The Paraconsistent Logic of Quantum Superpositions
aa r X i v : . [ qu a n t - ph ] J un The Paraconsistent Logic ofQuantum Superpositions
N. da Costa and C. de Ronde
1. Universidade Federal de Santa Catarina - Brazil2. Instituto de Filosof´ıa ”Dr. A. Korn”Universidad de Buenos Aires, CONICET - ArgentinaCenter Leo Apostel and Foundations of the Exact SciencesBrussels Free University - Belgium
Abstract
Physical superpositions exist both in classical and in quantum physics.However, what is exactly meant by ‘superposition’ in each case isextremely different. In this paper we discuss some of the multipleinterpretations which exist in the literature regarding superpositionsin quantum mechanics. We argue that all these interpretations havesomething in common: they all attempt to avoid ‘contradiction’. Weargue in this paper, in favor of the importance of developing a newinterpretation of superpositions which takes into account contradic-tion, as a key element of the formal structure of the theory, “rightfrom the start”. In order to show the feasibility of our interpretationalproject we present an outline of a paraconsistent approach to quantumsuperpositions which attempts to account for the contradictory proper-ties present in general within quantum superpositions. This approachmust not be understood as a closed formal and conceptual scheme butrather as a first step towards a different type of understanding regard-ing quantum superpositions.
Keywords: quantum superposition, para-consistent logic, interpretation of quan-tum mechanics.PACS numbers: 02.10 De
There is an important link in the history of physics between the interpre-tation of theories and their formal development. Relativity would have not1een possible without non-euclidean geometry nor classical physics with-out infinitesimal calculus. In quantum mechanics, the formal scheme waselaborated —mainly by Schr¨odingier, Heisenberg, Born, Jordan and Dirac—almost in parallel to the orthodox interpretation. However, still today, theinterpretation of quantum mechanics remains controversial regarding most ofits non-classical characteristics: indeterminism, holism, contextuality, non-locality, etc. There is, still today, no consensus regarding the meaning ofsuch expressions of the theory. In this paper we shall be concerned with aspecific aspect of quantum mechanics, namely, the principle of superpositionwhich, as it is well known, gives rise to the so called quantum superposi-tions. Physical superpositions exist both in classical and in quantum physics.However, what is exactly meant by “superposition” in each case is extremelydifferent. In classical physics one can have superpositions of waves or fields.A wave (field) α can be added to a different wave (field) β and the sum willgive a ‘new’ wave (field) µ = α + β . There is in this case no weirdness forthe sum of multiple states gives as a result a new single state. In quantummechanics on the contrary, a linear combination of multiple states, α + β isnot reducible to one single state, and there is no obvious interpretation ofsuch superposition of states. As a matter of fact, today quantum superposi-tions play a central role within the most outstanding technical developmentssuch as quantum teleportation, quantum cryptography and quantum com-putation [21, 26]. The question we attempt to address in this paper regardsthe meaning and physical representation of quantum superpositions. Thereare many interpretations of quantum mechanics each of which provides ananswer to this question. In the following we shall review some of these pro-posals. We shall then argue in favor of the possibility to develop a newinterpretation which considers contradictory properties as a main aspect ofquantum superpositions and present an outline of a formal approach basedon paraconsistent logic which attempts to consider contradiction “right fromthe start”. We must remark that we do not understand this approach asa closed formal and conceptual scheme but rather as a first step towards adifferent type of understanding regarding quantum superpositions.Paraconsistent logics are the logics of inconsistent but nontrivial theories.The origins of paraconsistent logics go back to the first systematic studiesdealing with the possibility of rejecting the principle of noncontradiction.Paraconsistent logic was elaborated, independently, by Stanislaw Jaskowskiin Poland, and by the first author of this paper in Brazil, around the middleof the last century (on paraconsistent logic, see, for example: [5]). A theory T founded on the logic L , which contains a symbol for negation, is called2nconsistent if it has among its theorems a sentence A and its negation ¬ A ;otherwise, it is said to be consistent. T is called trivial if any sentence ofits language is also a theorem of T ; otherwise, T is said to be non-trivial.In classical logics and in most usual logics, a theory is inconsistent if, andonly if, it is trivial. L is paraconsistent when it can be the underlying logicof inconsistent but non trivial theories. Clearly classical logic and all usuallogics are not paraconsistent. The importance of paraconsistent logic is notlimited to the realm of pure logic but has been extended to many fieldsof application such as robot control, air traffic control [24], control systemsfor autonomous machines [25], defeasible deontic reasoning [23], informationsystems [1] and medicine.In the following, we attempt to call the attention to the importance ofextending the realm of paraconsistent logic to the formal account of quantumsuperpositions. Firstly, we shall discuss the very different meanings of theterm ‘superposition’ in both classical and quantum physics. In section 3, weshall present some of the very different interpretations of the meaning of aquantum superposition which can be found in the literature. In section 4,we shall argue in favor of the importance of considering an interpretationof superposition in terms of paraconsistent logic. In section 5, we present aformal scheme in terms of paraconsistent logic which attempts to account forthe inner contradictions present within a quantum superposition. Finally,in section 6, we argue in favor of considering contradiction “right from thestart”. In classical physics, every physical system may be described exclusively bymeans of its actual properties , taking ‘actuality’ as expressing the preexistent mode of being of the properties themselves, independently of observation —the ‘pre’ referring to its existence previous to measurement. Each systemhas a determined state characterized mathematically in terms of a point inphase space. The change of the system may be described by the change ofits actual properties. Potential or possible properties are considered as thepoints to which the system might arrive in a future instant of time. Theoccurrence of possibilities in such cases merely reflects our ignorance aboutwhat is actual. Contrary to what seems to happen in quantum mechanics,statistical states do not correspond to features of the actual system, butquantify our lack of knowledge of those actual features ([9], p. 125).3lassical mechanics tells us via the equation of motion how the state ofthe system moves along the curve determined by initial conditions in thephase space. The representation of the state of the physical system is givenby a point in phase space Γ and the physical magnitudes are representedby real functions over Γ. These functions commute between each other andcan be interpreted as possessing definite values independently of physicalobservation, i.e. each magnitude can be interpreted as being actually preex-istent to any possible measurement. In the orthodox formulation of quantummechanics, the representation of the state of a system is given by a ray inHilbert space H . But, contrary to the classical scheme, physical magnitudesare represented by operators on H that, in general, do not commute. Thismathematical fact has extremely problematic interpretational consequencesfor it is then difficult to affirm that these quantum magnitudes are simul-taneously preexistent . In order to restrict the discourse to different sets ofcommuting magnitudes, different Complete Sets of Commuting Operators(CSCO) have to be chosen. The choice of a particular representation (givenby a CSCO) determines the basis in which the observables diagonalize andin which the ray can be expressed. Thus, the ray can be written as differentlinear combinations of states: α i | ϕ B i > + α j | ϕ B j > = | ϕ B q > = β m | ϕ B m > + β n | ϕ B n > + β o | ϕ B o > (1)The linear combinations of states are also called quantum superpositions.As it was clearly expressed by Dirac [6]: “The nature of the relationshipswhich the superposition principle requires to exist between the states of anysystem is of a kind that cannot be explained in terms of familiar physicalconcepts. One cannot in the classical sense picture a system being partlyin each of two states and see the equivalence of this to the system beingcompletely in some other state.” The formal difference of using vectors in H instead of points in Γ seems to imply that in quantum mechanics —apartfrom the ‘possibility’ which is encountered in classical mechanics— there isanother, different realm which must be necessarily considered and refers,at each instant of time, to contradictory properties . To see this, considerthe following example: given a spin 1 / | ↑ z > , welet it interact with a magnetic field in the z direction. All outcomes thatcan become actual in the future are potential properties of the system, inan analogous manner as all possible reachable positions of a pendulum arein the classical case. But at each instant of time, for example at the initial4nstant, if we consider the z direction and the projection operator | ↑ z >< ↑ z | as representing a preexistent actual property, there are other incompatibleproperties arising from considering projection operators of spin projectionsin other directions. For example, in the x direction, the projection operators | ↑ x >< ↑ x | and | ↓ x >< ↓ x | do not commute with | ↑ z >< ↑ z | and thus,cannot be considered to possess definite values simultaneously. Since Borninterpretation of the wave function, these properties are usually consideredas possible . However, this possibility is essentially different from the ideaof possibility discussed in classical physics which relates to the idea of a process . If we consider that the formalism of quantum mechanics providesa description of the world, a representation of what there is —and does notmerely make reference to measurement outcomes—, at each instant of timethe properties, | ↑ z >< ↑ z | , | ↑ x >< ↑ x | and | ↓ x >< ↓ x | must be taken intoaccount independently of their future actualization for they all provide nontrivial information about the state of affairs. In particular, the properties | ↑ x >< ↑ x | and | ↓ x >< ↓ x | , which constitute the superposition and must beconsidered simultaneously are in general contradictory properties .In the quantum logic approach one of the properties, namely, the onein which we can write the state of affairs as a single term, is consideredas ‘actual’ while the others are taken to be ‘potential’ properties. Potentialproperties can become actual. These properties, e.g. | ↑ x >< ↑ x | , | ↓ x >< ↓ x | , | ↑ y >< ↑ y | and | ↓ y >< ↓ y | in our example, are always part of superpositionswith more than one term and are constituted by contradictory properties.However, from a mathematical perspective, independently of their mode ofexistence, both potential and actual properties are placed at the same levelin the algebraic frame which describes the state of affairs according to quan-tum mechanics: the projections of the spin in all directions are atoms of thelattice and there is no formal priority of the actual over the potential prop-erties. This rises the question if one can consider quantum superpositionsas preexistent entities, independently of their future actualization.In the laboratory, it is precisely this contradictory potential realm whichis necessary to be considered by the experimentalist in the developmentswhich are taking place today regarding the processing of quantum infor-mation as quantum computing and quantum communication [21, 26]. Thisseems to point in the direction that these properties have an existence whichcannot be reduced to their becoming actual at a future instant of time. Super-positions correspond to possible outcomes which occur on an equal footingin the superposition of the final state, so that there is no sign that any oneof them is more real than any other ([9], p. 120). Taking these problems5nto account there are many interpretations which attempt to provide ananswer to the question: what is a quantum superposition? We shall discussin the next section some of these proposals. As we have seen above, the formal description of quantum mechanics seemsto imply a deep departure from the classical notion of possible or proba-ble. This was cleverly exemplified by Erwin Schr¨odinger in his famous catexperiment [31], in which a half dead and half alive cat seemed to laughof the idea of possessing a determined state. However, one can find in theliterature, there are many different interpretations of quantum mechanicsin general and of the meaning of a quantum superposition in particular. Inthis section we shall review some of these very distinct interpretations. Wedo not attempt to provide a complete review of interpretations but ratherto analyze instead their specific understanding of quantum superpositions.Although we must take into account the fact that ’state vector’ and ’cat’are two concepts in different levels of discourse ([8], p. 189). From a re-alist perspective, which considers physics as providing a description or anexpression of the world, the question still arises, if this formal or mathemat-ical representation given by superpositions, namely equation 1 —which al-low us to calculate the probability of the possible measurement outcomes—,can be related conceptually to a notion which can allow us to think, in-dependently of measurement outcomes, about the ‘superposition of statesin Hilbert space’ in an analogous manner as we think of a ‘point in phasespace’ (in the formal level) as describing an ‘object in space-time’ (in theconceptual level). What is describing a mathematical superposition? Canwe create or find adequate concepts which can provide a representationalrealistic account of a quantum superposition independent of measurementoutcomes? Of course, from a general empiricist perspective one is not com-mitted to answering these set of questions. The idea that the quantum wavefunction as related to a superposition is just a theoretical device with no on-tological content goes back to Bohr’s interpretation of quantum mechanics.The impossibility to interpret the quantum wave function in an ontologicalfashion can be understood in relation to his characterization of Ψ in terms ofan algorithmic device which computes measurement results. This position According to Bohr ([36], p. 338) the Schr¨odinger wave equation is just an abstract what there is . Superpositions are thus, a theoretical device throughwhich one can consider the actual observation hic et nunc . Empiricism canbe linked to probability in terms of the frequency interpretation which rests,contrary to the original conception of probability, not on the idea that prob-ability describes in terms of ignorance an existent state of affairs, but ratherin a set of empirical results found in a series of measurements. However, andindependently of the problems encountered within such empiricist stances, ifsuperpositions are considered just as a theoretical device , then the questionof interpretation seems to loose its strength. For why should we pursue aninterpretation if, like Fuchs and Peres remark, quantum mechanics does thejob and already provides an algorithm for computing probabilities for themacroscopic events? There are other reasons which one could put forwardto account for the importance of interpretation even from an empiricist per-spective (see for example van Fraassen [33]), however these reasons mustremain only secondary in the quest of science.On the contrary, from a realist position, there is need to provide ananswer to the link between the theory and its conceptual understandingof the world. To put it in a nutshell: what is quantum mechanics tellingus about the world? As noticed by Bacciagaluppi ([2], p. 74), the hiddenvariable program attempts to “restore a classical way of thinking about whatthere is .” In this sense, Bohm’s proposal seems to restore the possibility ofdiscussing in terms of a state of affairs described in terms of a set of definitevalued properties. In Bohmian mechanics the state of a system is given bythe wave function Ψ together with the configuration of particles X . Thequantum wave function must be understood in analogy to a classical fieldthat moves the particles in accordance with the following functional relation: dxdt = ∇ S , where S = ~ δ ( δ being the phase of ψ ). Thus, particles alwayshave a well defined position together with the rest of their properties andthe evolution depends on the quantum field. It then follows that, there are method of calculus and it does not designate in itself any phenomena. See also [3] fordiscussion.
7o superpositions of states, the superposition is given only at the level ofthe field and remains as mysterious as the superposition of classical fields.The field does not only have a dynamical character but also determines theepistemic probability of the configuration of particles via the usual Bornrule.A different approach, which starts from a particular interpretation ofquantum superpositions is the so called many worlds interpretation (MW),considered to be a direct conclusion from Everett’s first proposal in termsof ‘relative states’ [12]. Everett’s idea was to let quantum mechanics find itsown interpretation, making justice to the symmetries inherent in the Hilbertspace formalism in a simple and convincing way [7]. MW interpretations areno-collapse interpretations which respect the orthodox formulation of quan-tum mechanics. The main idea behind many worlds interpretations is thatsuperpositions relate to collections of worlds, in each of which exactly onevalue of an observable, which corresponds to one of the terms in the super-position, is realized. Apart from being simple, the claim is that it possessesa natural fit to the formalism, respecting its symmetries. The solution pro-posed to the measurement problem is provided by assuming that each oneof the terms in the superposition is actual in its own correspondent world.Thus, it is not only the single value which we see in ‘our world’ which getsactualized but rather, that a branching of worlds takes place in every mea-surement, giving rise to a multiplicity of worlds with their correspondingactual values. The possible splits of the worlds are determined by the lawsof quantum mechanics but each world becomes again ‘classical’. Quantumsuperpositions are interpreted as expressing the existence of multiple worlds,each of which exists in (its own) actuality. However, there are no superpo-sitions in this, our actual world, for each world becomes again a “classicalworld”. The many worlds interpretation seems to be able to recover theseislands of classicality at the price of multiplying the ‘actual realm’. In thiscase, the quantum superposition is expelled from each actual world andrecovered only in terms of the relation between the multiple worlds.The Geneva school to quantum logic and similar approaches such as thatof Foulis and Randall [16] attempt to consider quantum physics as relatedto the realms of actuality and potentiality in analogous manner to classicalphysics. According to the Geneva school, both in classical and quantumphysics measurements will provoke fundamental changes of the state of thesystem. Continuing Heisenberg’s considerations in the new physics, Con- What is special for a classical system, is that ‘observables’ can be described by func- definite experimental project (DEP) is an experimental proce-dure (in fact, an equivalence class of experimental procedures) consistingin a list of actions and a rule that specifies in advance what has to beconsidered as a positive result, in correspondence with the yes answer to adichotomic question. Each DEP tests a property. A given DEP is called certain (correspondingly, a dichotomic question is called true ) if it is surethat the positive response would be obtained when the experiment is per-formed or, more precisely, in case that whenever the system is placed ina measurement situation then it produces certain definite phenomenon tohappen. A physical property is called actual in case the DEPs which test itare certain and it is called potential otherwise. Whether a property is actualor potential depends on the state in which one considers the system to be.Though in this approach both actuality and potentiality are considered asmodes of being, actual properties are considered as attributes that exist , inthe EPR sense as elements of physical reality , while potential properties arenot conceived as existing in the same way as real ones. They are thought as possibilities with respect to actualization, because potential properties maybe actualized due to some change in the state of the system. In this case thesuperposition provides a measure —given by the real numbers which appearin the same term as the state— over the potential properties which couldbecome actual in a given situation. tions on the state space. This is the main reason that, a measurement corresponding tosuch an observable, can be left out of the description of the theory ‘in case one is notinterested in the change of state provoked by the measurement’, but ‘only interested inthe values of the observables’. It is in this respect that the situation is very different fora quantum system. Observables can also be described, as projection valued measures onthe Hilbert space, but ‘no definite values can be attributed to such a specific observablefor a substantial part of the states of the system’. For a quantum system, contrary to aclassical system, it is not true that ‘either a property or its negation is actual’. Einstein designed, in the by now famous EPR ‘paper’ [11], a definition of when a phys-ical quantity could be considered an element of physical reality within quantum mechanics.By using this definition Einstein, Podolsky and Rosen argued against the completeness ofthe quantum theory. For a general discussion see [15]. Quantum Superpositions and the ‘Contradiction’of Properties?
Although the interpretations we have discussed in the previous section fromboth their formal and metaphysical commitments have many differences,there is still something they all share in common: they all attempt to avoidcontradictions. Indeed ‘contradiction’ has been regarded with disbelief inWestern thought due to certain metaphysical presuppositions which go backto Plato, Aristotle, Leibniz and Kant. Even after the development of para-consistent logic in the mid XX century and the subsequent technical progressthis theory has allowed, the aversion towards contradiction is still presenttoday within science and philosophy. The famous statement of Popper thatthe acceptance of inconsistency “would mean the complete breakdown of sci-ence” remains an unfortunate prejudice within present philosophy of science(see [4], Chap. 5).Leaving instrumentalist positions aside, one of us has argued elsewhere[29] that one can find in the vast literature regarding the interpretation ofquantum mechanics, two main strategies which attempt to provide an an-swer to the riddle of ‘what is quantum mechanics talking about’. The firststrategy is to begin with a presupposed set of metaphysical principles andadvance towards a new formalism. Examples of this strategy are Bohmianmechanics, which has been discussed above, and the collapse theory pro-posed by Ghirardi, Rimini and Weber (also called ‘GRW theory’) [18], whichintroduces non-linear terms in the Schr¨odinger equation. The second strat-egy is to accept the orthodox formalism of quantum mechanics and advancetowards the creation and elucidation of the metaphysical principles whichwould allow us to answer the question: ‘what is quantum mechanics talkingabout’ ? Examples of this second strategy are quantum logic and its differentlines of development such as the just described Geneva School of Jauch andPiron, and the modal interpretation (see for example [10, 30, 35]). Fromthis perspective, the importance is to focus in the formalism of the theoryand try to learn about the symmetries, the logical features and structuralrelations. The idea is that, by learning about such aspects of the theorywe can also develop the metaphysical conditions which should be taken intoaccount in a coherent ontological interpretation of quantum mechanics.But even independently of the choice of this strategy, it seems quite clearthat technical developments which are taking place today regarding quan-tum mechanics have advanced quite independently of the commitments to10ny classical metaphysical background. Quantum computation makes useof the multiple flow of information in the superposition even considering (inprinciple) contradictory paths. Also quantum cryptography uses the rela-tion between contradictory terms in order to send messages avoiding classicalspies. At a formal level, the path integral approach takes into account themultiple contradictory paths within two points [14]. Thus, since both theformalism and experiments seem to consider ‘contradictory elements’ withinquantum mechanics, we argue that it can be of deep interest to advance to-wards a formalism which takes contradiction into account “right from thestart”. Evidently, such a formalism could open paths not only to con-tinue the technical developments just mentioned but also to understand themeaning of quantum superpositions from a new perspective. Our proposalis twofold, firstly, to call the attention of the importance of considering con-tradictory properties within the formalism and interpretation of quantumsuperpositions; and secondly, to show that paraconsistent logics can opena formal line of research. In the next section we make a first step in thissame direction, providing an outline to an approach based on paraconsistentlogic.
We bring into, now, a paraconsistent logical system ZF , that is a strong settheory, even stronger than common ZF (Zermelo Frenkel set theory). On ZF and related matters, see [5]. In what follows, we employ the terminol-ogy, notations and conventions of Kleene [19]The basic symbols of the langugae ZF are the following: 1) Proposi-tional connectives: implication ( → ), conjunction ( ∧ ), disjunction ( ∨ ) and(weak) negation ( ¬ ), equivalence ( ↔ ) is defined as usual. 2) Individual vari-ables: a demmuerable set of variables, that are represented by smal Latinletters of the end of the aplphabet. 3) The quantifiers ∀ (for all) and ∃ (thereexists). 4) The binary predicate symbols ∈ (membership) and = (identity).5) Auxiliary symbols: parenthesis.Syntactic notions, for example those of formula, closed formula or sen-tence, and free occurrence of a variable in a formula, are defined as custom-ary. Russell’s symbol for description ( ι ) is introduced by contextual defi- In an analogous fashion as D´ecio Krause has developed a Q-set theory which accountsfor indistinguishable particles with a formal calculus “right from the start” [20]. { x ; F ( x ) } , where F ( x ) is a formula and x a variable. Definition 5.1 A ◦ abbreviates ¬ ( A ∧ ¬ A ).Loosely speaking, A ◦ means that A is a well-behaved formula, i.e., thatit is not the case that one has A and ¬ A both true (or, what is the samething, that the contradiction A ∧ ¬ A is false). Definition 5.2 ¬ ∗ A abbreviates ¬ A ∧ A ◦ . ¬ ∗ functions like a strong negation, a kind of classical negation in ourlogic. On the other hand, ¬ is the weak (or paraconsistent) negation. Postulates of ZF a) Propositional postulates: A → ( B → A )2. ( A → B ) → ( A → ( B → C )) → ( A → C )3. A A → BB
4. ( A ∧ B ) → A
5. ( A ∧ B ) → B A → ( B → ( A ∧ B ))7. A → ( A ∨ B )8. B → ( A ∨ B )9. ( A → C ) → (( B → C ) → (( A ∨ B ) → C ))10. A ∨ ¬ A ¬¬ A → A B ◦ → (( A → B ) → (( A → ¬ B ) → ¬ A ))13. ( A ◦ ∧ B ◦ ) → ( A → B ) ◦ A ◦ ∧ B ◦ ) → ( A ∧ B ) ◦
15. ( A ◦ ∧ B ◦ ) → ( A ∨ B ) ◦ b) Quantificational Postulaes: C → A ( x ) C →∀ xA ( x ) ∀ xA ( x ) → A ( t )3. A ( t ) → ∃ xA ( x )4. A ( x ) → C ∃ xA ( x ) → C ∀ x ( A ( x )) ◦ → ( ∀ xA ( x )) ◦ ∀ x ( A ( x )) ◦ → ( ∃ xA ( x )) ◦ The preceding postulates are subject to the usual restrictions. In clas-sical logic, as well as in most usual logics, we are allowed to reletterbind variables and suppress vacuous variables, but it seems that this isnot probable in connection with our system. Therefore, we introducethe postulate:7. If B is a formula obtained from A by relettering bound variables or bythe suppression of void quantifiers, then A ↔ B is an axiom. Remark.
Postulates 1-15 constitute a propositional system of paraconsis-tent logic and adding the quantificational postulates we obtain a first-orderquantificational paraconsistent logic. c) Set-Theoretic Postulates:
They are all those of classical ZF in whose formulations the symbol ofnegation is replaced by the symbol of strong negation ¬∗ . The postulatescan be formulated supposing that ZF is a pure set theory or a theory thatcontains Urelemente (objects that are not sets). Our results don’t dependon the version employed of ZF . Moreover the existence of some sets thatcause problems (do not exist) in ZF , like Russell’s collection, could be pos-tulated as existing in ZF ; however, this possibility is here excluded. ( ZF is studied, for instance, in [22]).From now on, capital letters stand for formulas. We have (see [5]):13 efinition 5.3 ⊢ A stands for A is a theorem of ZF ; A is the negationof ⊢ A . Definition 5.4 A ∗ is the formula obtained from A by replacing any occur-rence of ¬ by an occurrence of ¬ ∗ . Theorem 5.5 In ZF : ⊢ A ∨ ¬ A, ⊢ ¬¬ A → A, ⊢ ( A ∧ ¬ ∗ A ) → B, ( A ∧ ¬ A ) → B, A → ¬¬ A, ¬ ( A ∧ ¬ A ) . Theorem 5.6 If A is provable in ZF , then A ∗ is probable in ZF ( ZF isincluded in ZF ). Theorem 5.7 ZF is inconsistent if and only if ZF is non trivial. ZF is, in a certain sense, contained in ZF . So, in ZF it is possible tosystematize extant classical mathematics; in consequence, ZF encompassesall mathematical analysis required for the treatment of standard quantummechanics, and this treatment is similar to the one with classical logic (andset theory) as the basic logic.In the study of a quantum system S in N F , we note that its statesbehave as in classical quantum mechanics, in the sense that ZF ∃ s ( s ∈ ˆ S ∧ ¬ ∗ ( s ∈ ˆ S )) (2)and some other similar formulas are (apparently) not probable, where ˆ S isthe set of the states of S included in a given superposition.When S is in the state of superposition of, say, the states s and s (classically inconsistent), we introduce in ZF the extra predicate K andexpand the system with the postulates K ( S, s ) and ¬ K ( S, s ) (3)as well as: K ( S, s ) and ¬ K ( S, s ) (4)Informally, for instance K ( S , s ) means that “ S has the superpositionpredicate associated to s ” (or the “paraconsistent predicate associated to s ”). In other words, superposition creates a contradictory situation, givingrise to contradictory relations. In ZF , we can not directly assume that the14inear combination of two classically incompatible states is an ‘inconsistent’state; this is so because the mathematics of usual quantum mechanics isclassical, and such kind of inconsistency would make our system trivial.To cope with this situation, we appeal to a new postulate: Postulate of inconsistency . Let S be a quantum system which is in thesuperposition of the (classically incompatible) states s and s . Under thehipothesys, we have: K ( S, s ) ∧ ¬ K ( S, s ) ∧ K ( S, s ) ∧ ¬ K ( S, s ) (5)This means that superposition implies contradiction. Similarly when super-position involves more than two states.Therefore, ZF constitutes the underlaying logic of an inconsistent, butapparently non trivial, quantum mechanics that we denote by QM ; usualquantum mechanics will be denoted by QM . Thus, QM is in a certain sense,contained in QM . But the details of the construction of a paraconsistentquantum mechanics will be left to a future series of technical works. Our proposal focuses on the idea that it would be worthwhile to develop anew interpretation of quantum superpositions which considers contradiction“right from the start”. We have provided an outline of a paraconsistentapproach to quantum superpositions which shows the possibility to considercontradictions also from a formal perspective. However, it should be clearthat we do not take paraconsistent logic to be the “true logic” which shouldreplace classical logic; in the same way as we do not regard quantum me-chanics as a theory that should replace classical mechanics [4, 30]. Fromour perspective we argue that physicists should recognize the possibility touse new forms of logic —such as paraconsistent logic— which might help usunderstanding features of different domains of reality; features which mightnot be necessarily accommodated by means of classical logic. We do notbelieve there is a “true logic”, but rather that distinct logical systems canbe of use to develop and understand complementary aspects of reality. Re-calling the words of Albert Einstein: “It is only the theory which can tell15ou what can be observed” it could be argued that only within a theory itis possible to consider and account for phenomena. From this standpoint thedevelopment of the formalism can be regarded not only as a merely technicalimprovement, but also as a way to open new paths of understanding andeven of development of new phenomena. Formal development is not under-stood here as going beyond the theory, as improving and showing somethingthat “was not there before” in the formalism —as it is the case of the GRWtheory or Bohmian mechanics. Rather, this development is understood astaking seriously the features which the theory seems to show us, exposingthem in all their strength, “right from the start”.We also have to stress that non relativistic quantum mechanics, basedon classical logic and on the common specific postulates, seems to be con-sistent in the strict logical meaning. So, a paraconsistent version of it has topostulate, in some way or other, the inconsistent character of determinablesituations. It appears to be that there is no possibility that someone can“deduce”, employing any one of a majority of extant logics, contradictoryconsequences of the specific axioms of non relativistic quantum mechanics.Evidently, we have to develop and to explore the ideas here sketched.One of the important points to take into consideration is that there arenumerous ways to obtain such kind of inconsistent quantum mechanics. Inaddition, we should verify if there are really important new results of QM which are not valid in QM . The philosophical meaning of QM also deservesdetailed analysis. Acknowledgments
The authors wish to thank an anonymous referee for his/her careful readingof our manuscript and useful comments.
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