Advances in Mathematical Physics

Volume 2016 (2016), Article ID 6830685, 12 pages

http://dx.doi.org/10.1155/2016/6830685

## Classical Logic and Quantum Logic with Multiple and Common Lattice Models

^{1}Department of Physics-Nanooptics, Faculty of Mathematics and Natural Sciences, Humboldt University of Berlin, Berlin, Germany^{2}Center of Excellence for Advanced Materials and Sensing Devices (CEMS), Photonics and Quantum Optics Unit, Ruđer Bošković Institute, Zagreb, Croatia

Received 4 May 2016; Revised 7 July 2016; Accepted 25 July 2016

Academic Editor: Giorgio Kaniadakis

Copyright © 2016 Mladen Pavičić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider a proper propositional quantum logic and show that it has multiple disjoint lattice models, only one of which is an orthomodular lattice (algebra) underlying Hilbert (quantum) space. We give an equivalent proof for the classical logic which turns out to have disjoint distributive and nondistributive ortholattices. In particular, we prove that both classical logic and quantum logic are sound and complete with respect to each of these lattices. We also show that there is one common nonorthomodular lattice that is a model of both quantum and classical logic. In technical terms, that enables us to run the same classical logic on both a digital (standard, two-subset, 0-1-bit) computer and a nondigital (say, a six-subset) computer (with appropriate chips and circuits). With quantum logic, the same six-element common lattice can serve us as a benchmark for an efficient evaluation of equations of bigger lattice models or theorems of the logic.

#### 1. Introduction: Is Logic Empirical?

In his seminal paper “Is Logic Empirical?” [1], Putnam argues that logic we make use of to handle the statements and propositions of the theories we employ to describe the world around us is uniquely determined by it. “*Logic is empirical*. It makes sense to speak of ‘physical logic.’ We live in a world with a nonclassical logic [of subspaces of the quantum Hilbert space which form an orthomodular (non-distributive, non-Boolean) lattice]. Certain statements—just the ones we encounter in daily life—do obey classical logic, but this is so because the corresponding subspaces of form a Boolean lattice” [1, Ch. V].

We see that Putnam, in effect, reduces the logic to lattices, while they should only be their models. “[We] just read the logic off from the Hilbert space ” [1, Ch. III]. This technical approach has often been adopted in both classical and quantum logic. In classical logic, it has been known as two-valued interpretation for more than a century. In quantum logic, it has been introduced by Birkhoff and von Neumann in 1936 [2] and it is still embraced by many authors [3]. Subsequently, varieties of relational logic formulations, which closely follow lattice ordering relations, have been developed, for example, by Dishkant [4], Goldblatt [5], Chiara [6], Nishimura [7, 8], Mittelstaedt [9], Stachow [10], and Pták and Pulmannová [11]. More recently, Engesser and Gabbay [12] made related usage of nonmonotonic consequence relation, Rawling and Selesnick [13] of binary sequent, Herbut [14] of state-dependent implication of lattice of projectors in the Hilbert space, Tylec and Kuś [15] of partially ordered set (poset) map, and Bikchentaev et al. [16] of poset binary relation.

Another version of Birkhoff-von-Neumann style of viewing propositions as projections in Hilbert space rather than closed subspaces and their lattices as in the original Birkhoff-von-Neumann paper has been introduced by Engesser et al. [17]. Recently, other versions of quantum logic have been developed, such as a dynamic quantum logic by Baltag and Smets [18, 19], exogenous quantum propositional logic by Mateus and Sernadas [20], a categorical quantum logic by Abramsky and Duncan [21, 22], and a projection orthoalgebraic approach to quantum logic by Harding [23].

However, we are interested in nonrelational kinds of logic which combine propositions according to a set of true formulas/axioms and rules imposed on them. The propositions correspond to statements from a theory, say classical or quantum mechanics, and are not directly linked to particular measurement values. Such kinds of logic employ models which evaluate a particular combination of propositions and tell us whether it is true or not. Evaluation means mapping from a set of logic propositions to an algebra, for example, a lattice, through which a correspondence with measurement values emerges, but indirectly. Therefore we shall consider a classical and a quantum logic defined as a set of axioms whose Lindenbaum-Tarski algebras of equivalence classes of expressions from appropriate lattices correspond to the models of the logic. Let us call such a logic an* axiomatic logic*. An axiomatic logic () is a language consisting of propositions and a set of conditions and rules imposed on them called axioms and rules of inference. We shall consider classical and quantum axiomatic logic.

We show that an axiomatic logic is wider than its relational logic variety in the sense of having many possible models and not only distributive ortholattice (Boolean algebra) for the classical logic and not only orthomodular lattice for the quantum logic. We shall make use of the PM classical logical system—Whitehead and Russell’s* Principia Mathematica* axiomatization in Hilbert and Ackermann’s presentation [24] in the schemata form and of Kalmbach’s axiomatic quantum logic [25, 26] (slightly modified by Pavičić and Megill [27, 28]—original Kalmbach axioms A1, A11, and A15 are dropped because they were proven redundant in [29]), as typical examples of axiomatic logic.

It is well-known that there are many interpretations of the classical logic, for example, two-valued, general Boolean algebra (distributive ortholattice) and set-valued ones [30, Ch. , ]. These different interpretations are tantamount to different models of the classical logic and in this paper and several previous papers of ours we show that they are enabled by different definitions of the relation of equivalence for its different Lindenbaum-Tarski algebras. One model of the classical logic is a distributive numerically valued, mostly two-valued, lattice, while the others are nondistributive nonorthomodular lattices, one of them being the so-called O6 lattice, which can also be given set-valuations [30, Ch. , ].

As for quantum logic, one of its models is an orthomodular lattice, while others are nonorthomodular lattices, one of them being again O6—the common model of both kinds of logic.

Within a logic we establish a unique deduction of all logic theorems from valid algebraic equations in a model and vice versa by proving the soundness and completeness of logic with respect to a chosen model. That means that we can infer the distributivity or orthomodularity in one model and disprove them in another by means of the same set of logical axioms and theorems. We can also consider O6 in which both the distributivity and orthomodularity fail; however, particular nondistributive and nonorthomodular conditions pass O6 only to map into the distributivity and orthomodularity through classical and quantum logic in other models of these kinds of logic.

We see that logic is at least not* uniquely* empirical since it can simultaneously describe distinct realities.

The paper is organised as follows. In Section 2 we define classical and quantum logic. In Section 3 we introduce distributive (ortho)lattices and orthomodular lattices as well as two nondistributive (one is O6) and four nonorthomodular ones (one is again O6), all of which are our models for classical and quantum logic, respectively. In Section 4, we prove soundness and completeness of classical and quantum logic with respect to the models introduced in Section 3. In Section 5, we discuss the obtained results.

#### 2. Kinds of Logic

In our axiomatic logic () the propositions are well-formed formulae (wffs), defined as follows.

We denote elementary, or primitive, propositions by ; we have the following primitive connectives: (negation) and (disjunction). is a wff for ; is a wff if is a wff; is a wff if and are wffs.

Operations are defined as follows.

*Definition 1 (conjunction). *One has

*Definition 2 (classical implication). *One has

*Definition 3 (Kalmbach’s implication). *One has

*Definition 4 (quantum equivalence). *One has

*Definition 5 (classical Boolean equivalence). *One has

Connectives bind from weakest to strongest in the order , , , , .

Let be the set of all propositions, that is, of all wffs. wffs containing and within logic are used to build an algebra . In , a set of axioms and rules of inference are imposed on . From a set of axioms by means of rules of inference, we get other expressions which we call theorems. Axioms themselves are also theorems. A special symbol is used to denote the set of theorems. Hence iff is a theorem. The statement is usually written as . We read this as follows: “ is provable,” meaning that if is a theorem, then there is a proof of it. We present the axiom systems of our propositional logic in the schemata form (so that we dispense with the rule of substitution).

*Definition 6. *For one says that is derivable from and writes or just if there is a finite sequence of formulae, the last of which is , and each of which is either one of the axioms of or is a member of or is obtained from its precursors with the help of a rule of inference of the logic.

##### 2.1. Classical Logic

In the classical logic , the sign will denote provability from the axioms and the rule of , but we shall omit the subscript when it is obvious from context as, for example, in the following axioms and the rule of inference that define .

*Axioms*

*Rule of Inference (Modus Ponens)*

We assume that the only legitimate way of inferring theorems in is by means of these axioms and the Modus Ponens rule. We make no assumption about valuations of the primitive propositions from which wffs are built but instead are interested in wffs that are valid, that is, true in all possible valuations of the underlying models. Soundness and completeness will show that those theorems that can be inferred from the axioms and the rule are exactly those that are valid.

##### 2.2. Quantum Logic

Quantum logic () is defined as a language consisting of propositions and connectives (operations) as introduced above and the following axioms and a rule of inference. We will use to denote provability from the axioms and the rule of and omit the subscript when it is obvious from the context, for example, in the list of axioms and the rule of inference that follow.

*Axioms*

*Rule of Inference (Modus Ponens)*

Soundness and completeness will show that those theorems that can be inferred from the axioms and the rule of inference are exactly those that are valid.

#### 3. Lattices

For the presentation of the main result it would be pointless and definitely unnecessarily complicated to work with the full-fledged models, that is, Hilbert space, and the new non-Hilbert models that would be equally complex. It would be equally too complicated to present complete quantum or classical logic of the second order with all the quantifiers. Instead, we shall deal with lattices and the propositional logic we introduced in Section 2. We start with a general lattice which contains all the other lattices we shall use later on. The lattice is called an* ortholattice* and we shall first briefly present how one arrives at it starting with Hilbert space.

A Hilbert lattice is a kind of orthomodular lattice which we define below. In any Hilbert lattice the operation* meet*, , corresponds to set intersection, , of subspaces of the Hilbert space ; the ordering relation corresponds to ; the operation* join*, , corresponds to the smallest closed subspace of containing ; and the* orthocomplement * corresponds to , the set of vectors orthogonal to all vectors in . Within the Hilbert space there is also an operation which has no parallel in the Hilbert lattice: the sum of two subspaces which is defined as the set of sums of vectors from and . We also have . One can define all the lattice operations on the Hilbert space itself following the above definitions (, etc.). Thus we have [33, p. ], where is the closure of , and therefore . When is finite dimensional or when the closed subspaces and are orthogonal to each other then [34, pp. ], [25, pp. , ], and [9, pp. ].

The projection associated with is given by for vector from that has a unique decomposition for from and from . The closed subspace belonging to is . Let denote a projection on , a projection on , and a projection on if , and let mean . Then corresponds to [9, p. ], to , to [9, p. ], and to . also corresponds to either or or . Two projectors commute iff their associated closed subspaces commute. This means that corresponds to . In the latter case we have and . ; that is, is characterised by [33, pp. ], [25, pp. , ], [9, pp. ], and [35, pp. ].

Closed subspaces as well as the corresponding projectors form an algebra called the Hilbert lattice which is an ortholattice. The conditions of the following definition can be easily read off from the properties of the aforementioned Hilbert subspaces or projectors.

*Definition 7. *An ortholattice, OL, is an algebra such that the following conditions are satisfied for any [36]:In addition, since for any , we define the greatest and the least element of the lattice:and the ordering relation () on the lattice:

*Definition 8 (Sasaki hook). *One has

*Definition 9 (quantum equivalence). *One has

*Definition 10 (classical equivalence). *One has

Connectives bind from weakest to strongest in the order , , , , and .

*Definition 11 (Pavičić, [37]). *An orthomodular lattice (OML) is an OL in which the following condition (orthomodularity) holds:

Every Hilbert space (finite and infinite) and every phase space is orthomodular.

*Definition 12 (Pavičić, [38]). * (The proof of the opposite claim in [37, Theorem ] is wrong.) A distributive ortholattice (DL) (also called a Boolean algebra) is an OL in which the following condition (distributivity) holds:

Every phase space is distributive and, of course, orthomodular since every distributive ortholattice is orthomodular.

The opposite directions of metaimplications in (27) and (28) hold in any OL.

*Definition 13 (Pavičić and Megill, [27]). *An OL in which either of the following conditions (weak orthomodularity) holdsis called a weakly orthomodular ortholattice, WOML.

*Definition 14 (Pavičić, this paper). *A WOML in which the following condition holdsis called a WOML1.

*Definition 15 (Pavičić, this paper). *A WOML1 in which the following condition holdsis called a WOML2.

*Definition 16 (Pavičić, this paper). *A WOML in which neither (27), (32), nor (31) hold is called a .

*Definition 17 (Pavičić and Megill, [27, 39]). *An OL in which the following condition (commensurability) holdsis called a weakly distributive ortholattice, WDL.

*Definition 18 (Pavičić and Megill, [27]). *A WOML in which the following condition (weak distributivity) holdsis called a weakly distributive ortholattice, WDL.

Definitions 17 and 18 are equivalent. We give both definitions here in order to, on the one hand, stress that a WDL is a lattice in which all variables are commensurable and, on the other, to show that in WDL the distributivity holds only in its weak form given by (34) which we will use later on.

*Definition 19 (Pavičić and Megill, [39]). *A WDL in which (28) does not hold is called a .

Any finite lattice can be represented by a Hasse diagram that consists of points (*vertices*) and lines (*edges*). Each point represents an element in a lattice, and positioning element above element and connecting them by a line means . For example, in Figure 1(a) we have . We also see that in this lattice, for example, does not have a relation with either or .