In 1936, G. Birkhoff and J. v. Neumann published an article with the title The logic of quantum mechanics'. In this paper, the authors demonstrated that in quantum mechanics the most simple observables which correspond to yes-no propositions about a quantum physical system constitute an algebraic structure, the most important proper ties of which are given by an orthocomplemented and quasimodular lattice Lq. Furthermore, this lattice of quantum mechanical proposi tions has, from a formal point of view, many similarities with (...) a Boolean lattice L8 which is known to be the lattice of classical propositional logic. Therefore, one could conjecture that due to the algebraic structure of quantum mechanical observables a logical calculus Q of quantum mechanical propositions is established, which is slightly different from the calculus L of classical propositional logic but which is applicable to all quantum mechanical propositions. This calculus has sometimes been called 'quan tum logic'. However, the statement that propositions about quantum physical systems are governed by the laws of quantum logic, which differ from ordinary classical logic and which are based on the empirically well-established quantum theory, is exposed to two serious objec tions: Logic is a theory which deals with those relationships between various propositions that are valid independent of the content of the respective propositions. Thus, the validity of logical relationships is not restricted to a special type of proposition, e. g. to propositions about classical physical systems. (shrink)

This paper is concerned with the problem of the validity of Leibniz's principle of the identity of indiscernibles in physics. After briefly surveying how the question is currently discussed in recent literature and which is the actual meaning of the principle for what concerns physics, we address the question of the physical validity of Leibniz's principle in terms of the existence of a sufficient number of naming predicates in the formal language of physics. This approach allows us to obtain in (...) a formal way the result that a principle of the identity of indiscernibles can be justified in the domain of classical physics, while this is not the case in the domain of quantum physics. (shrink)

The hypotheses of weak and strong objectification of quantum mechanical observables, as well as theoretical arguments and experimental evidence against these hypotheses, are systematically reviewed.

Es werden die Veränderungen der Wissenschaftssprache der Physik untersucht, die durch den Übergang von der klassich-relativistichen Physik zur Quantenphysik erfolgt sind. Die neuen und prinzipiellen Beschränkungen der Möglichkeiten der Überprüfung wissenschaftlicher Aussagen führen zu Reduktionen der hypothetischen Annahmen, die der Sprache der klassischen Physik zu Grunde liegen. Diese Reduktionen haben ihrerseits Abschwächungen der syntaktischen Strukturen zur Folge, die besonders in der formalen Logik und der Modallogik deutlich werden. Diese auf schwächeren Prämissen basierenden Strukturen sind die Quanten-Logik und die Quanten-Modallogik, die (...) damit auch einen weiteren Geltungsbereich besitzen als die entsprechenden klassischen formalen Systeme. (shrink)

Kant’s theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant’s theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant’s theory of arithmetic can (...) be picked up, improved by reconstruction and defended under a contemporary perspective: the theory of numbers as products of rule following construction presupposing successive synthesis in time and the theory of arithmetic equations, sentences or “formulas”—as Kant says—as synthetic a priori. In order to do so, two calculi in terms of modern mathematics are introduced which formalise Kant’s theory of addition as a form of synthetic operation. (shrink)

Reichenbach, Grünbaum, and others have argued that special relativity is based on arbitrary conventions concerning clock synchronizations. Here we present a mathematical framework which shows that this conventionality is almost equivalent to the arbitrariness in the choice of coordinates in an inertial system. Since preferred systems of coordinates can uniquely be defined by means of the Lorentz invariance of physical laws irrespective of the properties of light signals, a special clock synchronization—Einstein's standard synchrony—is selected by this principle. No further restrictions (...) conerning light signal synchronization, as proposed, e.g., by Ellis and Bowman, are required in order to refute conventionalism in special relativity. (shrink)

Kant's theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant's theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant's theory of arithmetic can (...) be picked up, improved by reconstruction and defended under a contemporary perspective: the theory of numbers as products of rule following construction presupposing successive synthesis in time and the theory of arithmetic equations, sentences or "formulas"—as Kant says—as synthetic a priori. In order to do so, two calculi in terms of modern mathematics are introduced which formalise Kant's theory of addition as a form of synthetic operation. (shrink)

Since the advent of Modern Physics in 1905, we observe an increasing activity of “interpreting” the new theories. We mention here the theories of Special Relativity, General Relativity and Quantum Mechanics. However, similar activities for the theories of Classical Physics were not known. We ask for the reasons for the different ways to treat classical physics and modern physics. The answer, that we provide here is very surprising: the different treatments are based on a fundamental misunderstanding of the theories of (...) classical physics. (shrink)

Classical mechanics in phase space as well as quantum mechanics in Hilbert space lead to states and observables but not to objects that may be considered as carriers of observable quantities. However, in both cases objects can be constituted as new entities by means of invariance properties of the theories in question. We show, that this way of reasoning has a long history in physics and philosophy and that it can be traced back to the transcendental arguments in Kant’s critique (...) of pure reason. (shrink)

Compound propositions which can successfully be defended in a quantumdialogue independent of the elementary propositions contained in it, must have this property also independent of the mutual elementary commensur-abilities. On the other hand, formal commensurabilities must be taken into account. Therefore, for propositions which can be proved by P, irrespective of both the elementary propositions and of the elementary commensur-abilities, there exists a formal strategy of success. The totality of propositions with a formal strategy of success in a quantum dialogue (...) form the effective quantum logic. The propositions of the effective quantum logic can be derived from a calculus Q eff which is — on the other hand — equivalent to a lattice L qi.Propositions about measuring results are above all time dependent propositions A(S;t). In a dialogue, different partial propositions will have in general different time values. If one can (accidentally) win a material dialogue, this dialogue can be related to a single time value. For the propositions of the effective quantum logic there exist formal strategies of success, independent of the elementary propositions contained in it. All partial propositions appearing in the dialogue are formally commensurable. Therefore the propositions of effective quantum logic which can be proved by formal dialogues can always be related to a single time. They present a description of the system S considered in which all partial propositions can be related jointly to the state of S.Therefore in the effective quantum logic we have — in the limit of equal time values — a situation which corresponds conceptually to the description of the system (S; ψ) in Hilbert space. Consequently, one would expect that also the lattice L qi — except from the tertium non datur 8 — agrees with the lattice L q of subspaces of Hilbert space. It has been shown that these lattices are in fact isomorphic. (shrink)

The main goal of quantum logic is the bottom-up reconstruction of quantum mechanics in Hilbert space. Here we discuss the question whether quantum logic is an empirical structure or a priori valid. There are good reasons for both possibilities. First, with respect to the possibility of a rational reconstruction of quantum mechanics, quantum logic follows a priori from quantum ontology and can thus not be considered as a law of nature. Second, since quantum logic allows for a reconstruction of quantum (...) mechanics, self-referential consistency requires that the empirical content of quantum mechanics must be compatible with the presupposed quantum ontology. Hence, quantum ontology contains empirical components that are also contained in quantum logic. Consequently, in this sense quantum logic is also a law of nature. (shrink)

For an ‘explanation' of physical facts by laws of nature, we have to establish a relation between physical facts and laws of nature. It is an open question, whether the laws of nature govern the facts with necessity or whether the laws are related to the facts merely by supervenience. In addition, it is not quite clear, whether the known laws of physics describe only artificially simplified cases, e.g. isolated situations, or whether the laws of physics actually grasp real facts. (...) Known solutions of these problems refer to situations where laws of classical physics are applied to phenomena of classical physics. However, if the same laws were applied to matter of facts of the domain of modern physics, then in many cases there would be no ‘explanation' in the sense mentioned. These new problems can be treated either by additional ‘interpretations' of the theories in question, or by a radical change of the ontological preconditions of classical physics. (shrink)

In modern physics, the constant "c" plays a twofold role. On the one hand, "c" is the well known velocity of light in an empty Minkowskian space—time, on the other hand "c" is a characteristic number of Special Relativity that governs the Lorentz transformation and its consequences for the measurements of space—time intervals. We ask for the interrelations between these two, at first sight different meanings of "c". The conjecture that the value of "c" has any influence on the structure (...) of space—time is based on the operational interpretation of Special Relativity, which uses light rays for measurements of space—time intervals. We do not follow this way of reasoning but replace it by a more realistic approach that allows to show that the structure of the Minkowskian space—time can be reconstructed already on the basis of a restricted classical ontology (Mittelstaedt, Philosophie der Physik und der Raum-Zeit, Mannheim: BI-Wissenschaftsverlag, 1988 and Mittelstaedt, Kaltblütig: Philosophie von einem rationalen Standpunkt, Stuttgart: S. Hirzel Verlag, pp. 221-240, 2003), and that without any reference to the propagation of light. However, the space—time obtained in this way contains still an unknown constant. We show that this constant agrees numerically with "c" but that it must conceptually clearly be distinguished from the velocity of light. Hence, we argue for a clear distinction between the two faces of "c" and for a dualism of space—time and matter. (shrink)

Usually, quantum mechanics is considered as the prototype of a probabilistic theory. In contrast to statistical mechanics, dice throwing, and roulette game, quantum mechanical probability statements cannot be reduced to causally determined individual events, whose explicit calculation is, however, too complicated for all practical purposes. Even hypothetically, one must not assume that quantum mechanical events were determined in principle and merely computationally intractable, since that assumption would lead to probabilistic predictions which contradict quantum mechanics. Hence, the title of this article (...) seems somewhat surprising at first glance, and in particular it seems difficult to connect a probability free quantum mechanics with the work of John von Neumann. (shrink)

In modern physics, the constant “c” plays a twofold role. On the one hand, “c” is the well known velocity of light in an empty Minkowskian space–time, on the other hand “c” is a characteristic number of Special Relativity that governs the Lorentz transformation and its consequences for the measurements of space–time intervals. We ask for the interrelations between these two, at first sight different meanings of “c”. The conjecture that the value of “c” has any influence on the structure (...) of space–time is based on the operational interpretation of Special Relativity, which uses light rays for measurements of space–time intervals. We do not follow this way of reasoning but replace it by a more realistic approach that allows to show that the structure of the Minkowskian space–time can be reconstructed already on the basis of a restricted classical ontology, and that without any reference to the propagation of light. However, the space–time obtained in this way contains still an unknown constant. We show that this constant agrees numerically with “c” but that it must conceptually clearly be distinguished from the velocity of light. Hence, we argue for a clear distinction between the two faces of “c” and for a dualism of space–time and matter. (shrink)

The logic of quantum physical propositions can be established by means of dialogs which take account of the general incommensurability of these propositions. Investigated first are meta-propositions which state the formal truth of object-propositions. It turns out that the logic of these meta-propositions is equivalent to ordinary logic. A special class of meta-propositions which state the material truth of object-propositions may be considered as quantum logical modalities. It is found that the logic of these modalities contains all the quantum logical (...) restrictions and thus differs essentially from the modal calculi of ordinary logic. (shrink)

The author investigates which methods of naming objects are possible in the language of physics on the basis of the real physical conditions and to which extend objects thereby can be identified. It is shown that in the language of classical physics naming by designation is always possible. But this implies only the temporal identity of objects, not the "trans - world" - identity, which is important for modalities. In the language of quantum physics naming by designation is no longer (...) applicable to individuals but only to classes of equivalent objects. And for these classes temporal identity as well as the "trans - world" - identity can be produced, which means that an adequate semantics can be specified for the concept of possibility necessary in this language. (shrink)