# A Pragmatic Interpretation of Quantum Logic

###### Abstract

Scholars have wondered for a long time whether quantum mechanics (QM) subtends a quantum concept of truth which originates quantum logic (QL) and is radically different from the classical (Tarskian) concept of truth. We show in this paper that QL can be interpreted as a pragmatic language of pragmatically decidable assertive formulas, which formalize statements about physical systems that are empirically justified or unjustified in the framework of QM. According to this interpretation, QL formalizes properties of the metalinguistic concept of empirical justification within QM rather than properties of a quantum concept of truth. This conclusion agrees with a general integrationist perspective, according to which nonstandard logics can be interpreted as theories of metalinguistic concepts different from truth, avoiding competition with classical notions and preserving the globality of logic. By the way, some elucidations of the standard concept of quantum truth are also obtained.

Key words: pragmatics, quantum logic, quantum mechanics, justifiability, decidability, global pluralism.

## 1 Introduction

The formal structure called quantum logic (QL) springs out in a natural way from the formalism of quantum mechanics (QM). Scholars have debated for a long time on it, wondering whether it subtends a concept of quantum truth which is typical of QM, and a huge literature exists on this topic. We limit ourselves here to quote the classical book by Jammer, which provides a general review of QL from its birth to the early seventies, and the recent books by Rèdei and Dalla Chiara et al., which contain updated bibliographies.

Whenever the existence of a quantum concept of truth is accepted, one sees at once that it has to be radically different from the classical (Tarskian) concept, since the set of propositions of QL has an algebraic structure which is different from the structure of classical propositional logic. Thus, a new problem arises, i.e. the problem of the “correct” logic to be adopted when reasoning in QM.

We want to show in the present paper that the above problem can be avoided by adopting an integrated perspective, which preserves both the globality of logic (in the sense of global pluralism, which admits the existence of a plurality of mutually compatible logical systems, but not of systems which are mutually incompatible) and the classical notion of truth as correspondence, which we consider as explicated rigorously by Tarski’s semantic theory. This perspective reconciliates non-Tarskian theories of truth with Tarski’s theory by reinterpreting them as theories of metalinguistic concepts that are different from truth, and can be fruitfully applied to QL. Indeed, we prove in this paper that QL can be interpreted as a theory of the concept of empirical justification within QM.

In order to grasp intuitively our results, let us anticipate briefly some remarks that will be discussed more extensively in Sec. 2.

First of all, it must be noted that QM usually avoids making statements about properties of individual samples of a physical system (physical objects). Rather, it is concerned with probabilities of results of measurements on physical objects (standard interpretation, as espounded in any manual of QM; see, e.g., Refs. 7, 8 and 9), or with statistical predictions about ensembles of identically prepared physical objects (statistical interpretation; see, e.g., Refs. 1, 10 and 11). Yet, QM also distinguishes between properties that are real (or actual) and properties that are not real (or potential) in a given state of the physical system that is considered (briefly, the property is actual in whenever a test of on any physical object in would show that is possessed by without changing ). This amounts to introduce implicitly a concept of truth that also applies to statements about individuals. Indeed, asserting that a property is actual in the state is equivalent to asserting that the statement that attributes to a physical object is true for every in the state . Moreover, according to QM, is true, for a given in the state , if and only if (briefly, iff) is actual in the state . Falsity is then defined by considering a complementary property of , so that is false for a given in the state iff is actual in . It follows in particular that is true (false) for a given in the state iff it is true (false) for every in , or, equivalently, iff it is certainly true (certainly false) in . This result explains the notion of true as certain introduced in some well known approaches to QM. More important, it shows that the notion of truth has very peculiar features in QM. Indeed, the truth and falsity of a statement about an individual are equivalent to the truth of two universally quantified statements. Both these statements may be false. In this case has no truth value, hence it is meaningless. The existence of meaningless statements implies, in particular, that no Tarskian set-theoretical semantics can be introduced in QM.

The quantum notion of truth and meaning pointed out above is typical of the standard interpretation of QM, and it is inspired by a verificationist position which identifies truth and verifiability, meaning and verifiability conditions. These identifications are rather doubtful from an epistemological viewpoint, yet it is commonly maintained in the literature that the standard quantum conception of truth has no alternatives, since it seems firmly rooted in the formalism of QM itself. The mathematical apparatus of QM would imply indeed the impossibility of defining an assignment function associating a truth value with every individual statement of the form by referring only to the property and the state of . The outcomes obtained in a concrete experiment whenever or are not actual in would depend on the set of observations that are carried out simultaneously, not only on (contextuality).

Notwithstanding the arguments supporting it, the standard viewpoint can be criticized, and an alternative SR interpretation of QM can be constructed based on an epistemological position (semantic realism, or, briefly SR) which allows one to define a truth value for every statement of the form according to a Tarskian set-theoretical model. Of course, all statements that are certainly true (equivalently, true) or certainly false (equivalently, false) according to the standard interpretation with its quantum concept of truth, are also certainly true or certainly false, respectively, according to the SR interpretation with its Tarskian concept of truth. The remaining statements are meaningless according to the former interpretation, while they have truth values according to the latter: these values, however, may change when different objects in the same state are considered, and cannot be predicted in QM (which is, in this sense, an incomplete theory).

Because of its intuitive, philosophical and technical advantages, we adopt the SR interpretation in the present paper. It is then important to observe that our definitions and reasonings take into account only statements that are certainly true (certainly false) in the sense explained above, hence they actually do not depend on the choice of the interpretation of QM (standard or SR). Thus, our reinterpretation of QL should be acceptable also for logicians and physicists who do not agree with our epistemological position. Of course, if the SR interpretation is not accepted one loses all philosophical advantages of the integrated perspective mentioned at the beginning of this section.

Let us come now to empirical justification. Whenever a statement is certainly true (certainly false), its truth (falsity) can be predicted within QM if the property and the state of are known, and can be checked (by means of nontrivial physical procedures, see Sec. 2.6). Hence, we can say that the assertion of () is empirically justified, since we can both deduce the truth of () inside QM and provide an empirical proof of it. More formally, one can introduce an assertion sign and say that is certainly true (certainly false) iff () is empirically justified. In this way a semantic notion (certainty of truth) is translated into a pragmatic notion (empirical justification). Now, we remind that a pragmatic extension of a classical language and some general properties of the concept of justification have been studied by Dalla Pozza and by the author and note that all results obtained in this research apply to the notion of empirical justification introduced above. Moreover, further results can be obtained which are typical of the case under consideration, since the notion of justification is now specified (empirical justification in QM). Thus, a pragmatic language can be constructed (Sec. 3) in which assertions of the form are taken as elementary assertive formulas (afs) and pragmatic connectives are introduced, for which a set-theoretical pragmatics is defined basing on the concept of empirical justification in QM. This pragmatics defines a justification value for every elementary or complex af of , yet not all complex afs of are pragmatically decidable, that is, such that an empirical procedure of justification exists (it obviously exists for all elementary afs of because of our arguments above). However, one can single out a subset of pragmatically decidable afs of and consider a sublanguage of which contains only afs in this subset. It is then easy to see that our set-theoretical pragmatics, when restricted to , endows it with the structure of QL.

The above result is highly interesting in our opinion. Indeed, it provides the desired reinterpretation of QL as a theory of the metalinguistic concept of empirical justification in QM, allowing us to place it within an integrationist perspective that avoids any conflict with classical logic (we stress again that this conclusion can be accepted also by scholars who want to maintain the standard interpretation of QM).

We conclude this Introduction by observing that our results suggest that the standard partition of properties in two subsets (actual properties and potential properties) should be substituted by a partition in three subsets, as follows.

Actual properties. A property is actual in the state iff the assertion , with in , is justified.

Nonactual properties. A property is nonactual in the state iff the assertion , with in , is justified.

Potential properties. A property is potential in the state iff both assertions and , with in , are unjustified.

## 2 Physical preliminaries

We introduce in this section a number of symbols, definitions and physical concepts that will be extensively used in Sec. 3 in order to supply an intuitive support and an intended interpretation for the pragmatic language that will be introduced there.

### 2.1 Basic notions and mathematical representations

The following notions will be taken as primitive.

Physical system .

Pure state of , and set of all pure states of (the word pure will be usually implied in the following).

Testable property of , and set of all testable properties of (the word testable will be usually implied in the following).^{1}^{1}1It must be noted that the physical properties considered here are first
order properties from a logical viewpoint. Higher order properties
obviously occur in physics and will be encountered later on (Sec. 2.6), but
they need not be considered here.

States and properties will be interpreted operationally as follows.

A state is a class of physically equivalent^{2}^{2}2The notion of physical equivalence for preparing or registering devices is
not trivial. We do not discuss it here for the sake of brevity.
preparing devices (briefly, preparations) which may prepare
individual samples of (physical objects). A physical
object is in the state iff it is prepared by a preparation .

A property is a class of physically equivalent ideal
dichotomic (outcomes 1, 0) registering devices (briefly, registrations) which may test physical objects.^{3}^{3}3Note that a registration may act as a new preparation of the physical object
, so that the state of may change after a test on it.

The above notions do not distinguish between classical and quantum mechanics. The mathematical representation of physical systems, states and properties are different, however, in the two theories. Let us resume these representations in the case of QM.

Every physical system is associated with a Hilbert space over the field of complex numbers (we use the Dirac notation in order to denote vectors of in the following).

Let us denote by the partially ordered set (briefly, poset) of all closed subspaces of (here denotes set-theoretical inclusion), and let be the set of all one-dimensional subspaces of . Then (in absence of superselection rules) a mapping

exists which maps bijectively the set of all states of onto ,^{4}^{4}4It follows easily that every state S can also be represented by any vector , which is the standard
representation adopted in elementary QM. Moreover, a state S is usually
represented by an (orthogonal) projection operator on in more
advanced QM. However, the representation introduced here is more
suitable for our purposes in the present paper. and a mapping

exists which maps bijectively the set of all properties of onto .^{5}^{5}5Equivalently, a property is often represented in QM as a pair ,
where is a self-adjoint operator on representing a
physical observable, and a Borel set on the real line. We
do not use this representation, however, in the present paper.

### 2.2 Physical Quantum Logic

The poset is characterized by a set of mathematical properties. In particular, it is a complete, orthocomplemented, weakly modular, atomic lattice which satisfies the covering law. We denote by , and orthocomplementation, meet and join, respectively, in , and remind that coincides with the set-theoretical intersection of subspaces of , while does not generally coincide with the set-theoretical complementation , nor coincides with the set-theoretical union . Furthermore, we denote the minimal element and the maximal element of by and , respectively. Finally, we note that obviously coincides with the set of all atoms of .

Let us denote by the order induced on , via the bijective representation , by the order defined on . Then, the poset is order-isomorphic to , hence it is characterized by the same mathematical properties characterizing . In particular, the unary operation induced on it, via , by the orthocomplementation defined on , is an orthocomplementation, and is an orthomodular (i.e., orthocomplemented and weakly modular) lattice, usually called the lattice of properties of . By abuse of language, we denote the lattice operations on by the same symbols used above in order to denote the corresponding lattice operations on .

Orthomodular lattices are said to characterize semantically orthomodular QLs in the literature. The lattice of properties has a less general structure in QM, since it inherits a number of further properties from . Therefore, will be called physical QL in this paper.

A further lattice, isomorphic to , will be used in the following. In order to introduce it, let us consider the mapping

,

where is the range
of , which generally is a proper subset of the power set of . The poset is
order-isomorphic to , hence to , since and are bijective, so that is
bijective and order-preserving. Therefore is
characterized by the same mathematical properties characterizing . In particular, the unary operation induced on it, via ,
by the orthocomplementation defined on , is an
orthocomplementation, and is an orthomodular
lattice. We denote orthocomplementation, meet and join on by the same symbols , , and ,
respectively, that we have used in order to denote the corresponding
operations on and , and
call the lattice of closed subsets of (the word closed refers here to the fact that, for
every , ). We also note that the operation
coincides with the set-theoretical intersection on
because of the analogous result holding in .^{6}^{6}6Whenever the dimension of is finite, the lattice and/or the lattice can be
identified with Birkhoff and von Neumann’s lattice of experimental
propositions, which was introduced in the 1936 paper that started the
research on QL. This identification is impossible, however, if is not finite-dimensional, since Birkhoff and von Neumann’s
lattice is modular, not simply weakly modular. The requirement of modularity
has deep roots in the von Neumann concept of probability in QM according to
some authors.

To close up, let us observe that the unary operation defined on can be extended to by setting, for every ,

min

(the symbol min obviously refers to the order defined on ). This extension will be needed indeed in Sec. 3.2.

### 2.3 Actual and potential properties

We say that a property is actual (nonactual) in the
state iff one can perform a test of on any physical object in
the state by means of a registration , obtaining outcome 1 (0)
without modifying .^{7}^{7}7One can provide an intuitive support to this definition by noticing that the
result obtained in a test of on a physical object in the state
can be attributed to only whenever is not modified by the test.
Moreover, only in this case the test is repeatable, i.e., it can be
performed again obtaining the same result.
It is well known that classical physics assumes that tests which do not
modify the state are always possible, at least as ideal limits of
concrete procedures, while this assumption does not hold in QM.

Basing on the above definition, for every state three subsets of can be introduced.

: the set of all properties that are actual in .

: the set of all properties that are nonactual in .

: the set (called the set of all properties that are indeterminate, or potential, in ).

By using the mathematical apparatus of QM, the sets and can be characterized as follows.

.

.

It can also be proved that () coincides with the set of all properties that have probability 1 (0), according to QM, for every in the state , and that the set (which is non-void in QM, while it would be void in classical physics) coincides with the set of all properties that have probability different from 0 and 1 for every in the state .

Further characterizations of the above sets can be obtained as follows.

Since the mapping is bijective, while every singleton , with , obviously is an atom of , one can associate a property (equivalently, ) with every . This property is an atom of , and is usually called the support of . The mapping thus induces a one-to-one correspondence between (pure) states and atoms of . Then, one can prove the following equalities.

.

.

and .

Finally, the following equality also follows from the above definitions.

### 2.4 Truth in standard QM

No mention has been done of truth values (true/false) in the foregoing sections. However, we will be concerned with logical structures in Sec. 3, hence it is natural to wonder what QM says about the truth of a sentence as “the physical object has the property ” (briefly, in the following).

We have already noted in the Introduction that QM usually avoids making explicit statements regarding individual samples of physical systems. Yet, a sentence as “the property is actual in the state ” (Sec. 2.3) intuitively means that all physical objects in the state have the property . Hence, it can be translated, in terms of truth, into the sentence “for every physical object in the state , is true”. This translation shows that QM is concerned also with truth values of individual statements. Moreover, by considering the literature on the subject, one can argue that QM more or less implicitly adopts the following verificationist criterion of truth.

EV (empirical verificationism). The sentence has truth value true (false) for a physical object in the state iff is actual (nonactual) in , while it is meaningless otherwise.

Criterion EV is obviously at odds with standard definitions in classical logic (CL), and is suggested by the fact that can be attributed (not attributed) to a physical object in the state S on the basis of an experimental procedure only when it is actual (nonactual) for (see Sec. 2.6). Hence, we say that is Q-true (Q-false) whenever its truth value is true (false) according to criterion EV, in order to stress the difference between the truth values introduced in QM and those introduced in CL.

Because of the foregoing translation, criterion EV implies the following proposition.

TF. The sentence is Q-true (Q-false) for a physical object x in the state iff it is Q-true (Q-false) for every physical object x in the state .

Loosely speaking, proposition TF can be rephrased by saying that is true (false) in the sense established by criterion EV iff it is certainly true (certainly false) in the same sense, which explains the intuitive terminology that we have adopted in the Introduction.

Furthermore, criterion EV implies that has a truth value in standard QM iff (of course, is Q-true iff , Q-false iff ). It is then important to observe that the characterizations of and provided in Sec. 2.3 show that, for every , one can deduce from theoretical laws of QM whether a property belongs to . In particular, belongs to () iff it has probability 1 (0) for every in the state . Hence, one can predict, for every and in the state , whether is Q-true, Q-false or meaningless. This result shows that standard QM is a semantically complete theory and, together with proposition TF, explains the definition of true as certain, or predictable, which occurs in some approaches to QM.

### 2.5 Nonobjectivity versus objectivity in QM

The position expounded in Sec. 2.4 about the truth value of sentences of the form , with , is sometimes summarized by saying, briefly, that physical properties are nonobjective in standard QM (to be precise, only the properties in should be classified as nonobjective for a given ).

Nonobjectivity of properties is supported by a number of arguments. Some of them are based on empirical results (e.g., the two-slits experiment), some follow from seemingly reasonable epistemological choices (e.g., the adoption of a verificationist position, together with the indeterminacy principle) and some take the form of theorems deduced from the mathematical apparatus of QM. These last arguments are usually considered conclusive in the literature. We remind here the Bell-Kochen-Specker and Bell’s theorems which seem to prove that it is impossible to assign classical truth values to all sentences of the form , with , without contradicting the predictions of QM.

However, all arguments which show that nonobjectivity of properties is an unavoidable feature of QM can be criticized (this of course does not make the claim of nonobjectivity wrong, but only proves that there are alternatives to it). In particular, one can observe that a no-go theorem as Bell-Kochen-Specker’s is certainly correct from a mathematical viewpoint, but rests on implicit assumptions that are problematic from a physical and epistemological viewpoint. Basing on this criticism, an alternative interpretation (semantic realism, or SR, interpretation) has been propounded by the author, together with other authors. As we have already observed in the Introduction, the SR interpretation adopts a Tarskian theory of truth as correspondence, and all properties are objective according to it (equivalently, the sentence has a truth value defined in a classical way for every physical object and property ). According to this interpretation is certainly true (certainly false) in the state , that is, it is true (false) in a classical sense for every is in the state , iff (), hence iff it is Q-true (Q-false) according to the standard interpretation.

The SR interpretation of QM has some definite advantages. Firstly, it makes QM compatible with a realistic perspective without requiring any change of its mathematical apparatus and preserving all statistical predictions following from the standard interpretation, hence it provides a solution of the quantum measurement problem. Secondly, it rests on a classical conception of truth and meaning. Thirdly, it leads one to consider QM as an incomplete theory, and provides some suggestions about the way in which a more general theory embodying QM could be constructed.

Also within the SR interpretation one can deduce from theoretical laws of QM whether (), for a given . Moreover, for every , the sentence obviously is certainly true, hence true (certainly false, hence false) iff (). On the contrary, no prediction of the truth value of can be done if . Thus, the difference between the standard and the SR interpretation reduces to the fact that, whenever , is meaningless within the former, while it has a truth value that cannot be predicted by QM within the latter.

### 2.6 Empirical proof in QM

The results at the end of Secs. 2.4 and 2.5 show that, whenever is in the state , the truth value of the sentence can be predicted (or theoretically proved) iff , both in the standard and in the SR interpretation. One is thus led to wonder whether and when the truth value of can be determined empirically. At first glance, it seems sufficient to test by means of a registering device belonging to (Sec. 2.1). This is untrue according to the standard as well as the SR interpretation. Indeed, both interpretations maintain that a single test modifies, whenever , the state of the physical object , so that its result refers to the final state after the test, which is different from (moreover, within the standard interpretation, has no truth value whenever ). Thus, a test of is physically meaningful iff , since only in this case it does not modify the state . It follows that an empirical proof of the truth value of can be given iff a theoretical proof of this value exists, and it consists in checking whether or . Then, the characterizations of and in Sec. 2.3 suggest the empirical procedures to be adopted. Indeed, they show that () iff (), or, equivalently, iff (). Hence, one can get an empirical proof that is Q-true (Q-false) within the standard interpretation, or equivalently, that is certainly true, hence true (certainly false, hence false) within the SR interpretation, by checking whether the state of belongs to the set (). The empirical procedure required by this check is rather complex, since it does not reduce to a test of on the physical object , but consists in testing a huge number of physical objects in the state by means of registrations belonging to , in order to show that all of them yield outcome 1 (0) (it has been proven elsewhere that this procedure actually tests a quantified statement, or a second order physical property).

We conclude by noticing that truth and empirical provability of truth coincide within the standard interpretation of QM, which expresses the verificationist position that characterizes this interpretation. On the contrary, within the SR interpretation of QM the concepts of truth and empirical provability of truth are different, in accordance with the well known distinction between truth and epistemic accessibility of truth in classical logic.

## 3 QL as a pragmatic language

We aim to show in this section that physical QL can be recovered as a pragmatic language in the sense established in Ref. 27. It is noteworthy that, by weakening slightly the assumptions introduced in Ref. 27, one could perform this task without choosing between the standard and the SR interpretation of QM (see footnotes 8 and 9). We adopt however the SR interpretation in this section, since we maintain that the verificationist attitude of the standard interpretation is epistemologically and philosophically doubtful, but we point out by means of footnotes the simple changes to be introduced in order to attain the same results within the standard interpretation.

### 3.1 The general pragmatic language

Let us summarize briefly the construction of the general pragmatic language introduced in Ref. 27.

The alphabet of contains as descriptive signs the propositional letters , , ,…; as logical-semantic signs the connectives , , , and ; as logical-pragmatic signs the assertion sign and the connectives , , , and ; as auxiliary signs the round brackets . The set of all radical formulas (rfs) of is made up by all formulas constructed by means of descriptive and logical-semantic signs, following the standard recursive rules of classical propositional logic (a rf consisting of a propositional letter only is then called atomic). The set of all assertive formulas (afs) of is made up by all rfs preceded by the assertive sign (elementary afs), plus all formulas constructed by using elementary afs and following standard recursive rules in which , , , and take the place of , , , and , respectively.

A semantic interpretation of is then defined as a pair , where is an assignment function which maps onto the set of truth values (1 standing for true and 0 for false), following the standard truth rules of classical propositional calculus.

Whenever a semantic interpretation is given, a pragmatic interpretation of is defined as a pair , where is a pragmatic evaluation function which maps onto the set of justification values following justification rules which refer to and are based on the informal properties of the metalinguistic concept of proof in natural languages. In particular, the following justification rules hold.

JR. Let ; then, iff a proof exists that is true, i.e., that (hence, iff no proof exists that is true).

JR. Let ; then, iff a proof exists that is unjustified, i.e., that .

JR. Let , ; then,

(i) iff and ,

(ii) iff or ,

(iii) iff a proof exists that whenever ,

(iv) iff and .

Furthermore, the following correctness criterion holds in .

CC. Let ; then, implies

Finally, the set of all pragmatic evaluation functions that can be associated with a given semantic interpretation is denoted by .

### 3.2 The quantum pragmatic language

The quantum pragmatic language that we want to introduce here is obtained by specializing syntax, semantics and pragmatics of . Let us begin with the syntax. We introduce the following assumptions on .

A. The propositional letters , , … are substituted by the symbols , , …, with , , … .

A. The set of all rfs of reduces to the set of all atomic rfs of (in different words, if is a rf of , then , with ).

A. Only the logical-pragmatic signs , , and appear in the afs of .

The substitution in A aims to suggest the intended interpretation that we adopt in the following. To be precise, the rfs , , … are interpreted as sentences stating that the physical object has the properties , , …, respectively (Sec. 2.4).

The restriction in A aims to select rfs that are interpreted as testable sentences, i.e., sentences stating testable physical properties (Sec. 2.1), so that physical procedures exist for testing their truth values (which may not occur in the case of a rf of the form, say, ; note that a similar restriction has been introduced in Ref. 27 when recovering intuitionistic propositional logic within ).

The restriction in A is introduced for the sake of simplicity, since only the pragmatic connectives , and are relevant for our goals in this paper.

Because of A, A and A, the set of afs of is made up by all formulas constructed by means of the following recursive rules.

(i) Let be a rf. Then is an af.

(ii) Let be an af. Then, is an af.

(iii) Let and be afs. Then, and are afs.

Let us come now to the semantics of . We introduce the following assumption on .

A. Every assignment function defined on is induced by an interpretation of the variable x that appears in the rfs into a universe of physical objects, hence and the values of on are consistent with (not necessarily determined by) the laws of QM within the intended interpretation established above.

Let us comment briefly on assumption A.

Firstly, note that the interpretation was understood in Sec. 2.1, when we introduced the informal expression “the physical object is in the state ”.

Secondly, observe that the requirement that be consistent with the laws of QM (briefly, QM-consistent) obviously follows from the fact that these laws, via intended interpretation, establish relations among the truth values of elementary rfs of whenever a specific physical object is considered. We denote by in the following the set of all QM-consistent assigment functions.

Thirdly, note that, since , there may be many interpretations of the variable x that lead to the same assigment function.

Finally, observe that the universe can be partitioned into
(disjoint) subsets of physical objects, each of which consists of physical
objects in the same state (different subsets corresponding to different
states). Thus, specifying the state of means requiring that the
interpretation of that is considered maps on a physical
object in the subset corresponding to the state , hence it singles out a
subclass of assigment functions. All functions
in assign truth value 1 (0) to a sentence whenever (), while
the truth values assigned by different functions in to
may differ if .^{8}^{8}8Assumption A can be stated unchanged whenever the standard
interpretation of QM is adopted instead of the SR interpretation. In this
case, however, for every , is defined only on a subset
of rfs, not on the whole (which requires a weakening of the
assumptions on if one wants to recover this case within the
general perspective in Sec. 3.1). Furthermore, reduces to a
singleton. Indeed, for every interpretation , a state
exists such that . Then, is defined on a rf iff (Sec. 2.4), and
does not change if is substituted by an interpretation such that .

Let us come now to the pragmatics of . We introduce the following assumption on .

A. Let a mapping be given which interpretes the variable in the rfs of on a physical object in the state . A proof that the rf is true (false) consists in performing one of the empirical procedures mentioned in Sec. 2.6 and showing that ().

Assumption A is obviously suggested by the intended interpretation discussed above. Taking into account A and JR in Sec. 3.1, it implies the following statement.

P. Let be a rf of , let be an interpretation of the variable on a physical object in the state , and let be defined as in Sec. 2.2. Then,

iff ,

iff .

The above result specifies on the set of all elementary afs of and shows that it depends only on the state . Hence, we write in place of in the following (for the sake of brevity, we also agree to use the intuitive statement “the physical object is in the state ” introduced in Sec. 2.1 in place of the more rigorous statement “the variable is interpreted on a physical object in the state ”).

Statement P provides the starting point for introducing a set-theoretical pragmatics for , as follows.

Firstly, we introduce a mapping

which associates a pragmatic extension with every assertive formula , defined by the following recursive rules.

(i) For every , .

(ii) For every , .

(iii) For every , , .

(iv) For every , , .

Secondly, we rewrite statement P above substituting to in it.

P. Let be an elementary af of and let be in the state . Then,

iff ,

iff .

Thirdly, we note that statement P defines the pragmatic evaluation function on all elementary afs of .

Finally, for every , we extend from the set of all elementary afs of to the set of all afs of bearing in mind JR and JR in Sec. 3.1, hence introducing the following recursive rules.

(i) For every , iff .

(ii) For every , , iff .

(iii) For every , , iff .

The above procedure defines, for every , a pragmatic evaluation function

which provides a set-theoretical pragmatics for , as stated.

### 3.3 On the notion of justification in

The notion of justification introduced in Sec. 3.2 is basic in our approach and must be clearly understood. So we devote this section to comments on it.

Whenever an elementary af of is considered, the notion of justification obviously coincides with the notion of existence of an empirical proof of the truth of because of assumption A and proposition P in Sec. 3.2, which fits in with JR in Sec. 3.1.

Whenever molecular afs of are considered, one can grasp intuitively the meaning of the notion of justification for them by considering simple instances. Indeed, let be a rf and let be in the state . We get

iff ,

which means, shortly, that it is justified to assert that cannot be asserted iff MQ entails that the truth value of is false for every in the state . This result, of course, fits in with JR in Sec. 3.1.

Furthermore, let and be rfs, and let be in the state . We get

iff ,

iff .

The first equality shows that asserting and conjointly is justified iff both assertions are justified. The second equality shows that asserting or asserting is justified iff one of these assertions is justified. Both these results, of course, fit in with JR in Sec. 3.1.

We add that

implies

and

implies

since . Nevertheless,

and iff ,

which shows that a tertium non datur principle does not hold for the pragmatic connective in (it has already been proved in Ref. 27 that this principle does not hold in the general language ).

It is also interesting to note that the justification values of different elementary afs, say and , must be different for some state , since if (Sec. 2.2), hence .

Finally, we remind that the general theory of associates
an assignment function with a set of pragmatic
evaluation functions (Sec. 3.1), hence this also occurs within . One may then wonder whether is necessarily
nonvoid and, if this is the case, whether it may contain more than one
pragmatic evaluation function. In order to answer these questions, let us
consider an interpretation of the variable that maps on a
physical object in the state . Then, determines a unique
assignment function and a unique pragmatic evaluation
function associated with it, that we have denoted by , for it
depends only on the state . Since every assigment function in
is induced by an interpretation because of A in Sec. 3.2, this
proves that is necessarily nonvoid for every . Moreover, note that an interpretation of may
exist within the SR interpretation of QM that maps on a physical object
in the state , with , yet such that . The pragmatic evaluation functions and are then different, but they are both
associated with the assignment function , so that they both belong to . Hence, may contain many pragmatic evaluation functions.^{9}^{9}9Assumption A in Sec. 3.2 can be stated unchanged if the standard
interpretation of QM is adopted instead of the SR interpretation. In this
case, however, it is impossible that a mapping exists such
that , with and , since and are defined on different domains ( and , respectively). Hence, an assigment function is associated with a unique state , and reduces to the
singleton .

### 3.4 Pragmatic validity and order in

Coming back to the general language , we remind that a notion of pragmatic validity (invalidity) is introduced in it by means of the following definition.

Let . Then, is pragmatically valid, or p-valid (pragmatically invalid, or p-invalid) iff for every and , ().

By using the notions of justification in , one can translate the notion of p-validity (p-invalidity) within as follows.

Let . Then, is p-valid (p-invalid) iff, for every , ().

The notion of p-validity (p-invalidity) can then be characterized as follows.

Let . Then, is p-valid (p-invalid) iff ().

The set of all p-valid afs plays in a role similar to the role of tautologies in classical logic, and some afs in it can be selected as axioms if one tries to construct a p-correct and p-complete calculus for . We will not deal, however, with this topic in the present paper.

Furthermore, let us observe that a binary relation can be introduced in the general language by means of the following definition.

For every , , iff a proof exists that is justified whenever is justified (equivalently, iff is justified).

The set-theoretical pragmatics introduced in Sec. 3.2 allows one to translate the above definition in as follows.

For every , , iff for every , implies .

The binary relation can then be characterized as follows.

For every , , iff .

The relation is obviously a pre-order relation on , hence it induces canonically an equivalence relation on , defined as follows.

For every , , iff and .

The equivalence relation can then be characterized as follows.

For every , , iff .

### 3.5 Decidability versus justifiability in

We have commented rather extensively in Sec. 3.3 on the notion of justification formalized in , for every , by the pragmatic evaluation function . It must still be noted, however, that the definition of on all afs in does not grant that an empirical procedure of proof exists which allows one to establish, for every , the justification value of every af of . In order to understand how this may occur, note that the notion of empirical proof is defined by A for atomic rfs of and makes explicit reference, for every , to the closed subset associated with by the function introduced in Sec. 2.2. Basing on this notion, the justification value of an elementary af can be determined by means of the same empirical procedure, making reference to the closed subset associated to by the function (Sec. 3.2). Yet, whenever is recursively defined on the whole , new subsets of states are introduced (as ) which do not necessarily belong to . If an af is associated by with a subset that does not belong to , no empirical procedure exists in QM which allows one to determine the justification value .

We are thus led to introduce the subset of all pragmatically decidable, or p-decidable, afs of . An af of is p-decidable iff an empirical procedure of proof exists which allows one to establish whether is justified or unjustified, whatever the state of may be.

Because of the remark above, the subset of all p-decidable afs of can be characterized as follows.

.

Let us discuss some criteria for establishing whether a given af belongs to .

C. All elementary afs of belong to .

C. If , then

Indeed, implies .

C. If , , then

Indeed, and imply , since because of known properties of the lattice (Sec. 2.2).

C. If , , then may belong or not to . To be precise, it belongs to iff or

Indeed, or, equivalently, , iff one of the conditions in C is satisfied.

It is apparent from criteria C and C that is closed with respect to the pragmatic connectives and , in the sense that implies , and , implies . On the contrary, is not closed with respect to , since it may occur that even if , . In order to obtain a closed subset of afs of , one can consider the set

the pragmatic connective does not occur in .

The set obviously contains all elementary afs of , plus all afs of in which only the pragmatic connectives and occur. We can thus consider a sublanguage of whose set of afs reduces to . This new language is relevant since all its afs are p-decidable, hence we call it the p-decidable sublanguage of and denote it by .

### 3.6 The p-decidable sublanguage

As we have anticipated in the Introduction, we aim to show in this paper that the sublanguage has the structure of a physical QL, hence it provides a new pragmatic interpretation of this relevant physical structure. However, this interpretation will be more satisfactory from an intuitive viewpoint if we endow with some further derived pragmatic connectives which can be made to correspond with connectives of physical QL. To this end, we introduce the following definitions.

D. We call quantum pragmatic disjunction the connective defined as follows.

For every , , .

D. We call quantum pragmatic implication the connective defined as follows.

For every , , .

Let us discuss the justification rules which hold for afs in which the new connectives and occur.

By using the function introduced in Sec. 3.2 we get (since the set-theoretical operation coincides with the lattice operation in , see Sec. 2.2),

.

Hence, for every ,

iff .

Let us come to the quantum pragmatic implication . By using the definition of , one gets

.

By using the function and the above result about , one then gets

.

It follows that, for every ,

iff .

Let us observe now that obviously inherits the notions of p-validity and order defined in (Sec. 3.4). Hence, we can illustrate the role of the connective within by means of the following pragmatic deduction lemma.

PDL. Let , . Then, iff for every , (equivalently, iff is p-valid).

Proof. The following sequence of equivalences holds.

For every ,