Classical and Quantum Logics with Multiple Non-Equivalent and Non-Isomorphic Models

Introduction

In [7] we show that there are non-isomorphic and non-equivalent models for quantum as well as for classical logic. In [6] we enlarge the number of such models. Models represent the semantics of logic and in textbook presentations they are divided in several interpretations. For instance, in [12 Ch. 8-10], they are grouped in numerically valued, set-valued, and topological interpretations. The set-valued interpretations include powerset, our O6 (called hexagon) [12 p. 272, 305], crystal, and Church’s diamond ones.

Yet, some authors and their referees are of the opinion that certain characterizations of our interpretations of quantum and classical logics are inappropriate, in particular Kowalski, Paoli and Giuntini [5] and Drozdowska [2]. Therefore, in this paper, we address the discrepancies between their characterizations of the considered interpretations and the characterizations accepted by others in the literature and by ourselves. In doing so, we have to start with notions, concepts, definitions, systems, theorems, procedures, and results we already put forward in several of our previous papers but that were misinterpreted by the aforementioned critics and possibly other readers. That being our path, we can only do our best to present those facts in a condensed juxtaposed manner, quantum vs. classical, aiming to pinpoint that there is nothing contradictory in having non-equivalent and non-isomorphic models of a single considered logic. Thereupon we explicitly address the criticisms in § 2.1.

The paper is organized as follows. In § 1.1 quantum and classical logic are presented in a comparative manner. In Sec. III various weakly orthomodular, orthomodular, weakly distributive, and distributive lattices are presented in a similar way. In § 2 we present mutually non-equivalent and non-isomorphic models of quantum and classical logical and proofs of soundness and completeness for all of them.

In Sec. V we present two criticisms of the fact that quantum and classical logics have mutually non-equivalent and non-isomorphic models recently put forward by several authors. We analyse each of their claims and show that they are either unfounded or provably wrong. In Sec. VI we summarize our arguments against the criticisms.

Quantum and Classical Logics

Logic $\mathcal{L}$ is a language defined by a set of axioms and rules imposed on propositions. We consider quantum $\mathcal{QL}$ and classical $\mathcal{CL}$ axiomatic logics.

Primitive (elementary) propositions/well-formed formulae (wffs) in $\mathcal{L}$ are denoted as $p_0,p_1,p_2,\ldots$; primitive connectives are $\neg$ (negation) and $\lor$ (disjunction). $p_j$ is a wff for $j=0,1,2,\ldots$; $\neg A$ is a wff if $A$ is a wff; $A\lor B$ is a wff if $A$ and $B$ are wffs. We denote the set of all wffs by $\mathcal{F}^{\circ}$. Further operations are defined as follows.

We do not give all the axioms here. The reader can find them in [6]. Notice that quantum logic requires a lot more axioms than classical logic but that is not of importance for the current presentation.

Quantum Logic : $\mathcal{QL}$

Axioms :

$\ldots$

Rule of Inference :

Classical Logic : $\mathcal{CL}$

Axioms :

$\ldots$

Rule of Inference :

Quantum and Classical Lattices

A lattice $L = \langle L^{\circ}, \cup, \cap\rangle$ is a partially ordered set such that for every pair of $a,b \in L^{\circ}$ there exist both $a \cap b \in L^{\circ}$ and $a \cup b \in L^{\circ}$. Properties of intersection and union in the algebra of sets are carried over into $L$ as expressions called conditions of $L$. The partial order of elements $a \le b$ acquires the forms $a \cup b = b$ and $a \cap b = a$. When the negation “$'$” is added we obtain an ortholattice $OL = \langle OL^{\circ}, ', \cup, \cap\rangle$ and when the weak-orthomodularity-like condition $w$-oml-like-c is added to an $OL$ we obtain a weakly orthomodular lattice WOML (in general a class of WOMLs). Notice that the greatest $(a \cup a')$ and the least $(a \cap a')$ elements of a lattice, usually denoted as 1 and 0, respectively, need not be numbers.

Again we do not give all the lattice conditions here. The reader can find them in [6].

Weakly Orthomodular Lattice WOML:

Conditions:

By adding the following conditions we obtain quantum and classical lattices. OML stands for orthomodular lattice, WDOL for weakly distributive ortholattice, and WD for distributive lattice (Boolean algebra). In general for classes of such lattices.

Quantum Lattices:

(orthomodularity)

Classical Lattices:

(distributivity)

A sublattice of $L$ which does not contain any stronger lattice or in our case any of of the considered stronger lattices we denote by $L^*$.

In [6] we provide some examples of these lattices (e.g., O6 and O7).

Lattices as Models of Logics

Soundness

To prove soundness means to prove that the axioms and rules of inference and consequently all theorems of $\mathcal{L}$ hold in the models of $\mathcal{L}$. The models of $\mathcal{QL}$ are O6, WOML*, WOML$_i^*$, and OML, and of $\mathcal{CL}$ are O6, WOML, WDL* and DL.

With the help of valuation $h$, given by Def. IV.1, which maps elements of logic to elements of lattice $L$: $a = h(A)$, $b = h(B)$, ... and operations $\neg$, $\vee$ to $\neg$, $\vee$, we can prove all mapped axioms and the rule of inference in any model $\mathcal{M} = L$, where $L$ can be WOML, WOML*, OML, WDOL, WDOL*, DL, etc.

Completeness

Completeness means that every lattice formula which has a valuation “1” that holds in all of the lattice models is provable in the logic.

For each of the models there is a corresponding instance of completeness. For example, for the weakly orthomodular lattice not containing the axiom of orthomodularity WOML*: $\Gamma\models_{\mathrm{WOML}^*} A \Rightarrow\Gamma\vdash_{\mathcal{QL}} A$, for the orthomodular lattice model OML: $\Gamma\models_{\mathrm{OML}} A \Rightarrow\Gamma\vdash_{\mathcal{QL}} A$, for the weakly distributive lattice not containing the axiom of distributivity model WDOL*: $\Gamma\models_{\mathrm{WDOL}^*} A \Rightarrow\Gamma\vdash_{\mathcal{CL}} A$, for the distributive lattice model DL: $\Gamma\models_{\mathrm{DL}} A \Rightarrow\Gamma\vdash_{\mathcal{CL}} A$, etc. Notice that two non-equivalent models WOML* and OML (non-orthomodular and orthomodular) generate the same quantum logic $\mathcal{QL}$ and that two non-equivalent models WDOL* and DL (non-distributive and distributive) generate the same classical logic $\mathcal{CL}$.

Technically, we carry the completeness proofs with the help of the Lindenbaum algebras. In order to carry them out we have to introduce some definitions and lemmas.

However, the relation $a=b$ on $F^\circ$ which defines a relation of equivalence $\approx$

would encompass all formulas WOML contains, including those from OML. Hence, we have to change the definition of the relation of equivalence so as to make it impossible for both $a\cup(b\cap(a'\cup b'))$ and $a\cup b$ to be in the same class of equivalence since that would amount to $a\cup(b\cap(a'\cup b'))=a\cup b$, i.e., to the orthomodularity. We achieve that by expanding the definition of the relation of equivalence by a term which verifies that the equation can pass a chosen WOML lattice. We have chosen lattice O6 in which the orthomodularity fails and all WOML equations and rules pass.

Since $o6(A\vee(B\wedge(\neg A\vee\neg B)))=x$ and $o6(A\vee B)=y$, meaning that the orthomodularity violates O6, the relation of equivalence, so defined, excludes the OML and confirms the WOML* as a model of quantum logic.

Proof. $f(A)=1$ is equivalent to $|A|=|B\vee\neg B|$ and therefore to

There are WOML conditions that are stronger than those given by Eq. (1), e.g., the one that defines WOML1 above or the following one which defines WOML2.

Here we should give Lemma 22 from [6] together with its proof because the proof from [6 Lemma 22] contains several typos.

Proof. The orthomodularity condition fails in O6, O7, and O8 from Figure 1. The condition which defines WOML2 (Eq. (8)) passes O6, but fails in O7 and O8. The condition which defines WOML1 passes O6 and O7, but fails in O8.

We also obtain

as well as

For classical logic we have [6]

as well as

This is all in addition to the standard models:

The theorems are given a graphical representation in Figure 2.

Remarks, Criticisms, and Responses

In our papers [6, 7] we made the following remark:

What we meant by this we explained as follows (ibid.):

we additionally have

which are traditionally called the orthomodularity and the distributivity.

Therefore, what we meant by Remark V.1 is explained by V.2. In the literature, the orthomodularity and distributivity are usually expressed in the equational form like in Eqs. (13), (14) and it is our opinion that the equivalence forms given by Eqs. (9), (10) should not be called the orthomodularity and distributivity, respectively. This is substantiated by the following conventions which, in effect, reduce quantum logic to an orthomodular lattice and classical logic to the Boolean algebra while we formulate them as genuine logics.

Historical Quantum Conventions. V.3.

“Quantum logic is a poset which possesses a least and a greatest element.” [10 p. 1] [A poset with unique least upper bound (join, $a \cup b$) and a unique greatest lower bound (meet, $a \cap b$) is a lattice.]

“An orthomodular poset is a quantum logic. . . Sometimes in the literature . . . quantum logic is understood [to be] a lattice.” [3 p. 73].

Historical Classical Conventions. V.4.

“[In classical logic] there is . . . a distributive law $X \wedge(Y \vee Z)$ eq. $(X \wedge Y) \vee(X \wedge Z)$ [where] “eq.” does not belong to [the] set of logical symbols.” [4 pp. 6, 7]

“[The distributivity] is expressed as [given by Eq. (12)]; $\equiv_c$ is defined by the truth table.” [1 pp. 29, 32]`

We see that directly or indirectly the authors in the field use the terms defined in the models of a logic, not in the logic itself. This seems to have misled Kowalski, Paoli and Giuntini [5] into the following claims.

Response V.6. to Claim V.5. Notice that the authors call the quantum logic (“the oldest and noblest,” p.186) orthomodular quantum logic (idem per idem argument in an attempt to prove that quantum logic itself is orthomodular). They further express their criticism as follows.

Response V.8. to Claim V.7. Apparently, the essence of the completeness proof of all QL models escapes the authors. To repeat, completeness means that every lattice formula which has a valuation “1” that holds in all of the lattice models is provable in the logic, meaning that all models WOML, $WOML^*$, $WOML1,\ldots OML$ yield the same theorems of QL (“indiscernible elements”) although these models are mutually non-isomorphic and non-equivalent as we show in the previous sections above and graphically present in Figure 2. It seem that the authors are not at ease with this result.

We can sum up our responses to the claims made by Kowalski, Paoli and Giuntini as follows:

• Kowalski, Paoli, and Giuntini’s claim that one should use equivalent algebraic semantics (model) which favours OML as the only acceptable model (semantics) of QL is unfounded since we rigorously proved that all other models/semantics of QL stand on equal footing with OML in spite of the fact that they might be mutually non-equivalent and non-isomorphic—see Figure 2 and Theorems IV.12-IV.18

• Kowalski, Paoli, and Giuntini apparently object (although it is not quite clear from the statements they put forward in their paper) the fact that soundness and completeness of $QL$ lead to the same set of theorems of $QL$. Such an objection would be unfounded (if really held) because it is the essence of soundness and completeness of $QL$ with respect to all of its models/semantics in the sense that all its non-equivalent and non-isomorphic models must yield all their theorems and rules of inference, nothing more and nothing less.

• Kowalski, Paoli, and Giuntini claim that $QL$ is an orthomodular quantum logic is unfounded since a logic which is sound and complete with respect to non-orthomodular models (WOML*, WOML1*, . . .) can hardly be called orthomodular, especially since the orthomodularity equation given by Eq. (15) cannot be expressed within the $QL$ formalism.

Another criticism comes from Drozdowska [2]. It exhibits misunderstandings and an error (which invalidates one of her two main claims, V.23), but since responses to her claims offer a good insight into the cardinal points of our results, we reckon that the best way to address them would be in the following Claim-and-Response manner. Drozdowska focuses on O6 and matrix semantics, adequate matrices, mapping of O6 to a two-element Boolean matrix, etc., but since O6 is a WOML, we make use of the algebraic model language to address her criticism. Her claim that “[the definitions of model and validity employed by Pavičić and Megill are matrix-theoretic in character” [2 p. 304] might only refer to O6 itself and/or O6-valuation within the equivalence relation of Lindenbaum algebra used to prove the completeness of $CL$ and therefore in our responses to her matrix-theoretic claims we keep to the abstract algebraic language.

Response V.18. to Claim V.17. Classical logic can hardly be called distributive since the distributivity equation Eq. (16) cannot be expressed within the $CL$ formalism and since non-distributive models exist.

Claim V.19. (Drozdowska) One possible meaning of distributivity of $CL$ concerns the syntactic side of the logic. In this sense, a logic is distributive if the distributivity laws $A \land(B \lor C) \equiv_{c} (A \land B) \lor(A \land C)$ and $A \lor(B \land C) \equiv_{c} (A \lor B) \land(A \lor C)$ are its theorems. [2 p. 285].

Response V.20. to Claim V.19. Such a definition of the distributivity law (given by Eq. (10)) does exist in the literature [11 p. 60], but that traditionally assumes that the only model of the classical logic is the Boolean algebra, i.e., that the theorems are evaluated by means of $\{\top,\bot\}$ truth-tables [11 p. 15]. This historical link of the classical logic to distributivity should not be taken as a base for calling classical logic distributive since by the same token we can call it orthomodular because the “orthomodular law” given by Eq. (9) holds in it or “modular law” $\vdash_{CL} (A\lor B)\land(A\lor C)\equiv_{q} (A\lor(B\land(A\lor C)))$ holds in it as well. On the other hand, since models/semantics of $CL$ can also be non-distributive lattices, to call the logic distributive is additionally not appropriate from a semantical point of view. A possible approach to this terminological awkwardness would be to restrain ourselves from attaching distributive, or orthomodular, or modular appellation to classical logic altogether and simply call $CL$ a classical logic, instead. The denominations distributivity, orthomodularity, and modularity which are ambiguous in the classical logic should be reserved for its models, i.e., lattices where they have unequivocal equational meaning given by Eq. (16), Eq. (15), and $(a\cup b)\cap(a\cup c)=(a\cup(b\cap(a\cup c)))$, respectively. (Notice that the weakly modular lattice is also a model of $CL$.)

Response V.22. to Claim V.21. Here, again, as Kowalski, Paoli and Giuntini do, Drozdowska finds it hard to accept the mathematical proof of the fact that WOML, and therefore also O6 as its special instance, is a legitimate model of CL. In particular, she hints that the fact that existence of such a model is not normal, meaning that Eq. (14) does not hold, might be a “problem” (in a paragraph underneath this statement she writes: “Secondly, another “problem” is that . . . ” [2 p. 301]).

Response V.24. to Claim V.23. This claim is wrong. It is based on an obvious error: $a \to_{\mathrm c} a$ is not equal to $a$ but to $1$: $a \to_{\mathrm c} a = a' \cup a = 1$—see Figure 1. Actually, we need not calculate the example to arrive at the result at all. Every equation of the form $f(a,b,\ldots)=1$ passes O6. Only the equations of the form $f(a,b,\ldots)=g(a,b,\ldots)$ can fail in O6. Consequently, the claim “that O6 does not fully preserve the inferential structure of CL” has not been proved by Drozdowska.

We can sum up our responses to the claims made by Drozdowska as follows:

• Drozdowska’s claim that the existence of non-distributive matrix models does not by itself provide evidence that classical logic $CL$ is non-distributive and that it is distributive because the distributivity law (Eq. (16)) is its theorem is unfounded because $CL$ also has the orthomodularity law as well as the modular law as its theorems. Hence, classical logic $CL$ could likewise be called an orthomodular or modular logic. In any case $CL$ is neither non-distributive nor distributive—it is just classical logic.

• Drozdowska’s claim that the equivalence relation $\equiv_c$ does not have an acceptable interpretation in $O_6$ because formulas that are logically equivalent are not semantically equivalent is unfounded because all models map lattice formulas with a valuation “1” to provable theorems in the logic. That some models, like a distributive lattice DL, have equational formulas which other models do not have, does not affect this mapping.

• Drozdowska’s claim that $O_6$ does not fully preserve the inferential structure of $CL$ is unfounded because it is based on an error in her calculation as explained in the Response of Claim V.23 and which boils down to the claimed failure of a formula which is equal to “1” in $O_6$. However, all formulas that are equal to “1” in any of the models pass $O_6$. Only those that are not, might fail in $O_6$, like, e.g., the lattice distributivity given by Eq. (16).

Conclusion

The goal of this paper and of revisiting of our previous results in it is to show that the criticism of the results according to which both quantum and classical logics have non-equivalent and non-isomorphic models [12 Chapters 8–11], [6, 7, 9], attempted by Kowalski, Paoli and Giuntini [5] and Drozdowska [2], are unfounded and/or erroneous.

In order to reach the goal, we start with a renewed side-by-side presentation of logics in Sec. II and non-equivalent and non-isomorphic lattices in Sec. III. Then, in Sec. IV we show that those non-equivalent and non-isomorphic lattices are models of the logics and prove soundness and completeness for all of them which we summarise in Figure 2, again in a side-by-side fashion. Notice that Secs. II-IV material are from Refs. [6, 7, 9], just presented in a renewed manner so as to make the main result that the same logics might have non-equivalent and non-isomorphic models more accessible and digestible to the reader and less likely to be misunderstood in the future, again.

We consider, analyse, and respond to the criticisms in Sec. V made by Kowalski, Paoli and Giuntini [5] and by Drozdowska [2]. The former one deals mostly with quantum logic and the latter mostly with classical logic.

Kowalski, Paoli and Giuntini try to legitimate the orthomodular lattice as the only semantics/model of quantum logic with the following argument: “whether or not [quantum logic] adequately represents a proper formalisation of the intrinsic logic of quantum mechanics, its algebraic counterpart can’t be anything else than the variety of orthomodular lattices.” [5 p. 188] Hence, they overlook the fact that all models be they mutually non-equivalent and non-isomorphic or not induce the same sets of theorems, axioms, and rules of inferences which represent quantum logic and that therefore the can’t-be-anything-else argument is unfounded. The fact that only one of the models underlies the structure of quantum mechanics has nothing to do with the existence of other models. On the other hand, their claim that quantum logic itself is orthomodular is unfounded since the lattice orthomodularity equation cannot be expressed within the logical formalism.

Drozdowska’s criticism boils down to three claims. The first one is an attempt, similar to the one of Kowalski, Paoli and Giuntini, to establish the distributivity trait via the non-equational distributivity law. However, this claim is unfounded since the classical logic also has the non-equational forms of the orthomodularity law as well as the modular law as its theorems.

The second claim that it is a “problem” that the formulas that are logically equivalent are not semantically equivalent is unfounded because all models map lattice formulas with a valuation “1” to provable theorems in the logic. The fact that the distributive lattice (Boolean algebra) has equational formulas which other models do not have, does not affect the “1” mapping of formulas of non-equivalent models to the logic. They all map to to the same set of logic theorems.

Third Drozdowska’s claim, i.e., her “proof” that O6 does not fully preserve the inferential structure of classical logic is simply wrong. The claim stemmed from an error she made in a calculation as explained in the Response to Claim V.23 in Sec. V. Actually, as it refers to a violation of a lattice formula equal to “1” in O6, Drozdowska should not have calculated anything at all since all such equations pass O6.

Both criticisms give an impression of being written in an indecisive manner as if the authors just were not sure how to proceed with moulding their dissatisfaction with and analogous results from the literature. So, we are drawn to the conclusion that it might be more productive to commit ourselves to solving open problems in the field of orthomodular lattices, instead. Fifteen of them are, for instance, given in [8 p. 789,90].

References