[36] Bennett, M.K. Ethics: Ethics is the art of relating to other people. [111] Döring, A. and Isham, C. J., 2011, “What is a thing’ Topos theory in the foundations of physics”, in New structures in physics, B. Coecke (Ed. [107] Döring, A. and Isham, C. J., 2008, “A topos foundation for theories of physics: I. 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. Moreover, the quantum state can be interpreted as assigning probabilities to the different possible ways in which the set of determinate quantities can have values, where one particular set of values represents the actual but unknown values of these quantities. 527-555, Kluwer Academic Publisher, Dordrecht. In this respect, the ongoing research in QL (section 6) can prove to be an important guide to try to advance our comprehension of the phenomena implied by these technologies. and Wilce A. There are other versions in higher-order logic, in which higher order variables appear. This course is required for all electronics engineering students and electronics majors and minors. [73] da Costa, N.C.A. 11-18. [202] Putnam, H., 1968, “Is Logic Empirical?”, Boston Studies in the Philosophy of Science V, 5, 199-215. A theory T founded on the logic L, which contains a symbol for negation, is called inconsistent if it has among its theorems a sentence A and its negation ¬A; otherwise, it is said to be consistent. ), pp. ), pp. 249-265, J.-Y. Email: hfreytes@gmail.com ), pp. [135] Freytes, H. and Ledda, A., 2009, “Categories of semigroups in quantum computational structures”, Mathematica Slovaca, 59, 413-432 An example of this are the papers by Erik P. Verlinde. [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. 655-781, Elsevier, Amsterdam. [128] Foulis, D.J. Dialectical logic in Physics "My right to speak, your right not to listen" Dialectical logic in Physics. 381-405, Kluwer Academic Publishers, Dordrecht. On the one hand, there is the question about its interpretation, already stressed by Schrödinger in a letter to Einstein: “It seems to me that the concept of probability is terribly mishandled these days. In simple terms, logic gates are the electronic circuits in a digital system. These are important digital devices that are mainly based on the Boolean function. ), Princeton University Press, Princeton. [223] Titani, S., 1999, “Lattice Valued Set Theory”, Archive for Mathematical Logic, 38, 395-421. Quantum Logic (QL) was developed as an attempt to construct a propositional structure that would allow for describing the events of interest in Quantum Mechanics (QM). In this frame, the question of “which measures on questions are to be regarded as states?” [179, p. 85] was answered by Mackey’s student Andrew Gleason: A measure on the closed subspaces means a function µ which assigns to every closed subspace a nonnegative real number, such that if {Ai} is a countable collection of mutually orthogonal subspaces having closed linear spam B, then µ(B) = Σµ(Ai). The different characters proposed for the representatives of the logical connectives completely changes the meaning of these connectives. [37] Bennett, M.K. [53] Cattaneo, G., Dalla Chiara, M.L., Giuntini, R. and Leporini, R., 2004, “An unsharp logic from quantum computation”, International Journal of Theoretical Physics, 43, 1803-1817. The “new language to speak about infinity” is obviously a new ‘set’ theory, since set theory is usually known as “the theory of the (actual) infinite.” For a discussion about the necessity of a new set theory see for example [171, 134, 170, 77]. [184] Matoušek, M. and Pták, P., 2014, “Orthomodular posets related to Zvalued states”, International Journal of Theoretical Physics, 53, 3323-3332 [185] Mittelstäedt; P., 1978, Quantum logic, Reidel, Dordrecht. [193] Paoli, F., Ledda, A., Spinks, Freytes, H. and Giuntini, R., 2011, “Logics from sqrt MV-algebras”, International Journal of Theoretical Physics, 50, 3882-3902. [14] Aerts, D., 2009, “Quantum particles as conceptual entities: a possible explanatory framework for quantum theory”, Foundations of Science, 14, 361-411. The state-attribution proposition [A,σ] is defined as: [A,σ] = {w : ϕ(w)(< A,σ >) = 1} and means ‘A must have value in σ’. This construction is shown to be nothing but an orthocomplemented non-distributive lattice [42, Annexe I]. Inasmuch as this picture of physical properties is confirmed by the empirical success of QM, this view means we must accept that the way in which physical properties actually hang together is not Boolean. ), Reidel, Dordrecht. [179] Mackey, G.W., 1963, The Mathematical Foundations of Quantum Mechanics, Benjamin, Amsterdam. Béziau and D. Jacquette (Eds. [140] Girard, J-Y, “Linear logic”, 1987, Theoretical Computational Science, 50, 1-102. Systematic Zoology, 17(4), 438-457. [236] Von Weizsäcker, C. F., 1985, “Heisenberg’s Philosophy”, In Symposium on the Foundations of Modern Physics 1985, P. Lathi and P. Mittelslaedt (Eds. [159, p. 83]. If w is a physical situation in which system X exists, then X has both a dynamic state ϕ and a value state λ, that is, w =< ϕ,λ >. Logic gates are an important concept if you are studying electronics. [PσA is the projector over the corresponding subspace.] [121] Finkelstein, D., “The physics of logic”, in Paradigms and Paradoxes: The Philosophical Challenge of the Quantum Domain, R.G. In this respect, in order to clarify the vast map of interpretations about QM and to discover the physical meaning of the theory, one can consider the strategies that different interpreters have taken. It organisms regular biannual meetings: Castiglioncello 1992, Prague 1994, Berlin 1996, Liptovsky Mikulas 1998, Cesenatico 2001, Vienna 2002, Denver 2004, Malta 2006. [216] Smets, S., 2001, The Logic of Physical Properties in Static and Dynamic Perspective, Doctoral dissertation, Brussels Free University, Brussels. Many candidates are facing problems in collecting Maths, Physics and Chemistry Topic wise notes collection for JEE(Joint Entrance Exam). An extension of the square to a case in which the underlying structure is replaced by the algebra of QL has been provided in [137] and it may be useful to identify paraconsistent negations in the structure of QM (see also for discussion [89]). Put your understanding of this concept to test by answering a few MCQs. (3) The ‘XOR’ gate: From ‘NOT’, ‘AND’ and  ‘OR’ gate. [115] Dvure?enskij A. and Pulmannova, S., 2000, New Trends in Quantum Structures, Kluwer Academic Publishers, 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]. Students will be introduced to basic concepts in analog and digital electronics, basic semiconductor theory, and analog communication systems (AM, FM, and PM). [217] Smets, S., 2005, “The Modes of Physical Properties in the Logical Foundations of Physics”, Logic and Logical Philosophy, 14, 37-53. [203] Putnam, H., 1974, “How To Think Quantum Logically”, Synthese, 29, 55-61. 7]). [206] Randall C.H. Thus constructed, the structure avoids KS-type theorems. Of course, the way in which MIs attack the problem rests on the distinction between the realms of possibility and actuality. Given the Hilbert state spaces H1 and H2 as representatives of two systems, the pure states of the compound system are given by rays in the tensor product space H = H1⊗ H2. Benjamin, Inc., Reading. [231, p. 173]. So, [A,σ] may be read as ‘necessarily, < A,σ >’. Wartofsky (Eds. They provide a new form of quantum logic strongly connected with the fuzzy logic of continuous t-norms [151]. [214] Sakurai, J. J. and Napolitano, J., 2010, Modern Quantum Mechanics, Addison-Wesley, London. Research on computational semantics is being developed in connection with epistemic logics by members of the Italian group. Dynamics of concepts as studied by cognitive science are also being considered with the aid of quantum structures. 1) is called exclusive OR gate or the XOR gate. [22] Amira, H., Coecke, B. and Stubbe, I., 1998, “How quantales emerge by introducing induction within the operational approach”, Helvetica Physica Acta, 71, 554-572. [86] de Ronde, C., 2015, “Modality, Potentiality and Contradiction in Quantum Mechanics”, New Directions in Paraconsistent Logic, pp. What we found via the analysis of possible families of valuations is that a theorem which we called, for obvious reasons, the Modal KS (MKS) theorem can be derived which proves that quantum possibility, contrary to classical possibility, is also contextually constrained [102]. Moreover, compound systems also find their place in the topos approach [111]. On the other hand, one faces the problem of its very definition: The Born rule was incorporated in the axiomatization of QM as a noncommutative measure over the lattice of events by von Neumann in the early thirties, but this measure needs a modular lattice to be well posed, while L(H) is an orthomodular one. 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). (d) As per the result the output X has to be 1, so all the inputs of AND gate should be 1. i.e. This kind of many valued logic may be extended to the successive levels of language. That is, either proposition p or its negation ¬p is true (excluded middle principle), both assertions p and ¬p cannot be simultaneously true (PNC), meanings are sharp and unambiguous, and the meaning of a compound expression is determined by the meanings of its parts. [84] de Ronde, C., 2013, “Quantum Superpositions and Causality: On the Multiple Paths to the Measurement Result”, Preprint. 127-226, Elsevier, Amsterdam. [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. As remarked by Coecke the meaning of the superposition principle might be the key to understand QM: Birkhoff and von Neumann crafted quantum logic in order to emphasize the notion of quantum superposition. 39-56, TUCS General Publication. The main reason for this is that in physical theories the information about the state of affairs is encoded in what is called “the physical state.” Both in classical and QM there are states of maximal knowledge, but the logical implications that may be grasped from each situation are not the same. [200] Pulmannová, S. and Wilce, A., 1995, “Representations of D-posets”, International Journal of Theoretical Physics, 34, 1689-1696. Varadarajan’s and Jauch’s books [230, 159] follow from this. 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. Here, ϕ is an eigenstate of A, with corresponding eigenvalue a, exactly if ϕ(< A,{a} >) = 1. Click ‘Start Quiz’ to begin! By CHARLES GALTON DARWIN. The groups in Firenze directed by Dalla Chiara, and Cagliari directed by Giuntini, have also developed different languages for quantum computation. [227] Van Fraassen, B.C., 1981, “A modal Interpretation of Quantum Mechanics”, in Current Issues in Quantum Logic, 229-258, E. G. Beltrametti and B. C. van Fraassen (Eds. ), pp. [150] Halmos, P., 1995, “Algebraic Logic I, Monadic Boolean algebras”, Compositio Mathematica, 12, 217-249. and Piron, C., 1969, “On the structure of quantal proposition systems”, Helvetica Physica Acta, 42, 842-848. [87] de Ronde, C., 2016, “Representational Realism, Closed Theories and the Quantum to Classical Limit”, in R. E. Kastner, J. Jeknic-Dugic and G. Jaroszkiewicz (Eds. 139-161, Vienna. Arenhart and Krause [23, 24, 25] have raised several arguments against the paraconsistent approach to quantum superpositions which have been further analyzed in [86]. In fact, the application to computational science and more broadly to information processing needs to manage composite systems, one of the profound difficulties that faces orthodox QL. [152] Heisenberg, W., 1958, Physics and Philosophy, Ruskin House, London. The MI proposed by Kochen and Dieks (K-D, for short), proposes to use the so called biorthogonal decomposition theorem (also called Schmidt theorem) in order to describe the correlations between the quantum system and the apparatus in the measurement process. [238] Wilce, A., 1998, “Perspectivity and Congruence in Partial Abelian Semigroups”, Mathematica Slovaca, 48, 117-135. Given a dynamical state represented by the atom |e> ∈ L(H), one constructs the sublattice D(|e>, R) with Kolmogorovian probabilities defined over alternative subsets of properties in the sublattice. [11] Aerts, D., 1984, “Construction of a structure which makes it possible to describe the joint system of a classical and a quantum system”, Reports in Mathematical Physics, 20, 421-428. ), Quantum Structural Studies, World Scientific, Singapore. One application of mathematical logic in physics is in demonstrating the that Theory A can be derived from Theory B given a set of assumptions, definitions, or other axioms. Email: cderonde@gmail.com But, if one chooses, for a given state |e>, a preferred observable R, these properties can be taken as determinate since the propositions associated with R, that is, with the projectors in which R decomposes, generate a Boolean algebra. [132] Foulis, D.J. However, for those readers who wish to engage more profoundly with the subject of QL, the article provides an extensive bibliography. Beziau, M. Chakraborty and S. Dutta (Eds. One input to EX-NOR is 1(i.e. Includes index 1 Notes. 4]. Smets together with Alexandru Baltag have proposed two axiomatizations of the logic of quantum actions [218]. [178]. In the introduction to the paper they make the point: One of the aspects of quantum theory which has attracted the most general attention is the novelty of the logical notions it presupposes. [105] Domenech, G., Freytes, H. and de Ronde, C., 2009, “Many worlds and modality in the interpretation of quantum mechanics: an algebraic approach”, Journal of Mathematical Physics, 50, 072108. Quantum Mechanics and Macrosystems, Springer-Verlag, Berlin. In the following section we shall review some of these advancements that have taken place in relation to QM. During its development, QL started to refer not only to a logic, but also to the multiple lines of research that attempted to understand QM from a logical perspective. [50] Caspers, M., Heunen, C., Landsman, N. and Spitters, B., 2009, “Intuitionistic quantum logic of an n-level system”, Foundations of Physics, 39, 731-759. Quantum logic as the logic of observables. [138] Freytes, H. and Domenech, G., 2013, “Quantum computational logic with mixed states”, Mathematical Logic Quarterly, 59, 27-50. ), Kluwer Academic Publishers, Dordrecht. It has also played an important role within the foundational debates about the theory. And there's a good reason for this. 303-322, Plenum, New York. [164] Kaye, P., Laflamme, R. and Mosca, M., 2007, An Introduction to Quantum Computing, Oxford University Press, New York. Studying the correct principles of reasoning is not the same as studying the psychology of reasoning. Classical logic, in turn, allows truth values for all propositions and thus it is not adequate for propositions about a quantum system, where the empirical content of propositions is relevant when applying the rules of logic. Download preview PDF. However, it can be restricted due to the given physical space in the device. 5.3] Due to the prescription that every notion should be defined in terms of operations, this line of research is called operationalism. At a distance of 1 m from the ground it keeps the body weight of 50 kg, attached to it completely. and Bueno, O., 2009, “Non Reflexive Logics”, Revista Brasilera de Filosofía, 232, 181-196. Rather, logic is a non-empirical science like mathematics. The difficulties with the definition of a “good measure” over the Hilbert lattice made von Neumann abandon the orthodox formalism of QM in Hilbert space, to which he himself had contributed a great deal, and face the classification of the factors and their dimension functions which led to the subject of von Neumann’s algebras. The truth table of a two-input AND basic gate is given as; In NOT gate the output of a NOT gate attains the state 1 if and only if the input does not attain the state 1. There is a great amount of work in progress in QL from new quantum structures, to the use of non-reflexive logics, paraconsistent logics, dynamical logics, etc. This makes it necessary to consider the action of material points on one another as action-at-a-distance. However, it can be restricted due to the given physical space in the device. and Randall, C.H., 1978, “Manuals, morphisms and quantum mechanics”, in Mathematical Foundations of Quantum Theory, A. Marlow (Ed. 1σ is the characteristic function of the set σ of values. [162] Kalmbach, G., 1983, Ortomodular Lattices, Academic Press, London. [136] Freytes, H., 2010, “Quantum computational structures: categorical equivalence for square roots QMV-algebras”, Studia Logica, 95, 63-80. and Zeilinger, A., 2001, The Physics of Quantum Information: Quantum Criptography, Quantum Teleportation, Quantum Computation, Springer, Berlin. Woods (Eds. [230] Varadarajan, V.S., 1985, Geometry of Quantum Theory, Springer, Berlin. As Miklos Rédei states: To see why von Neumann insisted on the modularity of quantum logic, one has to understand that he wanted quantum logic to be not only the propositional calculus of a quantum mechanical system but also wanted it to serve as the event structure in the sense of probability theory. The combination of contexts has more consequences than the ones that occur when they are used separately. The representation of physical quantities with arrows”, Journal of Mathematical Physics, 49, 053517. [90] DeWitt, B., 1973, “The Many-Universes Interpretation of Quantum Mechanics”, In Foundations of Quantum Mechanics, 167-218, Academic Press, New York. Clearly, no classical logic is paraconsistent. This approach has technological applications in connection with quantum computation and robotics [18, 19]. In the case of an infinite-dimensional Hilbert space the modular law is not satisfied. Or we can describe the attempts for defining a precise scientific language that corresponds to the mathematical scheme. [239] Zafiris, E. and Karakostas, V., 2013, “A categorial semantics representation of quantum events”, Foundations of Physics, 43, 1090-1123. part1, Page 1 and Foulis, D.J., 1997, “Interval algebras and unsharp quantum logics”, Advances in Mathematics, 19, 200-215. [163] Kauark-Leite, P., 2004, The Transcendental Approach and the Problem of Language and Reality in Quantum Mechanics, Doctoral dissertation, Centre de Recherche en Epistémologie Appliquée – École Polytechnique, Paris. [154] Heunen, C. and Spitters, B., 2009, “A Topos for Algebraic Quantum Theory”, Communications in Mathematical Physics, 291, 63-110. [65] Coecke, B., Duncan, R., Kissinger, A. and Wang, Q., 2012, “Strong Complementarity and Non-locality in Categorical Quantum Mechanics”, in Proceedings of the 27th Annual IEEE Symposium on Logic in Computer Science LiCS 2012, pp. [100] Domenech, G. and Freytes, H., 2005, “Contextual logic for quantum systems”, Journal of Mathematical Physics, 46, 012102-1 – 012102-9. [55] Chajda, I. and Kühr, J., 2012, “A generalization of effect algebras and ortholattices”, Mathematica Slovaca, 62, 1045-1062. [112] Dvure?enskij, A. and Pulmannová, S., 1994, “Difference posets, effects, and quantum measurements”, International Journal of Theoretical Physics, 33, 819-850. 151-169, World Scientific, Johensuu. Historically, logic originated with the ancient Greek philosopher Aristotle. [114] Dvure?enskij, A., 1995, “Tensor product of difference posets and effect algebras”, International Journal of Theoretical Physics, 34, 1337-1348. In spring 2005, Dov Gabbay, Kurt Engesser, Daniel Lehmann and Jane Spurr had an excellent idea—to ask experts on quantum logic and quantum structures to write long chapters for the Handbook of Quantum Logic and Quantum Structures. It is closely related to the virtue of temperance (reason applied to the appetites). Mathematically, this happens by representing the state of a system of mass m by a point (p, q) in its corresponding phase space Γ of positions q and momenta p. Newton’s law tells us how this point moves along the path determined by the initial conditions. ), pp. It is also connected to the interpretation of the ‘Sasaki hook’—namely, the quantum implication that is the closest to the classical one—which may be understood in terms of a dynamic modality instead of in terms of deduction—in fact, it does not satisfy the deduction theorem [60]. Logic was further developed and systematized by the Stoics and by the medieval scholastic philosophers. The Boolean expression is Y = Aˉ\bar{A}Aˉ, read as Y equals NOT A. [13] Aerts, D., 2009, “Quantum structure in cognition”, Journal of Mathematical Psychology, 53, 314-348. This means that, regardless of its use in the literature, quantum possibility is not classical possibility. LOGIC AND PROBABILITY IN PHYSICS. This is not the case in QM. Aristotle’s architectonic determined the fate of both classical and medieval physics, as well as metaphysics. However, it was clear from the very beginning that this new quantum possibility was something completely different from that considered in classical theories. ), pp. 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]). New York, NY: Basic Books. In case of a finite-dimensional Hilbert space H, the ortholattice L(H) is modular, that is, satisfies the following condition known as the modular law: The modular law is equivalent to the identity (x∧y)∨(y∧z) = y∧((x∧y)∨z). The study of the modal character of QM was explicitly formalized in the seventies and eighties by a group of physicists and philosophers of science. In any Hilbert lattice the meet operation ∧ corresponds to set theoretical intersection between subspaces, and the join operation ∨ corresponds to the smallest closed subspace of H containing the set theoretical union of subspaces. [124] Foulis, D.J. [71] da Costa, N.C.A. Conjunction and disjunction are not associated to the join and meet lattice operations. Urs, therefore, are in fact nothing but qubits. [210] Rédei, M., 1998, Quantum Logic in Algebraic Approach, Kluwer Academic Publishers, Dordrecht. [20] Aerts, D. Gabora, L. and Sozzo, S., 2013, “Concepts and their dynamics: a quantum-theoretic modeling of human thought”, Topics in Cognitive Science, 5, 737-772. It is the purpose of this paper to show that, in any separable Hilbert space of dimension at least three, whether real or complex, every measure on the closed subspaces is derived in this fashion. [62] Coecke, B., Moore, D.J. In classical physics, the rules of classical logic hold in every context, but they also hold when merging the contexts. During the eighties and nineties, in line with the neo-Kantian QL line of research, the French philosopher Michel Bitbol analyzed the different alternatives of the language of physical properties and their role in objectivity. 155-160 DOI: 10.1126/science.88.2277.155 . In fact, the difficulties with a rigorous definition of probability were well known to von Neumann [212]. The logic gates are building blocks at digital electronics. It is easy to see that such a measure can be obtained by selecting a vector v and, for each closed subspace A, taking µ(A) as the square of the norm of the projection of v on A. The LMP Graduate Student Conference will bring together philosophers of logic, mathematics, and physics for two days of presentations and discussions with some of the leaders in these fields. First theorem – It states that the NAND gate is equivalent to a bubbled OR gate. While the first group started from a set of (classical) metaphysical presuppositions and intended to change the formalism in order to fit QM into their desired metaphysical picture [for example, Bohmian mechanics, Ghirardi-Rimini-Weber theory], a second group concentrated their efforts—taking as a standpoint the orthodox formalism—on trying to understand the symmetries and characteristics of the formalism in order to derive a suitable interpretation of the theory. [158] Jammer, M., 1974, Philosophy of Quantum Mechanics, Wiley, New York. [226] Van Fraassen, B.C., 1974, “The Einstein-Podolsky-Rosen paradox”, Synthese, 29, 291-309. In 1937, Kodi Husimi [156] showed that a weaker law, the so called orthomodular law is satisfied in the ortholattice L(H). Interpreting these operations as the logical connectives, they represent and (∧), or (∨) and not (¬). 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. When observable R ≠ I, there is a set of handles {|eri>,i = 1…k} given by the nonzero projections of |e> onto the eigenspaces of R and the spines represented by all the rays in the orthogonal complement of the subspace generated by the handles. The operationalist perspective of Birkhoff and von Neumann was implicitly related to the measurement problem (MP). It was Aristotle who created classical logic and used it in order to develop his own physical and metaphysical scheme, providing an answer to the problem of movement and knowledge set down by the Heraclitean and Eleatic schools of thought. (It is true ? [62, 60, 218] (see Section 5.2). The standpoint of this approach is the observation that QL is essentially a dynamical logic, that it is about actions rather than propositions [30]. We can either ask which language concerning the atoms has actually developed among physicists in the thirty years that have elapsed since the formulation of QM. University of Rosario [21] Aerts, D. and Sozzo, S., 2014, “Quantum entanglement in concept combination”, International Journal of Theoretical Physics, 53, 3587-3603. [228, p. 281]. 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. It is true that sometimes states involving probabilities occur in classical physics: think of the probability distributions ρ in statistical mechanics. (Ed. It is known that there are systems for which such a finite a priori probability does not exist. 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. ), pp. When dealing with many degrees of freedom, a statistical description is useful. 245-254, IEEE Publisher. […] The analysis can now be carried further in two entirely different ways. [57] Coecke, B., 2000, “Structural characterization of compoundness”, International Journal of Theoretical Physics, 39, 581-590. Several kinds of compound propositions may be defined by specifying the dialog-game. 223-242, Philosophia Verlag, Munich. [205] Randall C.H. physics, biology, or psychology. Potential or possible properties are then only considered as the points to which the system might arrive in a future instant of time. [75] da Costa, N.C.A. ), 50-64, Springer-Verlag, Berlin. Practicing All Logic Gates - Basic Science Entrance Exam Questions and Answers in online helps you to improve your ability to attend the real time maths, chemistry, physics Entrance Exams. [149] Gudder, S., 2002, “Quantum Computational Logic”, International Journal of Theoretical Physics, 42, 39-47. [44] Bohr, N., 1985, Collected Works, vol. Within QL many different philosophical approaches and lines of research have been developed, discussed and addressed. [16] Aerts, D., 2011, “Quantum interference and superposition in cognition. This article provides a map of these multiple approaches in order to introduce the very different strategies and problems discussed in the QL literature. Moore and A. Wilce (Eds. [161] Kalman, J. Later on, also Holger Lyre would argue in favor of this possibility: In quantum theory in particular, this view has a lot of plausibility. A lattice (L,∨,∧,1,0) is said to be modular when for all elements x, y and z, if x ≤ z, then. Following this line of thought, George Mackey published in 1963 a monograph [179] in which he recovered von Neumann’s idea of “projections as propositions” [234, p. 247]. QL relates the two seemingly different disciplines of physics and logic. Related with both a property and its applications valued logic may be read Y... Each one of the theory the tradition introduced by Niels Bohr and his logic still remained the basis correct! Terms, logic originated with the abolition of the set σ of values one starts denying collapse. Probability measure over it with its elements considered as the main results of system... Uniqueness theorem that characterizes D ( |e >, R ) [ 47,,... 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