásztor Varga ◽  
Gábor Alagi
Keyword(s):  
Classical Logic ◽  
Propositional Logic ◽  
Matrix Method ◽  
Decision Problem ◽  
Modus Ponens ◽  
Deduction Theorem ◽  
Classical Propositional Logic ◽  
Consequence Operation ◽  
Semantic Consequence ◽  
Logical Calculi

Abstract In this paper an application of the well-known matrix method to an extension of the classical logic to many-valued logic is discussed: we consider an n-valued propositional logic as a propositional logic language with a logical matrix over n truth-values. The algebra of the logical matrix has operations expanding the operations of the classical propositional logic. Therefore we look over the Łukasiewicz, Post, Heyting and Rosser style expansions of the operations negation, conjunction, disjunction and with a special emphasis on implication. In the frame of consequence operation, some notions of semantic consequence are examined. Then we continue with the decision problem and the logical calculi. We show that the cause of difficulties with the notions of semantic consequence is the weakness of the reviewed expansions of negation and implication. Finally, we introduce an approach to finding implications that preserve both the modus ponens and the deduction theorem with respect to our definitions of consequence.

Deduction theorems within RM and its extensions

1999 ◽  
Vol 64 (1) ◽  
pp. 279-290 ◽  
Author(s):  
J. Czelakowski ◽  
W. Dziobiak
Keyword(s):  
Extension Property ◽  
Infinite Family ◽  
Relevant Logic ◽  
Congruence Extension Property ◽  
Deduction Theorem ◽  
Consequence Operation ◽  
The Family ◽  
Relative Congruence

AbstractIn [13], M. Tokarz specified some infinite family of consequence operations among all ones associated with the relevant logic RM or with the extensions of RM and proved that each of them admits a deduction theorem scheme. In this paper, we show that the family is complete in a sense that if C is a consequence operation with CRM ≤ C and C admits a deduction theorem scheme, then C is equal to a consequence operation specified in [13]. In algebraic terms, this means that the only quasivarieties of Sugihara algebras with the relative congruence extension property are the quasivarieties corresponding, via the algebraization process, to the consequence operations specified in [13].

Sequential method in quantum logic

1980 ◽  
Vol 45 (2) ◽  
pp. 339-352 ◽  
Author(s):  
Hirokazu Nishimura
Keyword(s):  
Quantum Logic ◽  
Proof Theory ◽  
Axiomatic System ◽  
Orthomodular Lattices ◽  
Predominant Role ◽  
Sequential Method ◽  
Von Neumann ◽  
Material Implication ◽  
Close Relationship ◽  
Logical Study

Since Birkhoff and von Neumann [2] a new area of logical investigation has grown up under the name of quantum logic. However it seems to me that many authors have been inclined to discuss algebraic semantics as such (mainly lattices of a certain kind) almost directly without presenting any axiomatic system, far from developing any proof theory of quantum logic. See, e.g., Gunson [9], Jauch [10], Varadarajan [15], Zeirler [16], etc. In this sense many works presented under the name of quantum logic are algebraic in essence rather than genuinely logical, though it is absurd to doubt the close relationship between algebraic and logical study on quantum mechanics.The main purpose of this paper is to alter this situation by presenting an axiomatization of quantum logic as natural and as elegant as possible, which further proof-theoretical study is to be based on.It is true that several axiomatizations of quantum logic are present now. Several authors have investigated the so-called material implication α → β ( = ¬α∨(α ∧ β)) very closely with due regard to its importance. See, e.g., Finch [5], Piziak [11], etc. Indeed material implication plays a predominant role in any axiomatization of a logic in Hilbert-style. Clark [4] has presented an axiomatization of quantum logic with negation ¬ and material implication → as primitive connectives. In this paper we do not follow this approach. First of all, this approach is greatly complicated because orthomodular lattices are only locally distributive.

Hermann Dishkant. The first order predicate calculus based on the logic of quantum mechanics. Reports on mathematical logic, no. 3 (1974), pp. 9–17. - G. N. Georgacarakos. Orthomodularity and relevance. Journal of philosophical logic, vol. 8 (1979), pp. 415–432. - G. N. Georgacarakos. Equationally definable implication algebras for orthomodular lattices. Studia logica, vol. 39 (1980), pp. 5–18. - R. J. Greechie and S. P. Gudder. Is a quantum logic a logic?Helvetica physica acta, vol. 44 (1971), pp. 238–240. - Gary M. Hardegree. The conditional in abstract and concrete quantum logic. The logico-algehraic approach to quantum mechanics, volume II, Contemporary consolidation, edited by C. A. Hooker, The University of Western Ontario series in philosophy of science, vol. 5, D. Reidel Publishing Company, Dordrecht, Boston, and London, 1979, pp. 49–108. - Gary M. Hardegree. Material implication in orthomodular (and Boolean) lattices. Notre Dame journal of formal logic, vol. 22 (1981), pp. 163–182. - J. M. Jauch and C. Piron. What is “quantum-logic”?Quanta, Essays in theoretical physics dedicated to Gregor Wentzel, edited by P. G. O. Freund, C. J. Goebel, and Y. Nambu, The University of Chicago Press, Chicago and London1970, pp. 166–181. - Jerzy Kotas. An axiom system for the modular logic. English with Polish and Russian summaries. Studia logica, vol. 21 (1967), pp. 17–38. - P. Mittelstaedt. On the interpretation of the lattice of subspaces of the Hilbert space as a propositional calculus. Zeitschrift für Naturforschung, vol. 27a no. 8–9 (1972), pp. 1358–1362. - J. Jay Zeman. Generalized normal logic. Journal of philosophical logic, vol. 7(1978), pp. 225–243.

1983 ◽  
Vol 48 (1) ◽  
pp. 206-208 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic ◽  
Propositional Calculus ◽  
Axiom System ◽  
Theoretical Physics ◽  
Reidel Publishing Company ◽  
Philosophical Logic ◽  
Orthomodular Lattices ◽  
Implication Algebras ◽  
The University

Impairment Questions and Answers

1999 ◽  
Vol 4 (4) ◽  
pp. 4-4
Keyword(s):  
Symptom Validity ◽  
Validity Testing ◽  
Symptom Validity Testing ◽  
Negative Results ◽  
Tunnel Vision ◽  
Questions And Answers ◽  
Blurry Vision ◽  
Fatigue Evaluation ◽  
Choice Testing ◽  
Rule Out

Abstract Symptom validity testing, also known as forced-choice testing, is a way to assess the validity of sensory and memory deficits, including tactile anesthesias, paresthesias, blindness, color blindness, tunnel vision, blurry vision, and deafness—the common feature of which is a claimed inability to perceive or remember a sensory signal. Symptom validity testing comprises two elements: A specific ability is assessed by presenting a large number of items in a multiple-choice format, and then the examinee's performance is compared with the statistical likelihood of success based on chance alone. Scoring below a norm can be explained in many different ways (eg, fatigue, evaluation anxiety, limited intelligence, and so on), but scoring below the probabilities of chance alone most likely indicates deliberate deception. The positive predictive value of the symptom validity technique likely is quite high because there is no alternative explanation to deliberate distortion when performance is below the probability of chance. The sensitivity of this technique is not likely to be good because, as with a thermometer, positive findings indicate that a problem is present, but negative results do not rule out a problem. Although a compelling conclusion is that the examinee who scores below probabilities is deliberately motivated to perform poorly, malingering must be concluded from the total clinical context.

Needed: A publication of negative results: Comment.

1959 ◽  
Vol 14 (9) ◽  
pp. 598-598 ◽  
Author(s):  
Leroy Wolins
Keyword(s):  
Negative Results

False negative results of real time strain elastography in thyroid nodular disease

2016 ◽  
Vol 37 (S 01) ◽  
Author(s):  
D Stoian ◽  
M Craciunescu ◽  
M Craina ◽  
S Pantea ◽  
F Varcus
Keyword(s):  
Real Time ◽  
False Negative ◽  
Strain Elastography ◽  
Negative Results ◽  
False Negative Results
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