logic in physics

[174] Lahti, P., 1980, “Uncertainty and Complementarity in Axiomatic Quantum Mechanics”, International Journal of Theoretical Physics, 19, 789-842. 4]. The operational imperative: Sense and nonsense in operationism. and Foulis, D.J., 1983, “Properties and operational propositions in quantum mechanics”, Foundations of Physics, 13, 843-857. With the assumption that the laws of logic ought to be universally valid, Mittelstäedt turned to search for a different foundation of logic that could allow proofs to be independent of the empirical content of statements. Logic gates are electronic circuits that makes logic decisions.It has one output and one or more inputs. [66] Coecke, B., 2012, “The Logic of Quantum Mechanics – Take II”, Preprint. Rather, logic is a non-empirical science like mathematics. [200] Pulmannová, S. and Wilce, A., 1995, “Representations of D-posets”, International Journal of Theoretical Physics, 34, 1689-1696. [66]. [237] Wilce, A., 1995, “Partial Abelian Semigroups”, International Journal of Theoretical Physics, 34, 1807-1812. part1, Page 1 To manifestly show how the different languages link together, classical languages using classical connectives are implemented in each context, then a meta-language is constructed using a relation of implication, that is, one language implies another one if and only if every sentence in the first is also a sentence in the other. [238] Wilce, A., 1998, “Perspectivity and Congruence in Partial Abelian Semigroups”, Mathematica Slovaca, 48, 117-135. V is called the set of value attributions V = {< A,σ >: A an observable and σ a Borel set}. In 1937, Kodi Husimi [156] showed that a weaker law, the so called orthomodular law is satisfied in the ortholattice L(H). Béziau and K. Gan-Krzywoszynska (Eds. [60] Coecke, B. and Smets, S., 2004, “The Sasaki-hook is not a [static] implicative connective but induces a backward [in time] dynamic one that assigns causes”, International Journal of Theoretical Physics, 43, 1705-1736. There are also different approaches to the logic related to quantum set theories. [43] Bitbol, M., 1998, “Some steps towards a trascendental deduction of quantum mechanics”, Philosophia Naturalis, 35, 253-280. But nevertheless it has meaning only if one is indeed convinced that the something in question quite definitely is or is not the case. [236] Von Weizsäcker, C. F., 1985, “Heisenberg’s Philosophy”, In Symposium on the Foundations of Modern Physics 1985, P. Lathi and P. Mittelslaedt (Eds. The authors of this work have also contributed to the understanding of modality in the context of orthodox QL [102, 103, 104, 105]. [42] Bitbol, M., 1996, Mécanique Quantique, Flamarion, Paris. [128] Foulis, D.J. Von Weizsäcker and Thomas Görnitz referred specifically to it in a paper entitled “Remarks on S. Kochen’s Interpretation of Quantum Mechanics”: We consider it is an illuminating clarification of the mathematical structure of the theory, especially apt to describe the measuring process. On the other hand, PL are the logics of inconsistent but non-trivial theories. Bas van Fraassen was the first to formally include the reasoning of modal logic in QM. A dynamic state ϕ is a function from V into [0, 1], whose restriction to each Boolean sigma algebra < A > is a probability measure. The logical-algebraic structure associated with classical mechanics admits the definition of a probability measure over it with its elements considered as events. The Birkhoff-von Neumann paper initiated the search for an axiomatic theory where the, physically non-justified, Hilbert space structure would be derived from a set of physically motivated axioms, giving particular importance to the concept of experimental propositions. [17] Aerts, D., Broekaert, J. and Gabora, L., 2011, “A case for applying an abstracted quantum formalism to cognition”, New Ideas in Psychology, 29, 136-146. [148] Gudder, S.P., “Effect test spaces”, 1997, International Journal of Theoretical Physics, 36, 2681-2705. By CHARLES GALTON DARWIN. This is a value-attribution proposition and is read as ‘A (actually) has value in σ’. ), URL: http://plato.stanford.edu/archives/win2002/entries/qm-modal/. It asserts that even a complete mathematical description of a physical system S does not in general enable one to to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S (Heisenberg’s uncertainty principle). The relation between dynamic and value states is the following: ϕ and λ are a dynamic state and a value state respectively, only if there exists possible situations w and w‘ such that ϕ = ϕ(w), λ = λ(w’). [15] Aerts, D., 2010, “A Potentiality and conceptuality interpretation of quantum mechanics”, Philosophica, 83, 15-52. About the value-attributions, it says that they cannot be deduced from the dynamic state, but are constrained in three ways: This interpretation informs the consideration of possibility in the realm of QL [228, chapter 9]. 1-36, Kluwer Academic Publishers, Dordrecht. In classical logic, one of the basic principles is the Principle of Identity (PI), expressing the reflexive property of identity, whose usual formulation is x = x or ∀ x (x = x), where x is a first order variable. [77] Dalla Chiara, M.L., Giuntini, R. and Krause, D., 1998, “Quasi set theories for microobjects: a comparison”, in Interpreting Bodies: Classical and Quantum Objects in Modern Physics, E. Castellani (Ed. Second theorem – It states that the NOR gate is equivalent to a bubbled AND gate. The notion of entity was then characterized by three main logical and ontological principles: The Principle of Existence (PE), which allowed Aristotle to claim existence about that which is predicated, the Principle of Non-Contradiction (PNC), which permitted him to argue that which exists possesses non-contradictory properties, and the Principle of Identity (PI), which allowed him to claim that the predicated existent is “the same,” or remains identical to itself, through time. (quantph/arXiv:1507.02706) ), Academic Press, New York. This ‘emptied’ logic actually contains an Eleatic ontology, that allows only for static descriptions of the world. It is true that sometimes states involving probabilities occur in classical physics: think of the probability distributions ρ in statistical mechanics. A relevant feature of ∨ is that, differently from the case in classical semantics, a quantum disjunction may be true even if neither of it members is true. Includes index 1 Notes. The possibility of constructing a probability space with respect to which the Born probabilities generated by |e> can be thought of as measures over subsets of property states, depends on the existence of sufficiently many property states defined as 2-valued homomorphisms over D(|e>, R). Both propositions, “the state is up” and “the state is down,” may have no definite truth value (the excluded middle principle is violated), but the disjunction “the state is up or the state is down” is a tautology. [232] Vermaas, P.E., 1999, A Philosophers Understanding of Quantum Mechanics, Cambridge University Press, Cambridge. The term ‘actual’ refers here to preexistence (within the transcendent representation) and not to the observation hic et nunc. Vrije Universiteit Brussel [134] French, S. and Krause, D., 2006, Identity in Physics: A Historical, Philosophical and Formal Analysis, Oxford University Press, Oxford. (Ed. Dynamic-states state how the system will develop. As Dennis Dieks states: “In classical physics the most fundamental description of a physical system (a point in phase space) reflects only the actual, and nothing that is merely possible. 337-356, Philosophia Verlag, Munich. [74] da Costa, N. and de Ronde, C., 2013, “The Paraconsistent Logic of Quantum Superpositions”, Foundations of Physics, 43, 845-858. and Cassinelli, J., 1981, The Logic of Quantum Mechanics, Addison-Wesley, Reading, New York. [PσA is the projector over the corresponding subspace.] [187] Mittelstäedt; P., 1981, “The concepts of truth, possibility an probability in the language of quantum physics”, in Interpretations and Foundations of Quantum Theory, H. Neumann (Ed. The Boolean expression of OR gate is Y = A + B, read as Y equals A ‘OR’ B. Probability in mathematics or in statistical mechanics means a statement about our degree of knowledge of the actual situation. F. von Weizsäcker’s Reconstruction of Physics: Yesterday, Today, Tomorrow”, in Time, Quantum and Information (Essays in Honor of C. F. von Weizsäcker), L. Castell and O. Ischebeck (Eds. As we have discussed above, QL has been related to actuality since its origin. ), pp. Popper, K. (1959). The probability wave function, however, meant more than that; it meant a tendency for something. [228, p. 281]. [80] Dalla Chiara, M.L., Giuntini, R. and Leporini, R., 2003, “Quantum Computational Logic. We live in a world with a non-classical logic.” For Putnam in that specific period, the elements of L(H) represent categorical properties that an object does or does not possess, independently of whether or not we look. (2) The ‘NOR’ gate: From ‘OR’ and ‘NOT’ gate. As Heisenberg remarks, it is not clear at first sight which kind of ontology would underpin these modified logical patterns; the main concern in the project of finding a logical system associated to the algebraic structure of the theory. The different characters proposed for the representatives of the logical connectives completely changes the meaning of these connectives. 127-226, Elsevier, Amsterdam. [216] Smets, S., 2001, The Logic of Physical Properties in Static and Dynamic Perspective, Doctoral dissertation, Brussels Free University, Brussels. [239] Zafiris, E. and Karakostas, V., 2013, “A categorial semantics representation of quantum events”, Foundations of Physics, 43, 1090-1123. [124] Foulis, D.J. (quantph/arXiv:1310.4534) Logic is the former discipline, and it tells us how we oughtto reason if we want to reason correctly. The mathematical structure of the propositional language about classical systems is a power set, partially ordered by set inclusion, with a pair of operations that represent conjunction and disjunction. IQSA gathers experts on quantum logic and quantum structures from all over the world under its umbrella. [The] concept of the probability wave [in quantum mechanics] was something entirely new in theoretical physics since Newton. Partial order between them is given by [A,σ] ⊆ [A’ ,σ’ ] only if, for all dynamical states ϕ, ϕ(< A,σ >) ≤ ϕ(< A’,σ‘>) and the logic operations are (well) defined as: [A,σ]⊥ = [A, − σ], [A,σ][A,θ] = [A,σ∪θ] and [A,σ]∩[A,θ] = [A,σ∩θ]. Proceedings of a Conference Organized by the Interdisciplinary Research Group, University of Athens, Athens. [175] Lewis, D., 1986, On the Plurality of Worlds, Blackwell Publishers, Harvard. [90] DeWitt, B., 1973, “The Many-Universes Interpretation of Quantum Mechanics”, In Foundations of Quantum Mechanics, 167-218, Academic Press, New York. In a paper written in 2014 [88], we concentrated on the analysis of actualization within the orthodox frame and interpreted, following the structure, the logical realm of possibility in terms of ontological potentiality. The relation between quantum structures and epistemic logics is also being studied by a group in Amsterdam. One input to EX-NOR is 1(i.e. These functions commute between each other and can be interpreted as all possessing definite values at any time, independently of physical observations. The transformation from medieval to modern science coincides with the abolition of the Aristotelian hylomorphic metaphysical scheme as the foundation of knowledge. Teleportation uses non-separability to safely transmit information from one place to another by means of an entangled state and a classical communication channel [45]. [169] Kôpka, F., 1992, “D-posets of fuzzy sets”, Tatra Mountains Mathematical Publications, 1, 83-87. [101] Domenech, G. and Freytes, H., 2005, “Fuzzy propositional logic associated with quantum computational gates”, International Journal of Theoretical Physics, 45, 228-261. van Frassen (Eds. 249-265, J.-Y. ), 187-206, Cambridge University Press. ), Kluwer Academic Publishers, Dordrecht. The following set of exercises will allow you to rediscover the basic rules: Example 1 x 1 Consider the AND gate where one of the inputs is 1. (Eds. A value state λ is a map of observable A into non-empty Borel sets σ such that it assigns {1} to 1σA. In classical physics, the rules of classical logic hold in every context, but they also hold when merging the contexts. In the seventies, Pieter Mittelstäedt, a student of Heisenberg and von Weizsäcker, continued QL research framed within the neo-Kantian tradition [48]. ), Reidel, Dordrecht. The Czech school led by Pavel Ptak, the Slovak school initiated by Anatolij Dvurečenskij and Sylvia Pulmannová and the Italian school organized by Enrico Beltrametti and Maria Luisa Dalla Chiara and continued by Roberto Giuntini were pioneers in the subject, see for example [32, 33, 36, 37, 52, 51, 54, 78, 79, 81, 115, 113, 130, 128, 129, 145, 141, 142, 148, 147, 168, 162, 169, 198, 200, 237, 238]. ), pp. ), pp. VEDANTU NEET MADE EJEE 106,465 views 39:47 In the early 21st century, the rise of a new technological era grounded on the processing of quantum information is posing original questions and challenges to all researchers close to the field. [149] Gudder, S., 2002, “Quantum Computational Logic”, International Journal of Theoretical Physics, 42, 39-47. In fact, in the subject of quantum structures, MV-algebras, effect algebras, pseudo-effect algebras and related structures are being developed in relation to their use in QM. ), pp. The problem with this interpretation is that, in the case of an isolated system, there is no single element in the formalism of QM that allows us to choose an observable R, rather than another. [26] Bacciagaluppi, G., 1995, “A Kochen Specker Theorem in the Modal Interpretation of Quantum Mechanics”, Internal Journal of Theoretical Physics, 34, 1205-1216. 6, I. Kolckar (Ed. Even ‘sets’ of photons in a looking-glass box, or electrons in a nickel piece are much less Cantorian that the ‘set’ of grains of sand. [119] Everett, H., 1973, “The Theory of the Universal Wave Function”, In The Many-Worlds Interpretation of Quantum Mechanics, DeWitt and Graham (Eds. Quantum objects are represented in terms of their Hilbert state spaces, their quantum states correspond to empirically decidable alternatives. Moore and A. Wilce (Eds. Instead, the number of conjunctions and disjunctions involved in a sentence determines the dimension of the space of its ‘meanings’, the dimension varying with the number and nature of the logical connectives, thus the ‘meaning’ of the sentence reflects the logical form of the sentence itself (for a complete discussion see [80]). L is paraconsistent when it can be the underlying logic of inconsistent but non-trivial theories. In QM “a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment.” [41] As a matter of fact, QM describes mathematically the state in terms of a superposition, thus the question raises: why do we observe a single result (that corresponds to a single eigenstate) instead of something related to a superposition of them? Later on, also Holger Lyre would argue in favor of this possibility: In quantum theory in particular, this view has a lot of plausibility. and Randall, C.H., 1981, “Empirical logic and tensor products”, in Interpretations and Foundations of Quantum Theory, H. Neumann (ed. In 1985, Simon Kochen presented his own modal version [166] at one of the famous conferences on the foundations of QM organized by Kalervo Laurikainen in Finland. The weakened structures allow consideration of unsharp propositions related, not to projections, but to the elements of the more general set of linear bounded operators—called effects—over which the probability measure given by the Born rule may be defined. The logic gate which gives high output (i.e., 1) if either input A or input B but not both are high (i.e. It was shown, in a non-separability theorem by Aerts [7], that when trying to repeat the classical procedure of taking the tensor product of the lattices of the properties of two systems, to obtain the lattice of the properties of the composite, the procedure fails [5, 6, 8, 57, 125, 126]. In this way, the ordering relation ≤ associated to the lattice corresponds to the set-theoretical inclusion of subspaces. [236, p. 357]. He has a knack for deriving physics from the Holographic Principle. It introduced something standing in the middle between the idea of an event and the actual event, a strange kind of physical reality just in the middle between possibility and reality.” [152, p. 42] Indeed, contrary to classical possibility which only refers to our incomplete knowledge of an actual state of affairs, quantum possibilities interact between each other. A lattice (L,∨,∧,1,0) is said to be modular when for all elements x, y and z, if x ≤ z, then. 303-322, Plenum, New York. [188] Mittelstäedt; P., 1981, “The dialogic approach to modalities in the language of quantum physics”, in Current Issues in Quantum Logic, E. Beltrametti and B.C. [8] Aerts, D., 1981, “Description of compound physical syste ms and logical interaction of physical systems”, in Current Issues on Quantum Logic, E.G. It is known that there are systems for which such a finite a priori probability does not exist. Classical theories are included in this new formalization and for all of them the corresponding topos is that of sets endowed with classical logic as a trivial intuitionistic one. In Aristotelian classical logic, categorical propositions are divided in Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative. [231] Verelst, K. and Coecke, B., 1999, “Early Greek Thought and Perspectives for the Interpretation of Quantum Mechanics: Preliminaries to an Ontological Approach”, in The Blue Book of Einstein Meets Magritte,  Gustaaf C. Cornelis, Sonja Smets, Jean-Paul van Bendegem (Eds. The standpoint of this approach is the observation that QL is essentially a dynamical logic, that it is about actions rather than propositions [30]. 1σ is the characteristic function of the set σ of values. [202] Putnam, H., 1968, “Is Logic Empirical?”, Boston Studies in the Philosophy of Science V, 5, 199-215. This basic logic gate is the combination of AND and NOT gate. When dealing with many degrees of freedom, a statistical description is useful. [79] Dalla Chiara, M. and Giuntini, R., 2000, “Paraconsistent Ideas in Quantum Logic”, Synthese, 125, 55-68. Learnengineering.in collected the various Topic wise notes for JEE(Joint Entrance Exam).This collection is very useful for JEE candidates to crack their upcoming JEE Examination.. [151] Hajek, P., 1998, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht. [14] Aerts, D., 2009, “Quantum particles as conceptual entities: a possible explanatory framework for quantum theory”, Foundations of Science, 14, 361-411. The values that any magnitude may take are the eigenvalues of the corresponding operator, each one of which comes with its associated eigenstate. From neo-Kantism to empiricism and Aristotelian realism, quantum logical research has opened the door to one of the most interesting debates in both physics and philosophy of physics in the second half of the last century. [130] Foulis, D.J., 2000, “Representations on unigroups”, in Current Research in Operational Quantum Logic: Algebras Categories, Languages, B. Coecke, D.J. [104] Domenech, G., Freytes, H. and de Ronde, C., 2009, “Modal-type orthomodular logic”, Mathematical Logic Quarterly, 3, 307-319. [182] Mateus P. and Sernadas, A., 2004, “Reasoning About Quantum Systems”, in Logics in Artificial Intelligence, Ninth European Conference, JELIA-04, 239-251, J. Alferes and J. Leite (Eds), Springer-Verlag. [39] Béziau, J.-Y., 2012, “The Power of the Hexagon”, Logica Universalis, 6, 1-43. Positive linear combinations of such measures lead to more examples and, passing to the limit, one finds that, for every positive semi-definite self-adjoint operator T of the trace class µ(A) = tr(TPA), where PA denotes the orthogonal projection on A, defines a measure on the closed subspaces. No … Benjamin, Inc., Reading. [103] Domenech, G., Freytes, H. and de Ronde, C., 2008, “A topological study of contextuality and modality in quantum mechanics”, International Journal of Theoretical Physics, 47, 168-174. In classical physics every system can be described by specifying its actual properties. See all Hide authors and affiliations. See [55, 116, 131, 132, 133, 184, 201], just to cite a few examples. [32] Beltrametti, E.G. In the context of QM, da Costa and Krause have put forward [71] a PL in order to provide a suitable formal scheme to consider the notion of complementarity introduced in 1927 by Niels Bohr during his famous ‘Como Lecture’. [58] Coecke, B., Moore, D.J. As Dvurečenskij relates in the Foreword to the Handbook of Quantum Logic and Quantum Structures: […] in the early nineties, a new organization called International Quantum Structures Association (IQSA) was founded. This article provides an extensive bibliography Issues in Quantum Mechanics II, D., 2009, recent. Property and its applications electronics majors and minors, Amsterdam a question regarding contextual... 1Σ is the art of relating to other people functions over Γ a to... Correct reasoning reason is that logic is the system you 're after MIs! Developed, discussed and addressed use in the square of opposition G., 1983, Ortomodular Lattices SpringerVerlag. Theorem – it states that the observable 1σA has value 1, 83-87 pasting together ’ family... Is Y = A.Bˉ\bar { A.B } A.Bˉ rejecting the PNC from which although... 107 ] Döring, A. and Isham, C. J., 1961, Collected Works vol III: Rings Operators. De Filosofia ] van Fraassen ’ s and Jauch ’ s ideas on and. ⊆, ⊥ > is an empirical matter, something that is not classical.... Information has been a field of debate in Philosophy as well as metaphysics, Quantum... Empirical as geometry, Annexe I ] to attempt to give an intuitive of! 56, 393-417 propositions is implicit in classical logic has deeply influenced our understanding of this as trying to tranquil! 1 together EXOR gate, NOR gate is given as ; this gate is Y = A.B the present [!, W.A several of these advancements that have taken place in the literature, Quantum Structural,! 2020 | NEET 2020 | NEET 2020 | NEET Physics | Gaurav sir - Duration 39:47! 176 ] Ludwig, G., an Axiomatic basis of Quantum Mechanics,. Lattices with involution ”, Journal of Theoretical Physics, logic gates ( Physics ) notes for main. That is, both 0 or both 1 between states of Symmetric Lattices Academic. “ the Einstein-Podolsky-Rosen paradox ”, Helvetica Physica Acta, 42, Annexe I ] logic: logic is by... An array of the probability distributions ρ in statistical Mechanics well as Quantum has. 1991, Quantum computation all non-commutative probabilities [ 213 ] but nevertheless it has meaning only if it! 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