The ManyValued Logic of Quantum Mechanics
Abstract
It is shown that any quantum logic in the Birkhoff  von Neumann sense that possesses an ordering set of probability measures can be isomorphically represented as a particular kind of infinitevalued Łukasiewicz logic with partially defined conjunction and disjunction. It is argued that such nonclassical features of BvN quantum logic as lack of distributivity or validity of the orthomodular law follow from properties of Łukasiewicz conjunction and disjunction.
Keywords
Quantum logic Manyvalued logic Łukasiewicz logic1 Introduction
In 1936 Garrett Birkhoff and John von Neumann in their seminal paper ‘The logic of quantum mechanics’ [2] argued that the structure of a set of dichotomic (‘yesno’) propositions about properties of quantum objects, usually called ‘quantum logic’, which is an orthomodular lattice, differs from the structure of a set of such propositions pertaining to classical objects, in which case it is a Boolean algebra. Nevertheless, since after experimental checking any such dichotomic proposition occurs to be either true or false, quantum logic in the BvN sense is generally treated as 2valued logic, nonclassical because of lack of distributivity which is unrestrictively guaranteed in Boolean algebras.
Among other nonclassical logics studied at that time, in general without references to quantum physics, were various kinds of manyvalued logics. Although there were several modest attempts at applying manyvalued (mostly 3valued) logic in description of quantum systems (see bibliography collected in [14]), they never reached mathematical maturity of Birkhoff and von Neumann’s proposal. As it was noticed by Max Jammer in his book ‘The Philosophy of Quantum Mechanics’ [4], this line of investigations ceased in the late 1950s.
The author’s way towards expressing BvN quantum logic as a particular kind of Łukasiewicz infinitevalued logic originated from Ma̧czyński’s functional representation theorem and fuzzy sets. In 1973 Maciej Ma̧czyński proved [8] that any orthomodular poset with an ordering set of probability measures S (only such orthomodular posets are physically interesting models of quantum logics) can be isomorphically represented as a family of functionals that map S into [0,1] and fulfill certain specific conditions. Since any such functional defines a fuzzy subset of S, it was natural to try to express Ma̧czyński’s conditions in the language of fuzzy set theory. This nontrivial task, undertaken by the author in 1987 [11], was accomplished in 1994 [12].
Since relations between fuzzy sets and infinitevalued logic are the same as relations between traditional (‘crisp’) sets and 2valued classical logic, fuzzy set representation of BvN quantum logic, i. e. an orthomodular lattice or poset with an ordering set of probability measures (states) S, allows to represent it further as a family of infinitevalued propositional functions defined on S with globally defined negation and partially defined conjunction and disjunction (see [13], all the results are collected in [14]). In this way BvN quantum logic can be seen as nonclassical logic mainly because of its manyvaluedness, with lack of distributivity and other nonclassical features being derived from it.
The paper is organized as follows: In Section 2 we give very general arguments why description of future experiments performed on quantum objects requires manyvalued logic. In Section 3 we recall the original ordertheoretic definition of BvN quantum logic, and Ma̧czyński’s functional representation theorem. Section 4 is devoted to the fuzzy set model of BvN quantum logic while in Section 5 we show that any BvN quantum logic with an ordering set of states can be isomorphically represented as a specific infinitevalued logic endowed with the standard manyvalued negation and partially defined Łukasiewicz conjunction and disjunction.
2 Quantum Mechanics Deals with Futura Contingentia, Therefore it Requires ManyValued Logic
The problem of futura contingentia, i.e., statements about future noncertain events has been present in European philosphy since Hellenic times. Aristotle considered the famous statement ‘There will be a sea battle tomorrow’ in De Interpretatione and it seems (cf. [6]) that he treated such statements as neither true nor false, i.e., according to the modern point of view, as not belonging to the domain of classical twovalued logic.
Since classical mechanics is a deterministic theory, the problem of futura contingentia does not concern it: In deterministic theories all the future is strictly determined, so any statement concerning future events is either true or false already at the moment of stating it, although it may not be known which case occurs. All but stubborn believers in nonprobabilistic hidden variable theories agree that quantum mechanics is indeterministic since in most cases the results of future experiments can be predicted only probabilistically, and this is a ‘fundamental’ fact not caused by lack of knowledge of some parameters, initial conditions, etc. In other words, quantum probabilities are ‘ontic’ not ‘epistemic’, i.e., they do not result from our lack of knowledge.
Jan Łukasiewicz, the founding father of modern theory of manyvalued logics, argued in his numerous papers [7] that statements about future noncertain events belong to the domain of manyvalued logic. Already in 1913 in his valuable but almost forgotten paper [5] he identified nonclassical, i.e., different from 0 or 1, truth values with probabilities that considered statements will turn out to be true (in classical sense) in the future.
This approach perfectly fits to statements concerning future noncertain events in quantum mechanics: Instead of saying ‘Probability that a photon will pass through a semitransparent mirror is 0.5’ one can, following Łukasiewicz, say ‘Truth value of a sentence “A photon will pass through a semitransparent mirror” is 0.5’. Of course when an experiment is completed, this sentence expressed in the past tense: ‘A photon passed through a semitransparent mirror’ occurs to be either true or false, which was the reason for treating BvN quantum logic as 2valued logic. Let us note that this ‘collapse of manyvalued logic onto 2valued logic’ that happens when the experiment is completed is no more strange than passing from probability that is different from 0 or 1 while concidering the future event ‘a photon will pass through a semitransparent mirror’ to probability that is either 0 or 1 while concidering the past event ‘a photon passed through a semitransparent mirror’.^{1} They are simply two different events and statements that describe them are two different statements.
When we agree that statements about future noncertain events should be analyzed with the use of manyvalued logic, we are still facing a problem which of the plethora of various models of manyvalued logics should be used. Fortunately in the case of statements concerning results of future experiments on quantum objects the answer is unique: There is exactly one rather specific model of infinitevalued logic endowed with globally defined negation and partially defined conjunction and disjunction such that any BvN quantum logic in the traditional ordertheoretic sense can be isomorphically represented as such logic. The rest of the paper is devoted to showing the details of this construction.
3 Birkhoff  von Neumann Quantum Logic and its Ma̧czyński’s Functional Representation
Since Birkhoff and von Neumann’s paper [2] the ‘standard’ model of quantum logic is the family \(\mathcal L (\mathcal H)\) of closed linear subspaces of a Hilbert space used to describe a studied quantum system or, equivalently, the family of operators of orthogonal projections onto these subspaces. These families from the ordertheoretic point of view are orthomodular lattices partially ordered, in the first case by the settheoretic inclusion, and in the second case by the relation P_{A} ≤ P_{B} iff P_{A}P_{B} = P_{A}. According to the modern approach to quantum logic theory the term ‘quantum logic’ usually denotes a little bit more general structure, namely orthomodular partially ordered set, and this position is adopted in the present paper.
By a quantum logic we mean an orthocomplemented σorthocomplete orthomodular poset, i.e. a partially ordered set L that contains the smallest element O and the greatest element I, in which the orthocomplementation map \(\perp :L\rightarrow L\) satisfying the conditions (a)–(c) exists:
 (a)

(a^{⊥})^{⊥} = a.
 (b)

If a ≤ b, then b^{⊥}≤ a^{⊥}.
 (c)

The greatest lower bound (meet) a ∧ a^{⊥} and the least upper bound (join) a ∨ a^{⊥} with respect to the given partial order exist in L and a ∧ a^{⊥} = O, a ∨ a^{⊥} = I.
Moreover, the σorthocompleteness condition holds:
 (d)

If \(a_{i}\leq a_{j}^{\perp }\) for i≠j (such elements are called orthogonal and are usually denoted a_{i} ⊥ a_{j}), then the join \( \bigvee _{i}a_{i}\) exists in L,
and so does the orthomodular identity:
 (e)

If a ≤ b, then b = a ∨ (a^{⊥}∧ b) = a ∨ (a ∨ b^{⊥})^{⊥}.
According to Birkhoff and von Neumann (cf. also [1, 10]) elements of a quantum logic represent experimentally verifiable dichotomic (‘yesno’) propositions about a studied physical system.
Elements a, b ∈ L are called compatible iff there exist in L pairwise orthogonal elements a_{1}, b_{1,}c such that a = a_{1} ∨ c and b = b_{1} ∨ c. Let us note that compatibility of elements in a Hilbertian quantum logic \(\mathcal L (\mathcal H)\) in its ‘projector’ version means that projectors are compatible according to this definition if and only if they are commuting (see, e.g., [1]).
Probability measure (also called state) on a quantum logic L is a mapping \( s:L\rightarrow [0,1]\) such that
 (i)

s(I) = 1,
 (ii)

\(s(\bigvee _{i}a_{i})={\sum }_{i} s(a_{i})\) for any sequence of pairwise orthogonal elements of L.
The reason for naming probability measures on quantum logics ‘states’ follows from the fact that in a Hilbertian quantum logic \(\mathcal L (\mathcal H)\) by Gleason Theorem these probability measures are in 1–1 correspondence with density operators on a Hilbert space \(\mathcal H\) that describes a studied quantum system, which in turn actually represent states of the studied quantum system.
A set of states S on a quantum logic L is called ordering (full, order determining) iff s(a) ≤ s(b) for all s ∈ S implies a ≤ b. Let us note that only quantum logics with ordering sets of states are physically interesting. This is caused by the fact that the only way in which one can establish experimentally the partial order relation between various propositions is to conduct experiments on a system prepared in various states. Therefore, throughout the rest of the paper we shall consider only quantum logics with ordering sets of states.
Maciej Ma̧czyński in [8] proved a theorem that later on occured to form a substantial part of a ‘bridge’ by which one can pass from BvN quantum logic in its ordertheoretic form to its fuzzy set or manyvalued representation. We quote this theorem in the form that appeared in [9].
Theorem 1
LetS be a nonempty set and letL be a set of mappings fromS into[0,1] thathas the following three properties:
 (i)

0 (thenull function) belongs to L
 (ii)

a ∈ Limplies1 − a ∈ L
 (iii)

for any (finite or countable) sequencea_{1}, a_{2},... ∈ Lsuchthata_{i} + a_{j} ≤ 1 fori≠j(suchfunctions were called in [8] pairwise orthogonal), we havea_{1} + a_{2} + ... ∈ L.
ThenL is a quantum logic with respect to the natural partial order of real functions, withorthocomplementationa^{⊥} = 1 − a.Every pointu ∈ Sinduces aprobability measurem_{u}on(L,≤,^{⊥}), wherem_{u}(a) = a(u) for alla ∈ L, and the family ofprobability measures {m_{u} : u ∈ S} isordering.
Conversely, if (L,≤,^{⊥}) isa quantum logic with an ordering setS of probability measures, then eacha ∈ Linduces afunction\(\overline {a}:S\rightarrow [0,1]\)where\(\overline {a}(m)=m(a)\)for allm ∈ S. The set of all suchfunctions\(\overline {L}=\{\overline {a} :a\in L\}\)has properties(i) – (iii) and\((\overline {L},\leq ,^{\perp })\)isisomorphic to (L,≤,^{⊥}).
4 Fuzzy Set Representation of BvN Quantum Logic
It was noticed already in [11] that Ma̧czyński’s functions that map a set S into [0,1] actually define membership functions of fuzzy subsets of S. Therefore, by Theorem 1 any quantum logic L with an ordering set of probability measures S can be expressed in the language of fuzzy set theory and then its condition (i) means that the empty set belongs to L, while condition (ii) means that if a set belongs to L, then its standard fuzzy complement also belongs to L. Also a part of condition (iii) can be expressed with the use of standard fuzzy operations: Two fuzzy sets A and B such that their Łukasiewicz intersection defined as:
When this fourth condition is added, Ma̧czyński’s theorem is fully expressible in the language of fuzzy set theory and takes the following form [12]:
Theorem 2
Any quantum logic L with an ordering set of probability measures S can be isomorphically represented in the form of a family \(\mathcal {L}(S)\) of fuzzy subsets of S satisfying the following conditions:
 (a)

\(\mathcal {L}(S)\) contains the empty set.
 (b)

\(\mathcal {L}(S)\) is closed with respect to the standard fuzzy set complementation.
 (c)

\(\mathcal {L}(S)\) is closed with respect to the countable Łukasiewicz unions of pairwise weakly disjoint sets.
 (d)

The empty set is the only set in \(\mathcal {L}(S)\) that is weakly disjoint with itself.
Conversely, any family of fuzzy subsets of an arbitrary universe \(\mathcal {U} \) satisfying conditions (a)–(d) is a quantum logic partially ordered by the inclusion of fuzzy sets, with the standard fuzzy set complementation as orthocomplementation, the orthogonality of the elements coinciding with their weak disjointness, and an ordering set of probability measures generated by points of the universe \(\mathcal {U}\) according to the formula
Let us note that it follows from the condition (d) that the only fuzzy subsets in the structure described above that have constant membership functions are the empty set and its complement  the universe. Indeed, if there is in the structure any set with constant membership function, then either this set, or its complement is weakly disjoint with itself which, by condition (d), means that its membership function either everywhere equals 0 or everywhere equals 1.
4.1 Two Pairs of Binary Operations
Although by Theorem 2 fuzzy set representation \(\mathcal {L}(S)\) of any BvN quantum logic L with an ordering set of probability measures S is characterized by fuzzy set operations, in particular Łukasiewicz intersection (1) and union (2), by the second part of this theorem it is still a partially ordered set, so it is still equipped with ordertheoretic operations of meet and join. Relations between these two pairs of operations were studied in [13] and are summarized in the following theorem (see also [14]).
Theorem 3
Let\(\mathcal {L}(\mathcal {U})\)bea family of fuzzy subsets of an arbitraryuniverse\(\mathcal {U} \)satisfyingconditions (a)–(d) of Theorem 2, and let\(A,B\in \mathcal {L}(\mathcal {U})\).Then\( A\sqcap B\in \mathcal {L}(\mathcal {U})\)iff\(A\sqcup B\in \mathcal {L}(\mathcal {U})\),and in this case A and B are compatible,A ⊓ B = A ∧ B,andA ⊔ B = A ∨ B.
It follows from this theorem that in general Łukasiewicz intersection (1) and union (2) are in \(\mathcal {L}(\mathcal {U})\) defined only partially, even if \(\mathcal {L}(\mathcal {U})\) is a lattice in which meet and join for any pair of elements exist. Indeed, if Łukasiewicz intersection (1) and union (2) were in any \(\mathcal {L}(\mathcal {U})\) defined globally, then all elements would be compatible, so any \(\mathcal {L}(\mathcal {U})\) would be a Boolean algebra. This also shows that the existence in \(\mathcal {L}(\mathcal {U})\) of meet or join of any elements is not a sufficient condition for the existence of their Łukasiewicz union or intersection, although the opposite is true.
It was noticed in [16] that by transposition of an implication: ‘if \(A\sqcap B,A\sqcup B \in \mathcal {L}(\mathcal {U})\), then A and B are compatible’, one obtains corollary:
Corollary 1
If\(A,B \in \mathcal {L}(\mathcal {U})\)arenot compatible, thenA ⊓ B, A ⊔ Barenot defined in\( \mathcal {L}(\mathcal {U})\).
This again shows that Łukasiewicz intersection (1) and union (2) are, in general, defined in \(\mathcal {L}(\mathcal {U})\) only partially. Let us stress that in view of wellknown interpretational difficulties yielded by the original BvN hypothesis that meets and joins are proper algebraic representations of conjunctions and disjunctions (cf. [1, 2]), the fact that Łukasiewicz operations are in quantum logics not globally defined is rather a virtue than a drawback (see discussion in [16]).
It is easy to show, e.g. by taking A(x) = B(x) = C(x) = 0.5, that Łukasiewicz intersection (1) and union (2) are, in general, not distributive, i.e., that in general
Lemma 1
If\(A\subseteq B\),thenB = A ⊔ (A^{′}⊓ B) = A ⊔ (A ⊔ B^{′})^{′}.
Proof
For any x we have: \([A\sqcup (A'\sqcap B)](x) = \min [A(x)+(A'\sqcap B)(x),1] = \min [A(x)+\max [A'(x) + B(x)1,0],1] =\min [A(x) + \max [1A(x) + B(x)  1 , 0], 1] = \min [A(x) + \max [B(x)  A(x), 0],1] = \min [A(x)+B(x)A(x),1] = B(x)\). The second equality follows from the fact that Łukasiewicz intersection (1) and union (2) together with the standard fuzzy complementation form a de Morgan triple, so the proof is finished.□
Let us now assume that \(\mathcal {L}(\mathcal {U})\) is a family of fuzzy subsets of an arbitrary universe \(\mathcal {U}\) satisfying conditions (b) and (c) of Theorem 2, and let \(A, B \in \mathcal {L}(\mathcal {U})\) be such that \(A\subseteq B\). This means that for any \(x \in \mathcal {U}\) we have A(x) ≤ B(x) ⇔ A(x)−B(x) ≤ 0 ⇔ A(x)+ 1−B(x) ≤ 1 ⇔ A(x)+B^{′}(x) ≤ 1 ⇔ A⊓B^{′} = ∅. This by condition (c) of Theorem 2 means that \(A\sqcup B' \in \mathcal {L}(\mathcal {U})\), which in turn by condition (b) of this theorem and de Morgan identity means that \((A\sqcup B')' = A'\sqcap B \in \mathcal {L}(\mathcal {U})\). Now let us note that because of associativity of Łukasiewicz intersection we have A ⊓ (A^{′}⊓ B) = (A ⊓ A^{′}) ⊓ B = ∅⊓ B = ∅, so by condition (c) of Theorem 2 \(A\sqcup (A'\sqcap B) \in \mathcal {L}(\mathcal {U})\). These observations together with Lemma 1 yield the following lemma:
Lemma 2
Let \(\mathcal {L}(\mathcal {U})\) be a family of fuzzy subsets of an arbitrary universe \(\mathcal {U}\) satisfying conditions (b) and (c) of Theorem 2. Then in \(\mathcal {L}(\mathcal {U})\) the formal counterpart of the orthomodular identity holds and all sets that appear in it belong to \(\mathcal {L}(\mathcal {U})\) .
5 BvN Quantum Logic as InfiniteValued Logic
Relations between fuzzy sets and manyvalued logics are exactly the same as relations between traditional (‘crisp’) sets and classical twovalued logic. For example, settheoretic complementation is related to logical negation, intersection of sets is related to conjunction, and union of sets is related to disjunction in the same way in both cases:
Before we shall pass to the manyvalued model of BvN quantum logic, let us stress that actually this logic does not deal with propositions but with propositional functions the domain of which is the set of states of a studied physical system. To show this let us consider a statement ‘A photon will pass through a linear polarizer’. Obviously the truth value of this statement depends on the polarization state of the photon, and various polarization states yield various truth values of this statement. It follows from the remark after Theorem 2 that there are only two constant propositional functions (i.e., actually, propositions) in the studied structure: the always false propositional function f and the always true propositional function v, which can be linguistically expressed, for example as: f = ‘The studied physical system does not exist’ and v = ‘The studied physical system exists’.
Let us call two propositional functions a(⋅) and b(⋅) exclusive if their Łukasiewicz conjunction (10) is always false, i.e., if for all x in their common domain
Theorem 4
Any quantum logicL with an ordering set of probabilitymeasuresS can be isomorphically represented as a familyΛ(S) ofpropositional functions defined onS and satisfying the following conditions:
 (a)

Λ(S) containsthe alwaysfalse propositional function f.
 (b)

Λ(S) isclosed with respect toŁukasiewicz negation (9).
 (c)

Λ(S) isclosed with respect toŁukasiewicz disjunction ofpairwise exclusive propositional functions, i.e. ifa(⋅)_{i} ⊓ a(⋅)_{j} = ffori≠j,then ⊔_{i}a(⋅)_{i} ∈Λ(S).
 (d)

The alwaysfalse propositional function f is the only propositional function inΛ(S) thatexcludes itself, i.e., for anya(⋅) ∈Λ (S),ifa(⋅) ⊓ a(⋅) = f,thena(⋅) = f.
Conversely, any family of manyvalued propositional functions defined on a common domain D and satisfying conditions (a) – (d) is a quantum logic in Birkhoff  von Neumann sense whenever we identify propositional functions that assume the same truth value for every argument in their common domain D. This family is partially ordered by pointwise partial order of propositional functions:
Let us note that formula (13) actually means that partial order between propositional functions is defined by Łukasiewicz implication defined by him (cf. [7]) as follows:
6 Conjunctions and Disjunctions in the ManyValued Logic of Quantum Mechanics
There are no doubts that in Boolean algebras, which are proper models of sets of dichotomic (‘yesno’) propositions about properties of classical objects, conjunctions and disjunctions of these propositions are modeled by ordertheoretic operations of meet and join. However, even Birkhoff and von Neumann  the founding fathers of quantum logic  in their historic paper [2] doubted whether this idea can be extended to sets of propositions about properties of quantum objects, where some properties are not simultaneously measurable. Discussions on this problem lasted for years (cf. [1, 16]) but the problem is still open.
Theorem 4 strongly suggests that conjunctions and disjunctions of statements concerning results of future experiments on quantum objects should be modeled by Łukasiewicz conjunction (10) and disjunction (11), not by ordertheoretic operations of meet and join. Of course it follows from Corollary 1 that these conjunctions and disjunctions can be formed only when elements of a quantum logic are compatible, i.e., respective properties of quantum objects are simultaneously measurable.
Since it follows from Theorem 3 that if Łukasiewicz operations (10) and (11) are defined they coincide, respectively, with meet and join, this explains why for many years meets and joins were treated as proper ordertheoretic models of conjunctions and disjunctions in quantum logics. The results of Section 4.1, when translated from the language of fuzzy sets into the language of manyvalued logic, show that such nonclassical features of BvN quantum logic as lack of distributivity or replacement of distributivity laws by a weaker orthomodular law may be seen as following from the manyvaluedness of this logic and from properties of Łukasiewicz negation, conjunction, and disjunction.
Let us finish with a remark that the problem of choosing proper mathematical representation of conjunctions and disjunctions is of utmost importance in studying situations of the EPR  Bell  GHZ  etc. types since in all these situations there appear conjunctions of the form: ‘The objectAhas propertyxAND the objectBhas propertyy’. Some investigations in this direction were published in [15].
Footnotes
 1.
Since the past is fully determined, probabilities of all events that happened in the past, i.e., truth values of all statements that concern past events are either 0 or 1.
 2.
These logical connectives are nowadays called Łukasiewicz conjunction and disjunction, although they were never studied by Łukasiewicz in his original papers (cf. [7]).
Notes
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