Does Quantum Physics Require a New Logic? - Springer Link

Does Quantum Physics Require a New Logic? Peter Mittelstaedt

1 Introduction In order to answer the question whether quantum physics requires a new logic, we will at first briefly discuss the more general problem of a justification of the laws of logic. Here, we are not interested in the most general form of logic and its applicability to various fields. Instead, we will investigate propositions which assign properties to physical systems, i.e. to material objects of the external reality. On the basis of this kind of elementary propositions a language of propositions can be established by introducing a concept of truth, several connectives, and relations between propositions. Within the framework of this language we can find the logic of this language as an internal structure, which is justified by this way of reasoning. The speaker of a scientific propositional language which is constructed in this way, is also an observer in the sense of physics who justifies or disproves propositions by measurement processes. The possibilities and abilities of the speaker-observer to verify or to falsify propositions by measuring the corresponding properties, have some influence on the most general rules of the syntax of this language, i.e., on the laws of logic. On the other hand, the possibilities of the speaker-observer to perform measurements will partly depend on the general structure of that domain of reality which is described by the scientific language. The observer is part of the world which he observes and subject to the law of the reality which he is investigating. Hence, the compatibility of the possibilities of the speaker-observer with the physical laws of the reality domain in question induces some unavoidable kind of self-referentiality into the physical theory. 1 These more general remarks will be illustrated in the present paper by the well-known classical logic, which can be applied to the reality domain of classical physics (Sect. 2) and by the "new" quantum logic, which refers to the domain of quantum physics (Sect. 4). The main problem of this paper is, whether we are confronted here with a plurality of logical systems or whether there is a hierarchy with a priority of one of these logical systems over the other one. We show that there is no pluralism of various logical systems corresponding to different fields of experience but a strict hierarchy, such 1

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that there is only one "true" logic of propositions about physical systems. The reasons for this result are, on the one hand, the most general features of the quantum physical reality, and on the other hand the universality, and self-referentiality of quantum physics (Sect. 3).

2 The Logic of Classical Physics 2.1 The Language We consider the object language Sc of classical physics, i.e., a propositional language with propositions A(S, t) which assign properties P(A) at time t to a physical system S. The domain of reality which is described by Sc is the macroscopic or classical physical reality. It is obvious that this language can also be applied to the apparatuses used by the speaker-observer. The classical physical reality is characterised by the complete determination of its objects. This means that for all elementary propositions A(S) it is objectively decided whether the property P(A) or the counter-property P(A) pertains to the system. These properties pertain to the system objectively and irrespective of the possibility of observation. However, we assume that for every property P(A) there exists a process of observation or measurement, such that in a finite number of steps we can decide whether P(A) or P(A) pertains to the system. The elementary propositions are logically independent and independent with respect to their verification and falsification. Elementary propositions pertain to an object irrespective of the tests of other elementary properties. Complete determination, finite decidability of elementary properties and their complete independence characterise the domain of classical reality. These most general "ontological" features have an essential influence on the language of classical physics, which is used for the description of this domain of reality.

2.2 Semantics and Syntax The ontological preconditions mentioned guarantee the objective decidedness and finite testability of elementary propositions A(S). Hence, A(S) will be called to be true, f- A(S), if and only if the system S possesses the property P(A). According to the preconditions the truth of A(S) can be shown by a finite proof procedure. In any case, either A or the counter-proposition A turns out to be true in this way. Hence, elementary propositions are valuedefinite. The semantics which is established by this concept of truth will be called "realistic" .2 2

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On the basis of the set S~) of elementary propositions we introduce the logical connectives by the possibilities to attack or to defence them, i.e., by the possibilities to prove or to disprove the connective. As an example we consider the two-place operation "sequential conjunction" connective An B

denotation

attacks

"A and then B"

l.A?, 2.B?

defences LA!, 2.A!

where A? means the challenge to prove A, and A! the successful proof. This attack-and-defence scheme can be illustrated most conveniently by a prooftree which is chronologically ordered. 3 The first branching point corresponds to the test of A at t 1, the second one corresponds to the B-test at t2.

Fig. 1. Proof-tree for An B in classical logic

The temporal order is fixed here, but the time difference t5t = t2 - t1 > 0 may assume arbitrary positive values. There is one branch of success. In a similar way the one place operation "negation" may be introduced by the proof tree for ·A (not A) with one branch for success and one branch for loss.

•A

Fig. 2. Proof-tree for ·A in classical logic

The sequential conjunction (and the other sequential connectives) refer to two instants of time t1 and t 2. The logical connectives refer to one common instant of time. The logical connective which corresponds to A n B is the logical conjunction, which is defined by the attack-and-defence scheme 3

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denotation A and B

attacks A?, B?

defences A!, B!

In contrast to the sequential conjunction we have here an arbitrary number of attacks and defences in arbitrary order. In this way it is guaranteed that the result of the A and B tests can be attributed to a common time value. If A 1\ B is proved, then A and B are simultaneously true. The assumed independent testability of propositions A and B implies that after a test of B the result of a preceding test of A is still valid and available without any restrictions. The unrestricted availability of the results of A- and B-tests implies that for the proof of A 1\ B we need only two steps, provided their time difference 8t = t2 - t1 is sufficiently small. There is no need to repeat the proofs.

Fig. 3. Proof-tree for A 1\ B in classical logic

In a similar way the other logical connectives, i.e. the disjunction A V B (A or B) and the material implication A-+ B (if A then B) can be defined by the possibilities to prove or to disprove them, or by the respective proof-trees. The full language Sc of classical physics can then inductively be defined by the sets2;'l of value-definite elementary propositions and by all finitely connected compound propositions A E Sc. The concept of truth can then be defined in the following way: A proposition A E Sc is said to be true if the proof-tree of A leads finally to a branch of success; it is called false if the proof-tree ends with a branch of loss. Furthermore, on the set of propositions we introduce two binary relations, 1. the value equivalence A= B, A is true if and only if B is true 2. the implication A :::; B ~ A = A 1\ B

The full language can then be formulated as


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AvB AvB

A-+B

A-+B Fig. 4. Proof-trees for A V B and A

---->

B in classical logic

2.3 Classical Logic

The semantics described here is a combination of a realistic semantics (with respect to elementary propositions) and a proof semantics (with respect to compound propositions). Hence, the truth of a compound proposition depends, on the one hand, on the connectives contained in it, and on the other hand on the elementary propositions and their truth values. This leads to the following question: Are there finitely connected propositions A E Sc which are true in the sense of the semantics described, irrespective of the truth values of the elementary propositions contained in it? Propositions of this kind will be called formally true. Examples for formally true propositions can easily be found. On account of the value definiteness of the elementary propositions we have the formally true proposition

A V -,A (tertium non datur) and on account of the unrestricted availability of propositions in a proof tree we have the formally true proposition

A----> (B----> A), which is true even without the assumption of value definiteness of elementary propositions. If both value definiteness and unrestricted availability of propositions are assumed, then we obtain the formally true proposition

A----> ((A 1\ B)

V

(A 1\ ·B)).

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There are many, even infinitely many formally true propositions. The totality of formally true propositions is called classical logic and the algorithm which provides propositions of this kind is the calculus of classical logic. Here we make use of a Bouwer-calculus for implications which turns out to be very convenient for our purpose. For the formulation of the calculus of the classical logic Lc we make use of two special propositions, -

the true proposition ( verum) V, such that f-- V and for all A E Sc we have A:::; V, and the false proposition (falsum) A = ---, V with A :::; A for all A E Sc.

By means of these two propositions we have f-- A {:} V :::; A and, in particular, f-- A -+ B {=::::} A :::; B. The calculus Lc can be formulated as a calculus of implications with "beginnings" ::::} A :::; B and rules like A :::; B ::::} C :::; D. The Lindenbaum-Tarski algebra of Lc is a complete, complemented, and distributed lattice LB (Boolean lattice). If it is freely generated by a finite number of elementary propositions, it is also atomic and fulfils the covering law. The calculus Lc of classical logic 1.1:::}A:::;A 1.2 A :::; B; B :::; C ::::} A :::; C 2.1 ::::} A 1\ B :::; A 2.2 ::::} A 1\ B :::; B

2.3 ::::} C :::; A; C :::; B ::::} C :::; A 1\ B 3.1 ::::} A:::; A VB

3.2 ::::} B :::; A VB 3.3 A :::; C; B :::; C ::::} A VB :::; C 4.1 ::::} A 1\ (A-+ B) :::; B 4.2 ::::} A 1\ C :::; B ::::} C :::; A -+ B 5.0 ::::} A :::; A,::::} A :::; V

5.1 ::::} A 1\ ·A :::; A 5.2 ::::} V :::; A

v ·A

The formal propositional logic does not depend on the elementary propositions which are contained in the formally true propositions. However, the


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logic depends on the general preconditions under which proof processes are possible. In the present case the most important preconditions are the finite decidability of elementary propositions and the unrestricted availability in a proof-process. Formally true propositions are not true in an absolute sense. Their truth follows from the pragmatic preconditions of proving or disproving propositions. Only in this transcendental sense they are a priori true. 4

3 The Quantum Physical Reality In the domain of quantum-physical reality the strong ontological preconditions of the classical language are no longer fulfilled.

3.1 Complementarity Niels Bohr made the important discovery that it is, in general, not possible to observe simultaneously two elementary properties P(A) and P(B) of the same system. 5 The reason for this phenomenon, the complementarity, is, according to Bohr, that the measuring devices for P(A) and P(B) are mutually exclusive. The strong complementarity relation can be relaxed by Heisenberg's uncertainty relation: Two complementary properties P(A) and P(B) can be measured jointly on the system, if both properties are measured unsharply and if the product of convenient uncertainty measures, for P(A) and P(B) is always larger than a universal bound given by Plancks constant n. 6 Here, we make use of the following terminology: Two properties P(A) and P(B) which can jointly and sharply be measured, will be called commensurable. If a simultaneous sharp measurement is not possible, then the two properties will be called incommensurable. There are various degrees of incommensurability. Properties which are maximal incommensurable will be denoted as complementary.

3.2 Nonobjectifiability

Complementarity and the uncertainty relation can be tested and confirmed experimentally. However, from these experiments it does not follow whether complementarity expresses merely the mutual exclusiveness of apparatuses or whether it refers to properties of the object system. If complementarity were merely describing some restrictions of measurement apparatuses, then it would still be possible after a measurement of P(A) to assume that the system possesses in addition to P(A) either P(B) or P(·B)- even if these 4

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assumptions cannot be confirmed by experiments. The property P(B) would then be objectively determined but subjectively unknown. Within the framework of classical physics this (ignorance) interpretation can be justified by means of the logic of the language of classical physics. Indeed, if P(A) was measured and thus the proposition A was shown to be true (in the sense of the realistic semantics), then for any other proposition B it follows from the logical equivalence A = (A 1\ B) V (A 1\ •B) that also (A 1\ B) V (A 1\ ·B) is true. Hence, together with the property P(A) the object system would also possess the property P(B) -or it would not possess this property. In this case we call the property P(B) "objective". The observer knows that an "objective" property is objectively decided even if it is subjectively unknown to him. It is one of the most important results of quantum physics that this is, in general, not the case. If a property P(A) is known to pertain to the object system, then one must not assume that, in addition, an arbitrary property P(B) pertains to the system or not. If P(B) is incommensurable with P(A), then P(B) cannot be tested by experiment. This is, however, less important. The essential point is that under the conditions described here the property P(B) is not only subjectively unknown to the observer but it is objectively undecided whether the system possesses P(B) or P(·B). 7 3.3 The Probability Argument The justification of this nonobjectification argument makes use of the probability structure of quantum mechanics. Let us assume that the system possesses property P(A). If, in addition, the complementary property P(B) or its counter-property P( ·B) would pertain to the system, then the system would be in a mixed state of these alternatives with conditional probabilities p(A; B) and p(A; ·B), respectively. The probability for any other property P(C) which is neither commensurable with P(A) nor with P(B) is then given by8

p(A; C)= p(A; B)p(B; C)+ p(A; ·B)p(•B; C). However, this formula does not hold in quantum mechanics. The correct formula in quantum mechanics reads

p(A; C)= p(A; B)p(B; C)+ p(A; •B)p(•B; C)+ Pint(A; B, C) and contains the nonvanishing interference term Pint(A; B, C). Hence, we find that the objectification assumption for property P(B) is in disagreement with the well-established statistics of P( C) outcomes. Experiments of this kind can easily be performed by a Mach-Zehnder interferometer. 9 7

8 9

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V ( C 1\ -.B)


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3.4 Universality

According to our present knowledge quantum physics is universally valid in all domains of the physical reality. Of course, this statement holds for the most general and most abstract laws of quantum mechanics but not for the large variety of special applications in nuclear physics, solid-state physics, etc. However, what is more important is that quantum mechanics is equally applicable to macroscopic physics as well as to microscopic phenomena. This somewhat surprising statement is confirmed by numerous macroscopic quantum effects. Consequently, quantum mechanics applies also to the macroscopic apparatuses which are used for the verification and falsification of quantummechanical propositions. Since the most general properties of the apparatuses belong to the pragmatic preconditions of our scientific language of physics, we arrive at the important conclusion that the laws of physics which describe quantum systems and their properties also determine the behaviour of apparatuses and hence the most general preconditions of the scientific object language of physics. 10 This result has far-reaching consequences for the construction of a scientific language. If the speaker-observer takes into account the possibilities and restrictions which are known from the domain of reality of object systems, then the language which is constructed in this way will not contain more structure in its syntax and logic than the domain of reality that is described by it. This is by no means obvious. In the language of classical physics the strong pragmatic preconditions of value definiteness and unrestricted availability of propositions lead to classical logic, in particular to the equivalence A= (AI\B)V(AI\·B). However, if this law is applied to quantum-mechanical propositions which are not commensurable, then one comes into conflict with the nonobjectifiability of propositions in quantum mechanics.

4 The Logic of Quantum Physics 4.1 Language and Reality in Quantum Physics

Since classical language and logic cannot consistently be applied to arbitrary quantum-physical situations we will make a new attempt at constructing a scientific language which can be applied to all quantum-physical situations. On account of the universality of quantum physics this language will be applicable also to macroscopic phenomena, which are usually treated within the framework of classical physics. For the construction of quantum language we will keep in mind the more general remarks about the interrelations between language and reality. Hence we make use only of those restricted ontological preconditions of the language which are compatible with the laws 10

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of quantum physics. On the basis of this new and weak quantum ontology a quantum pragmatics can be formulated which incorporates all restrictions for proving or disproving propositions which come from the incommensurability of quantum-physical properties. Since proof processes contain measurement processes, incommensurabilities restrict the possibilities of testing quantum-mechanical propositions. 11 4.2 The Syntax of Quantum Language For the constitution of a quantum language we begin again with elementary propositions A, which state that a property P(A) pertains to the object system. Accordingly, the proof of the elementary proposition A consists of a measurement of property P(A) with positive outcome. The possibilities for quantum measurements allow for the assumption that after the measurement of P(A) we obtain either a positive or negative result. 12 Hence, an elementary proposition A can either be proved (result A) or disproved (result A) and is thus value-definite. Furthermore, if after a successful proof of A a new proof attempt for A is made, then one obtains again the result A, if the applied measurement is repeatable. However, if after a successful proof of A another elementary proposition B is proved, then a new proof attempt for proposition A will, in general, not lead to the previous positive result. Hence, two propositions A and B are, in general, not simultaneously decidable. This is only the case if the corresponding properties P(A) and P(B) are commensurable. In this case we will also call the propositions A and B "commensurable". Elementary propositions A, B, ... are thus, in general, incommensurable, i.e. not simultaneously (jointly) decidable. If proposition A, say, was shown to be true, then after a proof attempt of B and irrespective of the result (B or B), a new proof attempt of A will, in general, not lead to the previous result. Instead, this result is available after the B-test only if A and B are commensurable. In a sequence of proofs the results are only restrictedly available, where the restrictions are given by incommensurabilities. For the definition of the connectives the restricted availability is very important. These restrictions do not invalidate the definitions of the negation ·A and the sequential conjunction A n B which are defined here by the same proof-trees as in classical language. The negation is defined by one proof attempt and the sequential conjunction by two subsequent proof attempts. In both cases the restricted availability does not matter since repeated proof attempts do not occur here. However, the restrictions do matter if one tries to define the other connectives. 13 11 12 13

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The logical conjunction A A B is defined here by the same attack-anddefence scheme as in classical language. Since unrestricted availability is no longer given here, the proof-tree for connective AAB

denotation AandB

attacks A?, B?

defences A!, B!

B consists of an infinite number of steps and cannot be reduced to two steps as in classical language. If, however, A and B were commensurable , A

A

then it would again be possible to reduce the proof-tree to one A-proof and one B-proof. In order to arrive generally at a finite proof-tree we make use of the commensurabil ity proposition k(A, B), which is defined to be true if and only if A and B are commensurable . The counter-propos ition is denoted here by k( A, B) .14 The logical conjunction A A B is then true if in addition to A and B also k(A, B) is shown to be true. Hence, we have a proof-tree with three subsequent tests at time values h, t2, t 3. Since the conjunction A A B is understood as a simultaneous connective, the time differences t 3 - t 2 and t2 - h must be sufficiently small.

Fig. 5. Proof-tree for the logical conjunction

The commensurabil ity propositions k(A, B) and k(A, B) are contingent propositions whose truth must be shown by a convenient sequence of measurements. We will not go into detail here. By means of the commensurabil ity propositions k(A, B) and k(A, B) one can also define the logical disjunction A V B and the material implication A ____, B by proof trees with a finite number of steps. Similarly as in classical language, we can define here binary B means that A relations between propositions. The proof equivalence A can be replaced in any proof-tree by B without thereby changing the result of the proof-tree. The binary relation of value equivalence A = B means that A is true if and only if B is true. 15 The relation of implication A ::; B can

=

14 15

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then be defined by A = A 1\ B. Finally, we mention that again A ---+ B is true if and only if A ::; B holds and that the commensurability proposition is true if and only if A ::; (A 1\ B) V (A 1\ ·B) holds.

A-+B

--cd---------~~~

Fig. 6. Proof-trees for the logical disjunction and the material implication.

The full language SQ of quantum physics can then inductively be defined by the set s~) of elementary propositions, the commensurability propositions k and k and the connectives mentioned. Together with the relations "=", "=",and "::;" the language SQ reads sQ

= {s~l; k, k; n, /\, v, ---+, •; =, =, ::;}.

4.3 Quantum Logic

Quantum logic is the formal logic of quantum language and its syntax. The reduced possibilities of proving propositions are, in particular, important for those propositions which are true, irrespective of the elementary propositions contained in them, i.e., for formally true propositions. It turns out that in quantum language there are less formally true propositions than in classical language. In order to make this more preliminary information more precise we will express the totality of all formally true propositions of quantum language by a calculus, the calculus of quantum logic. There are, first of all, many formally true .propositions of classical language which are also formally true in quantum language. The value definiteness of elementary propositions implies that also all finitely connected propositions are value-definite, i.e., the proposition A V ·A, the tertium non datur law, is formally true. The precondition that measurements are repeatable in principle implies that k(A, A) is always true and hence A---+ A, the law of identity,


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is formally true. In a si!fiilar way, it follows that -,(A 1\ -,A), the law of contradiction, is formally true in quantum logic. The three cases mentioned are not very surprising since these formally true propositions contain only one proposition A. Hence, commensurability problems cannot appear. There are, however, also formally true propositions in quantum logic which contain two ore more elementary propositions, where nothing is presupposed about their mutual commensurability. An example of this kind is the proposition (AI\(A---+ B))---+ B, the modus ponens law, which is formally true in quantum logic irrespective of the truth or falsity of the commensurability proposition k(A, B). More important for the characterisation of quantum logic are those propositions which are formally true in classical logic but not in quantum logic. The shortest and in addition most important proposition which is formally true in classical logic but not in quantum logic is the proposition A---+ (B ---+A). In classical logic the proof-tree for A---+ (B ---+ A) contains only branches of success.

~A·········· Fig. 7. Proof-tree for A---+ (B---+ A) in classical logic

In quantum logic the situation is more complicated since the proof-tree for the material implication contains also the test of commensurability propositions k(A, B). For this reason, the proof-tree for A ---+ (B ---+ A) contains 5 branches, but only 3 branches of success. Only if the commensurability of A and B were presupposed would the proof-tree contain only successful branches. This means that, in general, the proposition A---+ (B---+ A) is not true and thus not formally true. The totality of all propositions which are formally true even under the restrictions that are provided by the commensurability tests is called quantum logic. There are -as in classical logic- infinitely many propositions which are formally true in the sense of quantum logic. They can be summarised in a quantum-logical calculus Lq, which contains "beginnings" =} A ::; B and "rules" of the form A ::; B =} C ::; D. For the formulation of this calculus we make again use of the two special propositions V ( verum) and A (falsum) such that for all propositions A E Sq the relations A ::; A ::; V hold. If A ---+ (B ---+ A) is true then the relation A ::; (B ---+ A) holds. A ::; B ---+ A implies B ::; A ---+ B and vice versa and A ::; B ---+ A holds if and only if k(A, B) is true. Hence, in a calculus of quantum logic the commensurability

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-----------------0

Fig. 8. Proof-tree for A-+ (B-+ A) in quantum logic

propositions k(A, B) can be eliminated by this implication and will no longer appear in its final formulation. The calculus LQ of quantum logic reads: The calculus of quantum logic 1.1 =>A::::; A 1.2 A ::::; B; B

::::; C => A ::::; C

2.1 =>A A B ::::; A 2.2 => A A B ::::; B

2.3 C::::; A; C::::; B => C::::; A A B

3.1 => A ::::; A VB 3.2 => B ::::; A V B 3.3 A ::::; C; B ::::; C => A VB ::::; C 4.1 =>A A (A -7 B) ::::; B 4.2 A A C ::::; B => A -7 C ::::; A -7 B 4.3 A ::::; B -7 A => B ::::; A -7 B 4.4 B ::; A -7 B; C::::; A -7 C => B*C ::::; A -7 B*C * E {A,V,-7}

5.0 => A ::::; A, => A ::::; V 5.1 =>A A -,A ::::; A 5.2 A A C ::::; A => A -7 C ::::; -,A 5.3 A ::::; B -7 A => -,A ::::; B -7 -,A 5.4 => V ::::; A V -,A

Comparing classical logic and quantum logic we find that a proposition which is formally true in quantum logic is also true in classical logic. The inverse relation is, however, not true. There are infinitely many formally true propositions in classical logic which are not formally true in the sense of quan-

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tum logic. The reason for this important result is the restricted availability of quantum propositions or the general incommensurability of two arbitrary quantum propositions. Indeed, if the mutual commensurability of all pairs of propositions were presupposed, then the calculus LQ of quantum logic would agree with the calculus Lc of classical logic. Hence, the quantumphysical restrictions lead to a logical system which is weaker than classical logic but obviously more general. Quantum logic is applicable to classical propositions as well as to quantum propositions. On account of the commensurability of classical propositions the laws of classical logic which hold for classical propositions are completely reproduced also in quantum logic. For this reason, quantum logic is the most general, universal logic of physical propositions. The Lindenbaum-Tarski algebra of the calculus LQ is given by a complete orthomodular lattice LQ. Subsets of mutual commensurable propositions constitute Boolean sublattices L ~) ~ LQ of the lattice LQ. 16 From an algebraic point of view, the orthomodular lattice LQ is much more complicated than the corresponding Boolean lattice LB. The orthomodular lattice of quantum logic can be further specified. It is freely generated by a finite (or infinite) number of elementary propositions. Moreover, if the entire quantum language SQ refers to one quantum system, then the lattice LQ is atomic and the atoms which correspond to pure states provide a maximal information about the system. In addition, the covering law is alsofulfilled by the lattice LQ. 17 Furthermore, the Hilbert lattice LH of projection operators in Hilbert space 18 can be obtained from this lattice LQ by adding the Soler law, the operational meaning of which is, however, still open. 19

5 Conclusion Quantum logic is weaker than classical logic and process semantics is weaker than a realistic semantics. These new structures do not contain new experimental information. On the contrary, for the formulation of quantum logic we left out all those supposed empirical results of classical physics which cannot be justified by quantum physics. Hence, quantum logic could have been discovered even without any quantum-physical knowledge merely by very cautious argumentation. Of course, there are situations which allow for classical logic. This is the domain of classical physics. However, the application of classical logic requires a justification in every individual case. Neither the realistic semantics nor the classical logic may be considered as the normal case. Hence, the relaxation of this "normal case" must not be justified 16 17 18 19

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but the "normal case" itself requires a legitimation. Quantum logic is the most general structure which is free from special presuppositions of classical physics.

References 1. P. Busch, P. Lahti, P. Mittelstaedt: The Quantum Theory of Measurement

(Springer, Heidelberg 1991) (2nd edn. 1996) 2. N. Bohr: 'The Quantum Postulate and the Recent Development of Atomic Theory'. In: Atti del Congresso Internationale del Fisici, Como, 11-20, September 1927, (Zanichelli, Bologna 1928) pp. 565-8 3. P. Mittelstaedt: Philosphical Problems of Modern Physics (Reidel, Dordrecht 1976) 4. P. Mittelstaedt: Quantum Logic (Reidel, Dordrecht 1978) 5. P. Mittelstaedt: Spmche und Realitiit in der modernen Physik (BIWissenschaftsverlag, Mannheim 1986) 6. P. Mittelstaedt: The Interpretation of Quantum Mechanics and the Measurement Process (Cambridge University Press, Cambridge 1998) 7. J. von Neumann: Mathematische Grundlagen der Quantenmechanik (Springer Verlag, Berlin 1932) 8. G. Birkhoff, J.v. Neumann: 'The Logic of Quantum Mechanics'. Annals of Mathematics. 37 (1936) pp. 823-43 9. E.-W. Stachow: 'Logical Foundations of Quantum Mechanics.' Int. Journ. of Theoretical Physics. 19 (1980) pp. 251-304 10. E.-W. Stachow: 'Structures of a Quantum Language for Individual Systems'. In: Recent Developments in Quantum Logic ed. by P. Mittelstaedt and E.W. Stachow (BI-Wissenschaftsverlag, Mannheim 1984) 11. M.P. Soler: 'Characterisation of Hilbert Spaces by Orthomodular Lattices'. Communications in Algebra 23(1) (1995) pp. 219-243