Quantum logic

As quantum logic ( English quantum logic ) tests are referred to a logical system to formulate the principles of quantum mechanics is just.

The structures of quantum physics seem paradoxical and are sometimes difficult to understand. Questions such as whether Schrödinger's cat is alive challenge understanding. In the context of the mathematical structures of the Schrödinger equation and Heisenberg's uncertainty relation , a logic was sought that is based on interpretations of quantum mechanics such as the principle of complementarity or the principle of correspondence . To do this, the conventional logic had to be modified.

There are essentially three different approaches to quantum logic:

John von Neumann and Garrett Birkhoff were the first to discover a so-called orthomodular logic in the mathematical structures of quantum physics ( Hilbert space , Hamilton operator ), which deviated from the previously common Boolean algebra .
Hans Reichenbach and others developed a three-valued quantum logic from a probability logic with the truth values true, false and indefinite.
Peter Mittelstaedt , Ernst-Walther Stachow and Carl Friedrich von Weizsäcker developed the dialogical logic into a temporal logic of quantum processes.

In 1968, Hilary Putnam took quantum logic as an opportunity to question the a priori validity of logical laws as a whole, which sparked a debate about the status of logical and algebraic laws.

overview

Initial problem

Together with the theory of relativity, quantum physics, which emerged around 1900, acted like a revolution in physics. The double slit experiment raised a number of questions, in particular whether electrons or light quanta are particles or waves . In 1926, six works of published Erwin Schrödinger , which eventually led to a complex differential equation led that one Schrödinger equation is called. In 1927 Heisenberg's uncertainty principle was formulated. It states that two complementary properties of a particle cannot be determined at the same time with any precision. The world of elementary particles seemed to be very different from what was known from previous physics.

If one wants to develop a logic from quantum mechanics, the distributive law of the connection of and ( ) and or ( ) is violated. This emerges from the uncertainty relation: ${\ displaystyle \ wedge}$ ${\ displaystyle \ vee}$

Let p be the statement: "The electron is fast" (The measurement of the momentum gives a number in a certain interval.)
Let q be the statement: “The electron is in a left interval” and
Let r be the statement: “The electron is in a right interval”.

Let q and r be statements about two neighboring spatial intervals, which together guarantee the location of the electron even if there is uncertainty. Then it is true , but no longer necessarily according to the uncertainty principle . The distributive law, however, says that both expressions are identical. This leads to the rejection of the classical distributive logic , because the momentum and position of the electron cannot both be determined exactly at the same time. ${\ displaystyle p \ wedge (q \ vee r)}$ ${\ displaystyle (p \ wedge q) \ vee (p \ wedge r)}$

Progress

John von Neumann initially suggested interpreting statements about observable physical quantities as projections in a Hilbert space . He and Birkhoff developed a suitable calculus in their publication from 1936. An axiomatization of this system was undertaken by George Mackey . The quantum logic research was continued from 1963–1968 mainly in Switzerland by Josef-Maria Jauch and Constantin Piron . Saul Kripke's translation into a modal logic calculus formed the basis for the work of Bas van Fraassen and later Maria L. Dalla Chiara .

At the same time, Reichenbach had started to use trivalent logics to describe quantum mechanics. V. Weizsäcker's suggestion to develop a dialogical calculus has been implemented since 1970, especially in Cologne (Peter Mittelstaedt, Ernst-Walther Stachow). Other research centers since the 1970s have been Genoa (Enrico Beltrametti) and Amherst (Charles H. Randall, David J. Foulis). In 1976 the first international meeting of quantum logicians took place in Bad Homburg ; This was followed by colloquia in Erice / Sicily in 1979 and in Cologne in 1984, at which the wide range of philosophical, logical, linguistic , algebraic , geometric and probabilistic research on the subject of quantum logic became visible, which are now represented in various sections of the IQSA .

Mittelstaedt carried out extensions of the quantum logic to a relativistic quantum logic and developed a quantum ontology . Von Weizsäcker was looking for a quantum theory of the original alternatives in which the quantum logic is iterated and applied to the original alternatives.

Debate about the empiricality of logic

The research on quantum logic raised questions about the status of logic in general. Their deviations from classic logic call into question whether this corresponds to the physical context. This calls into question the validity of classical logic. It can even be doubted whether a single logical system can claim to provide a correct and adequate framework for a truth-preserving description of scientific processes and conditions. By Hilary Putnam was proposed prominently that only a quantum logic may be correct, the validity of which is not analytically us open up, making it an empirical will question what logic is correct. One reply comes mainly from Michael Dummett , who added the problem to the modern realism debate. Putnam's demand for the questioning of logic presupposes a realistic position, which in turn presupposes both the distributivity and the bivalence principle of classical logic. The question of the validity of logic for the world presupposes that the statements in which the world can be fundamentally described are true or false.

Approaches in detail

Propositional calculation according to Birkhoff and von Neumann

In a 1936 working paper, John von Neumann and Garrett Birkhoff suggested interpreting the operators of the Schrödinger equation as statements about the quantum system:

The elementary statements of logic correspond to the projection operators .

The eigenvalues 1 and 0 correspond to the truth values of this logic.

Quantum logic was born. However, it deviated from the conventional logic in some points. The algebraically formulated logic system of Boolean algebra had to be revised.

Corresponding to the algebraic relationships, there are relationships between the statements that form a calculus in which - contrary to classical propositional logic - the distributive law is replaced by the so-called orthomodularity and the tertium non datur only applies to a limited extent.

Quantum logic can be formalized in mathematical language analogously to the modular association . First nine basic rules are given here; the rule package is called the orthologic OL . The dash is a line of consequence , so the rule says that you can move from the statements above to the statements below:

No.	rule	designation
1	${\ displaystyle {\ frac {A} {A}}}$	From A follows A: reflexive conclusion
2	${\ displaystyle {\ frac {A \ wedge B} {A}}}$	${\ displaystyle \ wedge}$ - Elimination 1
3	${\ displaystyle {\ frac {A \ wedge B} {B}}}$	${\ displaystyle \ wedge}$ - Elimination 2
4th	${\ displaystyle {\ frac {\ lnot \ lnot A} {A}}}$	Duplex negatio affirmat
5	${\ displaystyle {\ frac {A} {\ lnot \ lnot A}}}$	Double negation introduction
6th	${\ displaystyle {\ frac {A \ wedge \ lnot A} {B}}}$	Ex contradictione sequitur quodlibet
7th	${\ displaystyle {\ frac {A \ rightarrow B, B \ rightarrow C} {A \ rightarrow C}}}$	transitive chain link
8th	${\ displaystyle {\ frac {A \ rightarrow B, A \ rightarrow C} {A \ rightarrow B \ wedge C}}}$	${\ displaystyle \ wedge}$ Introduction with premises A
9	${\ displaystyle {\ frac {A \ rightarrow B} {\ lnot B \ rightarrow \ lnot A}}}$	Counterposition
10	${\ displaystyle {\ frac {A \ wedge (\ lnot A \ vee (A \ wedge B))} {B}}}$	Orthomodularity

According to André Fuhrmann , the 10th axiomatic rule, orthomodularity, is notated in the form of junction logic, like the other nine rules of OL . It consists of the modularity law ( implied ) on the one hand and reflections on the other. These are so-called orthocomplements that take on the function of negation. ${\ displaystyle x \ leq b}$ ${\ displaystyle x \ vee (a \ wedge b) = (x \ vee a) \ wedge b}$

Logicians examine logics, among other things, to see whether they are decidable ; so this logic system has also been extensively studied. The rule of orthomodularity does not correspond to any framework condition that can be formulated in the first level of logic , which is why its decidability has not yet been proven.

Notation	Way of speaking
${\ displaystyle \ Diamond p}$	It is possible that p
${\ displaystyle \ Box p}$	It is necessary that p
${\ displaystyle \ Diamond p \ wedge \ Diamond \ neg p}$	p is contingent

In 1963 Saul Kripke was able to develop a model (Kripke frame K :) for the multitude of modal logic systems proposed up to then . The above-mentioned orthology OL can be mapped into the intuitionistic modal logic and fully characterized by a class of Kripke frames. On this axiomatic basis, Bas van Fraassen ( Toronto ) and Maria L. Dalla Chiara ( Florence ) used modalities in the context of quantum logic since the 1970s . Franz Josef Burghardt further developed the modal logic of quanta. ${\ displaystyle \ Box (A \ to B) \ to (\ Box A \ to \ Box B)}$

Trivalent logic

The statue Quantum Man (2006) by Julian Voss-Andreae shows the different view of one and the same real as attempted in quantum logic.

Since the classically presupposed commensurability condition does not have to be fulfilled in quantum mechanics , some scientists such as B. Paulette Destouches-Février, Hans Reichenbach and Bas van Fraassen tried to introduce a trivalent logic as quantum logic. However, this leaves the principle of two-valence .

Van Fraassen developed an exclusionary negation . If a physical variable m does not take on a certain value - for example 7 - this can mean in terms of exclusion negation not only that m is not 7, but also that the system is not in any state for which a value of m heard.

Hans Reichenbach claims that when assessing scientific statements one can only base oneself on considerations of probability. Science should not be expected to be certain. In the 1930s and subsequent years he worked on probabilistic problems . For the logical description of quantum mechanics, Reichenbach created a three-valued quantum logic from this probability logic with the truth values true, false and indefinite. It uses three types of negation (exclusive, diametrical, and full negation) and three types of implication (standard implication , alternative implication , quasi- implication ). After Ulrich Blau put a trivalent logic of natural language up for discussion, a parallel was drawn to the trivalent in Reichenbach, because everyday examples for the case of unfulfilled presuppositions suggest such an assessment.

The following truth tables with false (f), indefinite (u) and true (w) apply to the joiners and ( ) and or ( ): ${\ displaystyle \ wedge}$ ${\ displaystyle \ vee}$

a and b
b a	f	u	w
f	f	f	f
u	f	u	u
w	f	u	w

a or B
b a	f	u	w
f	f	u	w
u	u	u	w
w	w	w	w

The subjunction (also called implication: if-then) is not designed uniformly. The versions by Jan Łukasiewicz and Ulrich Blau as well as the alternative and quasi-implications by Reichenbach are shown here:

Łukasiewicz
b a	f	u	w
f	w	w	w
u	u	w	w
w	f	u	w

blue
b a	f	u	w
f	w	w	w
u	w	w	w
w	f	u	w

Alternative implication
b a	f	u	w
f	w	w	w
u	w	w	w
w	f	f	w

Quasi-implication
b a	f	w
f	u	u
w	f	w

Defenders of three-valued logic believe that logic must adapt to the uncertainty of the measurement statements of quantum physics and not the other way around.

Dialogic logic of temporal statements

Carl Friedrich von Weizsäcker (left) and Peter Mittelstaedt during the International Symposium on Quantum Logic, Cologne 1984

In 1955 Carl Friedrich von Weizsäcker suggested in Göttingen , that of Birkhoff and v. Neumann to derive the propositional calculus from fundamental epistemological considerations on quantum mechanics. Peter Mittelstaedt carried this out in the years 1958–1963 as far as it was possible with the mathematical means available at the time. The elaboration of a logic of temporal statements by Weizsäcker also echoed in Rudolf Carnap's late work . In 1959 Mittelstaedt borrowed Paul Lorenzen's dialogues for the semantic justification of compound statements about physical quantities ( observables ). A temporal quantum logic was researched from this dialogic logic.

In the dialogical logic of Lorenzen and Kuno Lorenz , the truth of a sentence is determined by a dialogue between proponent (P) and opponent (O), in which the dialogue partners refer to previous assertions and demonstrations. The proponent has won if he has defended an attacked no longer logically linked statement (elementary statement) or if the opponent (noted with O in the left column) does not defend an attacked elementary statement. The junction subjunction ( if-then) in the context used here is what Reichenbach means in the three-valued logic implication . There are two dialogues, one about the if-sentence and then one about the next-sentence. The previous lines are attacked with the question mark. ${\ displaystyle \ rightarrow}$

${\ displaystyle O}$	${\ displaystyle P}$	comment
	${\ displaystyle A \ rightarrow A}$	Compound statement: if A then A.
${\ displaystyle A?}$		The if-sentence is asserted and thereby the overall statement is attacked.
	${\ displaystyle?}$	Proof or presentation is required.
${\ displaystyle [A]}$		The asserted A is shown or proven.
	${\ displaystyle A}$	According to the rule, the next sentence must be asserted as a defense . ${\ displaystyle \ rightarrow}$
${\ displaystyle?}$		Proof or presentation is required.
	${\ displaystyle [A]}$	The asserted A is shown or proven. P won, the overall statement is true.

This is where Mittelstaedt and Weizsäcker come in. The basic rules of dialogical logic can be designed in such a way that the evidence for a statement made at the beginning is no longer available after a certain time.

Von Weizsäcker puts forward the following consideration: For example, let m the concrete statement: “The moon can be seen” (substituted for A). The proponent claims as in the diagram . ${\ displaystyle A \ rightarrow A}$

“The opponent replaces A with m. When asked to prove it, he says: 'Here you see the moon, just above the horizon!' The proponent recognizes the evidence. Now asked to prove himself, he says: 'Here you see the moon, just above the horizon!' The opponent must acknowledge the evidence and with it his defeat. - But in this example the proponent has to make sure that it reacts quickly enough. Otherwise the opponent, who had just shown him the moon, could refuse to accept the second proof: the moon has since set. "

- Carl Friedrich von Weizsäcker : The unity of nature . P. 245.

In the usual formal, non-temporal logic, this overall statement is immediately true in terms of formal logic, because the proponent can simply take over the position of A from the opponent. In temporal logic, material truth is dependent on evidence or presentation. ${\ displaystyle A \ rightarrow A}$

Peter Mittelstaedt has shown that the law does not apply in quantum logic for these reasons . There are four other laws that quantum logic violates. Mittelstaedt justifies the violation of these laws by the application of the uncertainty principle: one set for A, the statement "This electron has momentum p" and B "This electron has the place q". The opponent now measures the momentum of the electron and finds p, then measures the location and finds q. Now the proponent repeats the momentum measurement, but unfortunately it does not find the value p again. The law does not apply, the proponent can no longer prove the second A (momentum p) by measurement. ${\ displaystyle A \ rightarrow (B \ rightarrow A)}$ ${\ displaystyle A \ rightarrow (B \ rightarrow A)}$

This pointed subjunction characterized by Mittelstaedt is also called Sasaki-hook . In 1952, the Japanese scientist Sasaki Usa developed a quantum projection, which Richard Joseph Greechie developed into a non-standard quantum logic.

Current assessment

Wolfgang Stegmüller has critically examined quantum logic. He considers Ernest Nagel's criticism to be justified, according to which Reichenbach's suggestion of a trivalent logic is based on too narrow an application of the principle of empiricism . Erhard Scheibe claims that it is possible to build up quantum theory while maintaining classical logic if one chooses an epistemic formulation for the contingent behavior of a system that relates directly to our experimental findings and not to claims about the existence of properties. Andreas Kamlah asks critically whether dialogical quantum logic is an analytical theory.

After 2000, the merits of quantum logic as a contribution to linguistic research were increasingly recognized, for example by Brigitte Falkenburg .

According to a modern, more formalistic understanding of logic, it can be assumed that the three different approaches no longer compete with each other: In dialogic logic and other logic systems of the Gentzen type , various sets of frameworks are offered, each of which leads to a specific logic, including quantum logic. The same applies to the systems of axioms of the Hilbert type. With these options, a logical set of rules can be used to decide which rule package you want to choose. So you don't have to play all of the rules off against each other.

Maria Luisa Dalla Chiara and Roberto Giuntini assess the situation as follows: Quantum logics and their formal properties give no indication of real properties or mechanisms between the observables of quantum theory. A realism in this regard that pioneers like v. Neumann, Reichenbach and v. Weizsäcker should therefore be rejected, as should Putnam's suggestion that the validity of a certain logic is a question of empiricism. Rather, they state that there are not only different logics, but also different quantum logics. It is therefore questionable whether there is a clear quantum logic.

literature

Garrett Birkhoff, John von Neumann: The logic of quantum mechanics . In: Ann. of Math. 37, 1936 ( PDF ; 761 kB).
Ulrich Blau: The logic of indefiniteness and paradoxes . Heidelberg 2008, pp. 191-290.
Enrico Beltrametti, Bas van Fraassen (ed.): Current Issues in Quantum Logic (= Ettore Majorana International Science Series . Vol. 8). New York / London 1981, ISBN 0-306-40652-7 .
André Fuhrmann: Quantum Logic. In: Jürgen Mittelstraß : Encyclopedia Philosophy and Philosophy of Science. Second edition. Volume 6, Metzler 2016, ISBN 978-3-476-02105-2 , pp. 532-533.
Peter Mittelstaedt, Ernst-Walther Stachow (ed.): Recent Developments in Quantum Logic (= Basics of the exact natural sciences, Vol. 6). Mannheim / Vienna / Zurich 1985, ISBN 3-411-01695-7 .
Peter Mittelstaedt: Quantum Logic (= Synthesis Library . Vol. 126). Doordrecht 1978, ISBN 90-277-0925-4 .
Peter Mittelstaedt: Are the Laws of Quantum Logic Laws of Nature? . In: Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 43 (2), 2012, pp. 215–222.
Ewald Richter : Quantum Logic. In: Joachim Ritter et al. (Ed.): Historical dictionary of philosophy . Volume 7, Basel 1989, ISBN 978-3-7965-0698-7 , pp. 1782-1785.
Carl Friedrich von Weizsäcker: Complementarity and Logic . In: Die Naturwissenschaften 42, 1955, pp. 521-529 u. 545-555.
Carl Friedrich von Weizsäcker: The unity of nature . Studies, Hanser, Munich 1971.

Web links

DJ Foulis: A Half Century of Quantum Logic - What Have We Learned? University of Massachusetts, Amhest 1995.
Maria Luisa Dalla Chiara, Roberto Giuntini: Quantum Logics. Florence 2008.
Mladen Pavičić: Bibliography on Quantum Logics and related structures. Zagreb 1992.
Alex Wilce: Quantum Logic and Probability Theory. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy . Stanford 2002/2012/2017.

Individual evidence

↑ Friedrich Hund: History of the quantum theory . 3rd edition 1984.

↑ Klaus Mainzer : Quantum Theory . In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. Second edition. Volume 6, Metzler 2016, p. 538.

^ Peter Forrest: Quantum logic . In: Edward Craig (Ed.): Routledge Encyclopedia of Philosophy . Vol. 7, 1998, pp. 882ff.

↑ John von Neumann: Mathematical foundations of quantum mechanics . 1932.

^ George Mackey: Mathematical Foundations of Quantum Mechanics . 1963.

^ Peter Mittelstaedt: Relativistic Quantum Logic . In: Int. Journal of Theor. Physics 22, 1983, pp. 293-314.

^ Carl Friedrich von Weizsäcker: Structure of Physics. Carl Hanser Verlag, 1985, Chapter Eight Reconstruction of the abstract quantum theory.

↑ Peter Schroeder-Heister: Logic, polyvalent . In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. Second edition. Volume 5, Metzler 2013, p. 62.

↑ Hilary Putnam: Is Logic Empirical? In: Boston Studies in the Philosophy of Science . tape 5 . D. Reidel, Dordrecht 1968, p. 216-241 .

↑ Michael Dummet: Is Logic Empirical? In: Contemporary British Philosophy . tape 4 , 1976.

^ John von Neumann, Garrett Birkhoff: The logic of quantum mechanics . In: Annals of Mathematics 37, 1936, pp. 823-843.

↑ Carl Friedrich von Weizsäcker: The unity of nature . Studies, Hanser, Munich 1971, 2nd edition 1981, p. 242.

^ Peter Mittelstaedt: Quantum Logic . Pp. 6-26. On the tertium non datur in quantum logic, see Peter Mittelstaedt and Ernst-Walther Stachow: The principle of excluded middle . In: Journal of Philosophical Logic 7, 1978, pp. 181-208.

↑ ^a ^b ^c ^d André Fuhrmann: Quantum Logic. In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. Second edition. Volume 6, Metzler 2016, ISBN 978-3-476-02105-2 , p. 532.

^ Maria Luisa Dalla Chiara, Roberto Giuntini: Quantum Logics. Florence 2008, p. 36.

^ Saul A. Kripke: Semantical Analysis of Logic I. Normal propositional Calculi. In: Journal for Mathematical Logic and Fundamentals of Mathematics. 9, 1963, pp. 67-96.

^ Bas van Fraassen: Meaning Relations and Modalities. In: Nous. 3, 1969, pp. 155-167. ML Dalla Chiara: Quantum Logic and Physical Modalities. In: Journal of Philosophical Logic. 6, 1977, pp. 391-404.

^ Franz Josef Burghardt: Modalities and Quantum Mechanics . In: Int. Journal of Theor. Physics 23, 1984, pp. 1171-1196, with further literature.

^ Bas van Fraassen: The Labyrinth of Quantum Logics . In: Cohen, Wartofsky: The Logico-Algebraic Approach to Quantum Mechanics (= The University of Western Ontario Series in Philosophy of Science . Vol. 5a). Pp. 577-607.

^ Bas van Fraassen: The Labyrinth of Quantum Logics . Pp. 577-607.

^ Martin Carrier : Reichenbach, Hans . In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. First edition. Volume 3, Metzler 1995/2004, p. 542.

^ Hans Reichenbach: Collected works . Volume 5: Philosophical foundations of quantum mechanics and probability . P. 182f.

↑ Ewald Richter: Quantum Logic. 1989, p. 1784.

↑ Werner Stelzner: Logic, multi-valued . In: Hans Jörg Sandkühler (Ed.): Encyclopedia Philosophy . 2nd, revised and expanded edition. Meiner, Hamburg 2010, Vol. 2, pp. 1462ff.

↑ Peter Schroeder-Heister : Logic, polyvalent . In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. Second edition. Volume 5, Metzler 2013, p. 62.

↑ Peter Schroeder-Heister: Logic, polyvalent . In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. Second edition. Volume 5, Metzler 2013, p. 63.

↑ Carl Friedrich von Weizsäcker: Complementarity and logic . In: Die Naturwissenschaften 42, 1955, pp. 521-529 u. 545-555.

^ Peter Mittelstaedt: Quantum Logic . In: Advances in Physics 9, 1961, pp. 106-147.

↑ At the end of his last book Philosophical Foundations of Physics (New York 1966, German edition, Introduction to Philosophy of Natural Science . Munich 1969, 2nd ed. 1974, p. 286) Carnap comments on the work of Birkhoff and von Neumann: “Here we are touching on deep-seated, as yet unsolved problems. [...] It is difficult to predict how the language of physics will change. But I am convinced that two tendencies which over the last half century have led to great improvements in the language of mathematics will equally sharpen and clarify the language of physics; the application of modern logic and set theory and the use of the axiomatic method in its modern form, which presupposes a formalized language. In today's physics, where […] the whole concept of physics is discussed, both methods could prove extremely useful. "

^ Peter Mittelstaedt: Quantum Logic . In: Advances in Physics 9, 1961, pp. 106–147, here pp. 124–128; also in the first edition by Peter Mittelstaedt: Philosophical Problems of Modern Physics . Mannheim 1963, pp. 127-133. Now in detail in this: Quantum Logic . Pp. 48-98.

↑ Peter Mittelstaedt: Time dependent propositions and quantum logic . In: Journal of Phil. Logic 6, 1977, pp. 463-472. Carl Friedrich von Weizsäcker: In what sense is quantum logic a temporal logic? In: Jürgen Nitsch, Joachim Pfarr, Ernst-Walther Stachow: Fundamental problems of modern physics. Festschrift for Peter Mittelstaedt on his 50th birthday. Mannheim 1981, ISBN 3-411-01600-0 , pp. 311-317.

↑ For the availability of a statement see: Kuno Lorenz : Logic, dialogische . In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. Second edition. Volume 5, Metzler 2013, p. 24.

↑ Peter Mittelstaedt: Philosophical Problems of Modern Physics . Mannheim 1986.

^ Maria Luisa Dalla Chiara, Roberto Giuntini: Quantum Logics. Florence 2008, p. 25.

↑ Weizsäcker: The unity of nature . Munich 1981, p. 246.

^ Maria Luisa Dalla Chiara, Roberto Giuntini: Quantum Logics. Florence 2008, p. 25.

↑ Sasaki Usa: Lattice theoretic characterization of affine geometry of arbitrary dimensions . In: Journal of Science . Hiroshima Univ. Series A, 16, Hiroshima 1952, pp. 223-238.

↑ Richard Joseph Greechie: A non-standard quantum logic with a strong set of states . In: EG Beltrametti, Bas van Fraassen (ed.): Current Issues in Quantum Logic (= Ettore Majorana International Science Series . Vol. 8). Plenum, New York 1981, pp. 375-380.

↑ Wolfgang Stegmüller: Main currents of contemporary philosophy . Stuttgart 1975, ISBN 3-520-30905-X , pp. 208-220.

↑ Wolfgang Stegmüller: Scientific explanation and justification . Berlin / Heidelberg / New York 1969, p. 506.

↑ Erhard Scheibe: The contingent statements of physics . 1964.

↑ Ewald Richter: Quantum Logic . 1989.

↑ Andreas Kamlah: Is the Mittelstaedt-Stachow quantum dialogue an analytical theory? In: Peter Mittelstaedt, Joachim Pfarr: Fundamentals of quantum theory . (= Basics of the exact natural sciences . Volume 1). Mannheim 1980, pp. 73-92.

^ Brigitte Falkenburg: Language and Reality. Peter Mittelstaedt's contribution to the Philosophy of Physics . In: Foundations of Physics 40, 2010, pp. 1171-1188.

^ Maria Luisa Dalla Chiara, Roberto Giuntini: Quantum Logics. Florence 2008, pp. 96-97.

[1] Friedrich Hund: History of the quantum theory . 3rd edition 1984.

[2] Klaus Mainzer : Quantum Theory . In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. Second edition. Volume 6, Metzler 2016, p. 538.

[3] Peter Forrest: Quantum logic . In: Edward Craig (Ed.): Routledge Encyclopedia of Philosophy . Vol. 7, 1998, pp. 882ff.

[4] John von Neumann: Mathematical foundations of quantum mechanics . 1932.

[5] George Mackey: Mathematical Foundations of Quantum Mechanics . 1963.

[6] Peter Mittelstaedt: Relativistic Quantum Logic . In: Int. Journal of Theor. Physics 22, 1983, pp. 293-314.

[7] Carl Friedrich von Weizsäcker: Structure of Physics. Carl Hanser Verlag, 1985, Chapter Eight Reconstruction of the abstract quantum theory.

[8] Peter Schroeder-Heister: Logic, polyvalent . In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. Second edition. Volume 5, Metzler 2013, p. 62.

[9] Hilary Putnam: Is Logic Empirical? In: Boston Studies in the Philosophy of Science . tape 5 . D. Reidel, Dordrecht 1968, p. 216-241 .

[10] Michael Dummet: Is Logic Empirical? In: Contemporary British Philosophy . tape 4 , 1976.

[11] John von Neumann, Garrett Birkhoff: The logic of quantum mechanics . In: Annals of Mathematics 37, 1936, pp. 823-843.

[12] Carl Friedrich von Weizsäcker: The unity of nature . Studies, Hanser, Munich 1971, 2nd edition 1981, p. 242.

[13] Peter Mittelstaedt: Quantum Logic . Pp. 6-26. On the tertium non datur in quantum logic, see Peter Mittelstaedt and Ernst-Walther Stachow: The principle of excluded middle . In: Journal of Philosophical Logic 7, 1978, pp. 181-208.

[AFuhrmann-14] André Fuhrmann: Quantum Logic. In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. Second edition. Volume 6, Metzler 2016, ISBN 978-3-476-02105-2 , p. 532.

[15] Maria Luisa Dalla Chiara, Roberto Giuntini: Quantum Logics. Florence 2008, p. 36.

[16] Saul A. Kripke: Semantical Analysis of Logic I. Normal propositional Calculi. In: Journal for Mathematical Logic and Fundamentals of Mathematics. 9, 1963, pp. 67-96.

[17] Bas van Fraassen: Meaning Relations and Modalities. In: Nous. 3, 1969, pp. 155-167. ML Dalla Chiara: Quantum Logic and Physical Modalities. In: Journal of Philosophical Logic. 6, 1977, pp. 391-404.

[18] Franz Josef Burghardt: Modalities and Quantum Mechanics . In: Int. Journal of Theor. Physics 23, 1984, pp. 1171-1196, with further literature.

[19] Bas van Fraassen: The Labyrinth of Quantum Logics . In: Cohen, Wartofsky: The Logico-Algebraic Approach to Quantum Mechanics (= The University of Western Ontario Series in Philosophy of Science . Vol. 5a). Pp. 577-607.

[20] Bas van Fraassen: The Labyrinth of Quantum Logics . Pp. 577-607.

[21] Martin Carrier : Reichenbach, Hans . In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. First edition. Volume 3, Metzler 1995/2004, p. 542.

[22] Hans Reichenbach: Collected works . Volume 5: Philosophical foundations of quantum mechanics and probability . P. 182f.

[23] Ewald Richter: Quantum Logic. 1989, p. 1784.

[24] Werner Stelzner: Logic, multi-valued . In: Hans Jörg Sandkühler (Ed.): Encyclopedia Philosophy . 2nd, revised and expanded edition. Meiner, Hamburg 2010, Vol. 2, pp. 1462ff.

[25] Peter Schroeder-Heister : Logic, polyvalent . In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. Second edition. Volume 5, Metzler 2013, p. 62.

[26] Peter Schroeder-Heister: Logic, polyvalent . In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. Second edition. Volume 5, Metzler 2013, p. 63.

[27] Carl Friedrich von Weizsäcker: Complementarity and logic . In: Die Naturwissenschaften 42, 1955, pp. 521-529 u. 545-555.

[28] Peter Mittelstaedt: Quantum Logic . In: Advances in Physics 9, 1961, pp. 106-147.

[29] At the end of his last book Philosophical Foundations of Physics (New York 1966, German edition, Introduction to Philosophy of Natural Science . Munich 1969, 2nd ed. 1974, p. 286) Carnap comments on the work of Birkhoff and von Neumann: “Here we are touching on deep-seated, as yet unsolved problems. [...] It is difficult to predict how the language of physics will change. But I am convinced that two tendencies which over the last half century have led to great improvements in the language of mathematics will equally sharpen and clarify the language of physics; the application of modern logic and set theory and the use of the axiomatic method in its modern form, which presupposes a formalized language. In today's physics, where […] the whole concept of physics is discussed, both methods could prove extremely useful. "

[30] Peter Mittelstaedt: Quantum Logic . In: Advances in Physics 9, 1961, pp. 106–147, here pp. 124–128; also in the first edition by Peter Mittelstaedt: Philosophical Problems of Modern Physics . Mannheim 1963, pp. 127-133. Now in detail in this: Quantum Logic . Pp. 48-98.