A Sound and Complete Tableaux Calculus for Reichenbach’s Quantum Mechanics Logic

https://doi.org/10.1007/s10992-023-09730-7 A Sound and Complete Tableaux Calculus for Reichenbach’s Quantum Mechanics Logic Pablo Caballero1 ·Pablo Valencia2 Received: 28 October 2022 / Accepted: 26 October 2023 / © The Author(s) 2023 Abstract In 1944 Hans Reichenbach developed a three-valued propositional logic (RQML) in order to account for certain causal anomalies in quantum mechanics. In this logic, the truth-value indeterminate is assigned to those statements describing physical phenomena that cannot be understood in causal terms. However, Reichenbach did not develop a deductive calculus for this logic. The aim of this paper is to develop such a calculus by means of First Degree Entailment logic (FDE) and to prove it sound and complete with respect to RQML semantics. In Section 1we explain the main physical and philosophical motivations of RQML. Next, in Sections 2and 3, respectively, we present RQML and FDE syntax and semantics and explain the relation between both logics. Section 4introduces Qcalculus, an FDE-based tableaux calculus for RQML. In Section 5we prove that Qcalculus is sound and complete with respect to RQML three-valued semantics. Finally, in Section 6we consider some of the main advantages of Qcalculus and we apply it to Reichenbach’s analysis of causal anomalies. Keywords Causal anomaly ·First degree entailment ·Indeterminate ·Reichenbach’s quantum mechanics logic ·Three-valued logic ·Semantic tableaux 1 Introduction In 1944 Reichenbach developed a three-valued logic in order to account for causal anomaliesthatariseinthestandardinterpretationofquantummechanics.A“causallaw BPablo Caballero [email protected] Pablo Valencia v[email protected] 1Department of Philosophy and Logic and Philosophy of Science, Universidad de Sevilla, Camilo José Cela St., Sevilla 41018, Spain 2Department of Philosophy and Anthropology, Universidad de Santiago de Compostela, Mazarelos Sq., Santiago de Compostela 15782, Spain 123 Published online: 18 November 2023 Journal of Philosophical Logic (2024) 53:223–245 of nature”, according to Reichenbach, is defined under two conditions (Reichenbach [19], 117): a) “it is required that the cause determine the effect univocally [...]; b) it is demanded that the effect spread continuously through space, following the principle of action by contact.” The type of causality that satisfies a) and b) is called “normal". Non-compliance with b) constitutes a causal anomaly. Reichenbach considers that the main problem of quantum mechanics can be summed up in the following question: is there an exhaustive interpretation —that is, one that gives value assignments to all physical quantities in all physical states (Reichenbach [21])— of quantum mechanics that satisfies the requirements of “normal causality"? (Reichenbach [22]). Since an exhaustive interpretation provides a value for every physical quantity in all physical states, it includes a complete description of interphenomena, that is, of unobservable entities that do not obey the usual classical laws –such as the relative motion of an electron (see Turquette [26]). However, since an exhaustive interpretation assigns certain values to interphenomena, it leads to causal anomalies. For example, in the case of the double slit experiment (the explanation of which can be found in Albert [1], Norsen [12]orMaudlin[9]), it is possible to modify the distribution pattern of the particles that reach the screen through the first slit by changing the state of the second slit (opening or closing it), located at a distance from these particles. This is a clear example of action at a distance that implies the breach of normal causality in any exhaustive interpretation of quantum mechanics. Therefore, given the definition of “exhaustive interpretation”, and since interphenomena intervene in quantum mechanics –which present actions at a distance–, it follows that there is no exhaustive interpretation of quantum mechanics without causal anomalies (Reichenbach [20,22]). One way out of the bind is to admit only statements about phenomena and to regard all statements about interphenomena as “nonsense”. This is the interpretation that Reichenbach associates with Bohr and Heisenberg (Reichenbach [22]). Naturally, the Bohr-Heisenberg interpretation lies in moving from an exhaustive interpretation to a restrictive one in which only statements about phenomena are allowed. This implies that the causal anomalies are dissolved, precisely because the pretence of giving definite values to all physical quantities is abandoned (see Reichenbach [19]). Nevertheless, the idea that all statements about interphenomena are “nonsense”, as Reichenbach points out, means giving up knowledge about such entities because no verification conditions are assigned to their corresponding statements (Reichenbach [19]). On the contrary, Reichenbach’s proposal consists of changing the laws of classical logic by adding a third truth-value, the indeterminate (i) (Reichenbach [19]). Any statement that expresses something about interphenomena must have value i. Thus, a statement will make sense if, and only if (iff onwards), it is true, false or indeterminate. Sinceiis a value analogous to the others, we substitute“nonsense” for “indeterminate” or currently “neither verified nor falsified”, in such a way that statements about interphenomena make sense again (Reichenbach [19]; Hardegree [5]). It is not necessary to give up the statement “the particle travels through the first slit”: it is possible that 123 224 P. Caballero, P. Valencia it has a meaning, even though its truth value is not “true”, and thus the corresponding causal anomaly is suppressed (see Reichenbach [19] and Section 6of this paper for further details). Although the analysis of the physical and philosophical aspects in Reichenbach’s proposal is not our main objective,1it is necessary to present the issues developed above as motivating Reichenbach’s logic, since the latter attempts to explain causal anomalies without allowing meaningless statements. Reichenbach provides a semantics for this logic, including new connectives and using the method of truth-tables. However, he does not develop a strictly deductive calculus, nor, by extension, the soundness and completeness theorems for it, something which has been implicitly or explicitly criticised, for instance, by Hempel [6], Nagel [10], Feyerabend [4] or Nilson [11]. For this reason, the aim of this paper is to implement the method of semantictableaux for First Degree Entailment logic (FDE)2to Reichenbach’s logic in order to obtain the corresponding tableaux-based deductive calculus (Q) and to prove it sound and complete. In Section 2, we present the formalism of quantum logic as expounded by Reichenbach [19]. We establish the relationship between Reichenbach’s quantum mechanics logic and FDE in Section 3. In Section 4, we elaborate the tableau rules for RQML that we must add to the ones for FDE. This allows us to obtain Qcalculus and to achieve our main goal (Section 5): to develop a deductive calculus which is sound and complete with respect to RQML semantics. Finally, in Section 6we consider some of the main advantages of Qcalculus over natural deduction calculi3for Reichenbach’s analysis of causal anomalies. 2 Reichenbach’s Three-valued Quantum Logic The following rule defines the concept of a well-formed-formula (WFF) in RQML4: WFF ::= p|¬A|∼A|A∨B We include two negations as primitive monary connectives: ¬,theclassical negation, and ∼,thecyclical negation (Reichenbach [19]). The set {¬,∼,∨} is functionally complete, as proved in Corolary 5.2 (Appendix A). We define A∧Bas ¬(¬A∨¬B). Secondly, Reichenbach points out that a statement with truth value iis obviously not true (it is neither false nor true). It follows that an inference from a true statement to an indeterminate statement, which is, therefore, not true, involves moving from a statement where the property of “being true” holds to one where it does not. Consequently, inferences from true statements to indeterminate statements do not guarantee 1SeeHempel[6], Nagel [10] or Feyerabend [4] for some the most important criticisms. 2For further references on FDE logic, see Section 3. 3In Section 6we consider two natural deduction systems for a logic equivalent to RQML. 4Although Reichenbach takes every connective introduced in this section as primitive, we only take ¬,∼ and ∨as primitive, since they are sufficient to define the remaining connectives (see Appendix A,where functional completeness of the set {¬,∼,∧} is proved). 123 225A Sound and Complete Tableaux Calculus for Reichenbach’s... Table 1Truth-table for ¬,∼ and AA¬A∼A A 10 ii ii 01 01 1 1 truth preservation, and so cannot be valid. Only 1 is a designated value5in RQML (Reichenbach [19]). We now present RQML semantics by means of truth-tables. First of all, we define a new monary connective, the complete negation (A), as follows (Reichenbach [19]): A=def ∼A∨∼∼A. The truth-table (Table 1) corresponds to the three negations. On the other hand, for the order 0 <i<1, v(A∨B)=max[v(A), v(B)]and v(A∧B)=min[v(A), v(B)]. Also, among the possible implications in RQML, Reichenbach uses the following three conditionals, all of which are defined in the following page: the standard conditional (⊃), the alternative implication6(→) and the quasi-implication () (Reichenbach [19]). Finally, we introduce the classical equivalence (≡) and the alternative equivalence (∼ =), also as defined symbols (Reichenbach [19]). The semantics of these connectives is given in Table 2. Although Reichenbach does not provide a definition for →,⊃and (only for ∼ = and ≡) (Reichenbach [19]), we define them in the following way7: A→B=def ∼¬(A∨B) AB=def (A∧B)∨∼A A∼ =B=def (A→B)∧(B→A)∧(¬A→¬B)∧(¬B→¬A) A⊃B=def (¬A∨B)∨(A∼ =B) A≡B=def (A⊃B)∧(B⊃A) Inthe followingsectionwe introduceFDEsyntaxandsemantics.Itsrelation toRQML, the logic explained in this section, will be clarified in Section 3.1. 3 First Degree Entailment In classical propositional calculus, an interpretation is a function from formulas to the truth values 1 and 0. But this involves that every formula is either true or false –never neither, and never both. By contrast, FDE is a logic where interpretations are formulated as relations rather than functions. Thus, a formula may relate to 1; it may relate to 0; it may relate to both; or it may relate to neither. Therefore, the fact that a propositional parameter is untrue does not mean that it is false; and the fact that a 5Remember that a designated truth value is the one preserved in valid inferences. 6As a special feature, we remark in Appendix Bthat deduction theorem (Theorem 6) only holds for this conditional. 7Note that ⊃corresponds to Łukasiewicz three-valued conditional (Łukasiewicz, [7]; Łukasiewicz [8]). RQML could then be regarded as an extension of Ł3 (however, see footnote 10 on some differences between RQML and both Ł3 and K3). 123 226 P. Caballero, P. Valencia Table 2Truth-table for ∧,∨,→,,⊃,∼ =and ≡ ABA∧BA∨BA→BABA⊃BA ∼ =BA≡B 11111 1 1 1 1 1ii 10 ii0i 10010 0 0 0 0 i1i11 i10i iii i 1i111 i00 i1ii0i 01011 i100 0i0i1i10i 00001 i111 propositional parameter is false does not mean that it is untrue (hereafter, we mainly follow Priest [18]; for further developments of FDE, Belnap-Dunn semantics and its extensions, see Routley & Routley [24], Rivieccio [23], Albuquerque, Pˇrenosil & Rivieccio [2], Omori & Wansing [14] and Pˇrenosil [17]). Hence, an FDE interpretation is a relation ρbetween propositional parameters and the values 1 and 0, that is, ρ⊆P×{1,0}, where Pis the set of propositional parameters. For example, pρ1 means that prelates to 1. Given an interpretation ρ,this can be extended to a relation between all formulas and truth values by the following recursive clauses (Priest [18]) –we take ¬and ∨as primitive connectives in FDE, and define A∧Bas ¬(¬A∨¬B)8: A∨Bρ1iff Aρ1orBρ1 A∨Bρ0iff Aρ0 and Bρ0 ¬Aρ1iff Aρ0 ¬Aρ0iff Aρ1 These clauses are the same as the classical truth conditions, stripped of the assumption that truth and falsity are exclusive and exhaustive. Let us now consider the tableaux for FDE. Each entry of a tableau has the form A,+or A,−, where A,+means that Ais related to 1 (Ais true) and A,−means that it is not. So, ¬A,+means that Ais related to 0 (Ais false) and ¬A,−that it is not. Hence, if both A,−and ¬A,−occur on a branch, it means that Ais indeterminate (Priest [18]). To test the inference A1, ..., AnFDE B, we start with an initial list of the form A1,+ . . . An,+ B,− 8While the language of FDE, as expounded by Priest, contains the connectives ∧,∨and ¬as primitive (Priest [18]), we define ∧in terms of ¬and ∨. 123 227A Sound and Complete Tableaux Calculus for Reichenbach’s... A branch of a tableau closes if it contains nodes of the form A,+and A,−.The basic FDE rules given by Priest [18] are as follows –although ∧is taken as a defined connective here, we introduce the corresponding tableau rules for FDE (as Priest does) for practical reasons: Box 1 FDE BASIC RULES ¬¬A,+ A,+ ¬¬A,− A,− A∧B,+ A,+ B,+ A∨B,− A,− B,− A∧B,− A,−B,− A∨B,+ A,+B,+ ¬(A∧B), + ¬A∨¬B,+ ¬(A∧B), − ¬A∨¬B,− ¬(A∨B), + ¬A∧¬B,+ ¬(A∨B), − ¬A∧¬B,− 3.1 First Degree Entailment and Reichenbach’s Three-valued Logic For any formula Aand any interpretation ρin FDE, there are four possibilities: Ais true and not false; Ais false and not true; Ais both true and false; Ais neither true nor false. It is then possible to think of FDE as a four-valued logic. However, consider an FDE interpretation satisfying the following constraint: Exclusion:For no A,Aρ1and Aρ0 No formula is both true and false in this FDE-interpretation, while a propositional parameter can still be neither true nor false. Our thesis is that the logic defined in terms of truth preservation over all interpretations satisfying this constraint could be RQML. However, RQML uses more primitive connectives than FDE, so we cannot say that RQML semantics strictly corresponds to an FDE interpretation satisfying the previous constraint. In fact, ∼Ais not definable in terms of ¬and ∨, given that they preserve value i. Hence, since RQML connectives must also be interpretable in terms of FDE relational semantics, we introduce cyclical negation as a primitive symbol in FDE and give its corresponding FDE interpretation: ∼Aρ1iff ¬Aρ1iff Aρ0 ∼Aρ0iff A ρ1 and ¬A ρ1iff A ρ1 and A ρ0 Since both logics now share the same syntax it is possible to use FDE formalism to give a calculus for RQML (remember that the definitions of the RQML conditionals and equivalences, that is, →,,⊃,≡and ∼ =, are in Section 2). Onwards, we will write FDE* for FDE satisfying the exclusion constraint and including ¬,∼and ∨as primitive constants. 123 228 P. Caballero, P. Valencia As we have seen in Section 4, RQML is a three-valued logic in which, for any given formula A, there are three possibilities: v(A)=1(Ais true); v(A)=0(Ais false); v(A)=i(Ais indeterminate). These are the same possibilities as in FDE*: Aρ1(A is true); Aρ0(Ais false); A ρ1 and A ρ0(Ais indeterminate). In fact, in RQML, as in FDE*, there can be gaps, but not gluts.9 Now,we canusetheFDE-tableauxformalism(A,+;A,−;¬A,+;¬A,−)tomake a proper calculus for RQML. We must pay attention that the exclusion constraint adds a closing rule: a branch of a tableau also closes if it contains nodes of the form A,+ and ¬A,+. Given the new connectives, in Section 4we will introduce new closing and tableau rules for RQML in addition to the rules already introduced for FDE in this section (Box 1). We first prove Theorem 1, which allows to connect FDE* relational semantics with RQML truth-functional semantics, but previously we give the following definition: Definition 1 We write FDE* for validity and logical consequence according to FDE* semanticsandRQML forvalidityandlogicalconsequenceaccordingtoRQMLsemantics, and define FDE* and RQML as follows: a)FDE* Aiff,ifBρ1 for all B∈, then Aρ1 b)RQML Aiff,ifv(B)=1 for all B∈, then v(A)=1 Theorem 1 10 FDE* AiffRQML A Proof From left to right, suppose FDE* A. Hence, for an FDE* interpretation ρ, if Bρ1 for every B∈, then Aρ1. But since Aρ1inFDE*iff v(A)=1inRQML A ρ1 and A ρ0inFDE*iff v(A)=iin RQML Aρ0inFDE*iff v(A)=0inRQML whenever v(B)=1 for every B∈, then v(A)=1 in RQML, which means that RQML A. Similarly, from right to left.  The above theorem depends on the introduction of ∼(together with ¬and ∨)asa primitive connective in FDE* and the definition of A,⊃,,→,≡and ∼ =in terms of the primitive connectives –it is clear that if FDE* and RQML did not share the same language,therelationshipstipulatedinTheorem1wouldnothold.Theaforementioned theorem will play an important role in proving Qsound and complete with respect to RQML semantics in Section 5. 9In other words, it can be the case that Ais neither true nor false, but not both (Priest [18]). 10 A similar proof of this theorem can be found in Section 3 of (Estrada-Gonzales & Cano-Jorge, [3]), in which Dunn’s semantics is applied to Reichenbach’s connectives to obtain a relational semantics which excludes the subset {0,1}of the set of truth values {0,1}(this is equivalent to our Exclusion constraint). Although both our proposal and that of the above paper use the same relational semantics, in order to define FDE* we take only ¬,∼and ∨as primitive connectives and define the remaining new connectives. By contrast, Estrada-Gonzales & Cano-Jorge take Reichenbach’s new negations and conditionals as primitive connectives, and regard Reichenbach’s logic as an expansion of the fragment {¬,∨,∧} of Kleene’s K3 logic by means of five new connectives. Note, however, that our set {¬,∼,∨} of primitive connectives is functionally complete (see Appendix A), while that of the primitive connectives of K3 is not. This suffices to state that K3 is not equivalent to RQML. The same applies to Ł3, containing ¬,∧,∨and ⊃as primitive connectives, which do not form a functionally complete set. Moreover, ∼is not definable with the connectives of Ł3. 123 229A Sound and Complete Tableaux Calculus for Reichenbach’s... Table 3Truth-table for ¬∼AA∼A¬∼A 1ii i01 01 0 4QCalculus: an FDE*–based Calculus for RQML In order to develop the table rules for Qcalculus, in this section we proceed in the following way. Firstly, according to RQML semantics, we elaborate the truth table corresponding to every connective. Next, we consider the cases in which the corresponding formula is true and the cases in which it is not: in both cases, this will allow us to check the truth-values to which the propositional parameters that are included in the formula relates. Finally, with this result, we develop the FDE* rules for the cases in which the formula relates to 1 and those in which it does not. For instance, to get the rule for ¬∼A,+, we first construct the table (Table 3). We check that when ¬∼A is true, Ais i. Hence, after applying the rule for ¬∼A,+, the nodes A,−and ¬A,− must be on the extended branch. Similarly for ¬∼A,−. Note that in the latter case, when ¬∼Ais not true, there are two possibilities: Ais true or false. Therefore, we must add two branches, one of them with A,+, and the other with ¬A,+,asshown in Box 4 below. We add the rules in Box 2 to 6 to the ones in Box 1 to develop Qcalculus. First, the closing rules for FDE* are the following: Box 2 CLOSING RULES A branch closes whenever one of these formulas or pairs of formulas appears on it: (1) A,+and A,− (2) A,+and ¬A,+ (3) ∼A,+ (4) ¬A,+ (1) is contradictory both in FDE with and without exclusion. Also, it immediately follows from the exclusion constraint for FDE* that (2) entails a contradiction. On the other hand, the reason why a branch closes if ∼A,+or ¬A,+occurs on it (rules (3) and (4), respectively) is the following: the truth-table for ∼A,+and ¬A,+(Table 4) show that they can never be true, so none of them relates to 1 in an FDE* interpretation. Table 4 Truth-table for ∼Aand ¬AA¬A∼A A ∼A¬A 10 ii0i ii 01i0 01 1 1i0 123 230 P. Caballero, P. Valencia Since the rules for ∧in Box 1, the ones for Ain Box 3 and all the rules in Box 5 and Box 6 correspond to defined connectives (see Section 2), they all could be omitted. However, we include them in Qfor practical reasons. Box 3 NON-CLASSICAL NEGATIONS ELIMINATION ∼A,+ ¬A,+ ∼A,− ¬A,− A,+ A,− A,− A,+ Box 4 COMBINED NEGATIONS ELIMINATION ¬A,− A,+A,− ¬∼A,+ A,− ¬A,− ¬∼A,− A,+¬A,+ Box 5 CONDITIONALS ELIMINATION A⊃B,+ ¬A,+B,+A,− ¬A,− B,− ¬B,− A→B,+ A,−B,+ AB,+ A,+ B,+ A⊃B,− A,+ B,− ¬A,− ¬B,+ A→B,− A,+ B,− AB,− A,−B,− 123 231A Sound and Complete Tableaux Calculus for Reichenbach’s... Lemma 3 Let b be an open completed branch of a tableau. Let A be any formula and ρthe interpretation induced by b. Then: if A,+occurs on b, then Aρ1 if A,−occurs on b, then A ρ1 if ¬A,+occurs on b, then Aρ0 if ¬A,−occurs on b, then A ρ0 Proof The proof is by induction on the complexity of A. Suppose Ais a propositional parameter: if A,+occurs on b, then Aρ1 by definition, and if A,−occurs on b, then A,+does not occur on b, since it is open. Hence, by definition, it is not the case that Aρ1,so A ρ1.Thecases for0aresimilar. Suppose∼A,+occurson thebranchb. Then ¬A,+occurs on b. By induction hypothesis, Aρ0. Hence, ∼Aρ1, as required. Now assume ∼A,−occurs on the branch. Then ¬A,−is on b.By induction hypothesis, ¬A ρ1, so ∼A ρ1, as required. For A:ifA,+occurs on b, then A,−occurs on b. By induction hypothesis, A ρ1. Therefore, Aρ1, as required. Now suppose ¬∼A,+ occurs on b. Then, A,−and ¬A,−occur on b. By induction hypothesis, A ρ1 and A ρ0. Hence, ¬∼Aρ1. Now assume A→B,+occurs on b. Then, A,−or B,+ occur in b. By induction hypothesis, A ρ1orBρ1. Therefore, A→Bρ1, as required. Similarly for the others rules.  Theorem 3 (Completeness) For finite ,ifFDE* A then QA. Proof Suppose that QA. Consider a completed open tableau for the inference, and choose an open branch. The interpretation that the branch induces makes all the members of true, and Afalse, by Lemma 3. Hence, FDE* A. Therefore, if FDE* A, then QA. Corollary 3.1 (Consistency)Qcalculus is consistent, that is, QA∧¬A, QA∧∼A and QA∧A for any formula A. Proof Suppose the opposite, i. e., that there exist some formula Asuch that either QA∧¬A,orQA∧∼AorQA∧A. Hence,byTheorem2,eitherFDE* A∧¬A, or FDE* A∧∼Aor FDE* A∧A. But none of these can be the case, since if Aρ1, then ¬A ρ1, ∼A ρ1 and A ρ1 in FDE*, so none of the three conjunctions above can be valid (and therefore cannot be a theorem, by Theorem 2). Hence, Qis consistent.13  Theorem 4 (Main Theorem) Qcalculus is sound and complete with respect to Reichenbach’s three-valued semantics, that is, for finite ,RQML AiffQA. Proof It follows directly from Theorems 1,2, and 3. 13 SeePost[16], where the consistency of the propositional axiomatic system in Principia Mathematica is proved by means of a similar method. 123 238 P. Caballero, P. Valencia 6 Logico-philosophical Advantages of QCalculus As explained in the introduction, one of the main goals of RQML is to provide an exhaustive interpretation14 of quantum mechanics which allows to avoid causal anomalies that arise, for instance, in the double slit experiment. Having shown that Qcalculus is sound and complete with respect to RQML semantics, we will now show how this calculus can be applied to Reichenbach’s solution to causal anomalies. Furthermore, we will compare Qwith other calculi developed for many-valued logics and present some main formal advantages of Q. First, we offer a brief outline of Reichenbach’s proposal to avoid the causal anomaly that arises in the classical interpretation of the double slit experiment –a detailed explanation of which can be found in (Estrada-González & Cano-Jorge [3]). For a system of two slits, let B1and B2respectively state that a given particle passes through slits 1 and 2, and let Crepresent the causal anomaly that arises in the classical interpretation of the double slit experiment. Now, let B1∼ =¬B2express that the particle passes through exactly one of the slits, and let (B1∨B2)→Cexpress that, if the particle is known to have passed through one of the slits, the causal anomaly obtains.15 As can be easily checked, Cfollows from the premises above if only classical inputs are considered (in this case, if we assume that B1∼ =¬B2, then B1∨B2is necessarily true even if no observation is made on the slits). Nevertheless, in RQML it is possible that v(B1∼ =¬B2)=v((B1∨B2)→C)=1 with v(B1)=v(B2)=iand either v(C)=ior v(C)=0, so B1∼ =¬B2, (B1∨B2)→CRQML C.16 It can be checked that the above solution makes use of the meaning of ias “neither verified nor falsifies”, and which is applied to both B1and B2in order to state that it is not known whether or not the particle has passed through any of the slits. Through Qcalculus, however, the proof that the causal anomaly does not follow in RQML can be deductively obtained by means of the following proof17 (from which the above countermodels can be easily inferred): 14 In the sense already indicated in the introduction. 15 A classical interpretation of the double-slit experiment implies that, if something is observed on the detection screen, then either B1is true or B2is true. 16 Roughly, this means that, even if something is observed on the detection screen, it does not follow that either B1or B2is true, since both B1and B2could be indeterminate, as indicated in the counter-model above. In this case, since neither B1nor B2can be asserted, neither can it be asserted that the state of slit 2 affects the passage of the particle through slit 1, simply because we cannot affirm that the particle has passed through slit 1. 17 Steps (a), (b), (c) and (d) follow from the definition of ∼ =. In steps (c) and (d) we would obtain ¬B1→ ¬¬B2,+and ¬¬B2→¬B1,+, although we have directly eliminated the double negations. 123 239A Sound and Complete Tableaux Calculus for Reichenbach’s... B1∼ =¬B2,+ (B1∨B2)→C,+ C,− B1→¬B2,+(a) ¬B2→B1,+(b) ¬B1→B2,+(c) B2→¬B1,+(d) C,+  B1∨B2,− B1,− B2,− ¬B2,+ ¬B2,−  B1,+  B1,− B1,+  ¬B2,− B2,+  ¬B1,− B2,−¬B1,−  As expected, from the only open branch we can read off the two following countermodels (by Theorem 1): v(B1)=v(B2)=i v(C)=i v(B1)=v(B2)=i v(C)=0 We now turn to the formal advantages of this analysis of Reichenbach’s solution. With regard to the development of deductive calculi for many-valued logics, although no tableaux-based calculus has yet been developed for RQML, in (Omori & Wansing [15]) two natural deduction systems are provided for a logic (called K34by the authors18) equivalent to RQML. The systems in question are provided in definitions 13 and 17 of the above-mentioned paper, and they are proved to be sound and complete with respect to the three-valued semantics in Theorem 27. Nevertheless, it must be noted that, contrary to tableaux-based calculus, natural deduction systems neither allow to sintactically prove that B1∼ =¬B2,(B1∨B2)→CC nor to infer a countermodel, which would provide the situation in which the causal anomalydoesnotobtain.Bycontrast,Qcalculusallowsbothtoprovethattheinference 18 K34is a variation of Kleene’s three-valued logic K3 which adds the negation ∼4(equivalent to Reichenbach’s ∼) as a primitive connective. 123 240 P. Caballero, P. Valencia from B1∼ =¬B2and (B1∨B2)→Cto Cis not valid19 and to read off from the open branch of the tableau a counter-model corresponding to Reichenbach’s solution to the causal anomaly that arises in the double slit experiment (as shown on the proof above). Hence, Qcalculus allows to obtain a proper counter-model (showing the expected physical situation, in which no observation is made on the slits and the causal anomaly does not obtain) which cannot be inferred from a natural deduction calculus. 7 Conclusion In this paper we have developed an FDE–based tableaux calculus (Q) for RQML and proved it sound and complete with respect to Reichenbach’s three-valued semantics. Theorem 1(Section 3) relates FDE* and RQML making it possible to develop Q calculus. Precisely, this theorem allows to move from FDE* relational semantics to RQML truth-functional semantics, and vice versa –otherwise, it would not have been possible to prove that Q(an FDE–based calculus) is sound and complete with respect to RQML semantics. Theorems 2and 3(Section 5), together with Theorem 1(Section 3), allow to prove Theorem 4(Section 5), the main objective of our paper. This theorem emphasises the relationship between RQML, FDE* and Qcalculus, and converts any consequence relation RQML Ain RQML into a deduction QAin Q, and vice versa. This makes it unnecessary to refer to FDE* relational semantics. Qcalculus and Reichenbach’s semantics are, therefore, sufficient to provide a sound and complete quantum logic system. Appendix A Definition 4 AsetSof truth-functional connectives is functionally complete in an n-valued logic L(for finite n)iff any truth-function fin Ldifferent from the ones in Scan be defined in terms of S(Theorem 5provides a functional completeness criterion). Theorem 5 20 If A ={0,i,1}, with order 0<i<1, the set containing the monadic functions f000,fiii,f111,f100,f010,f001 and the binary functions ∧and ∨is functionally complete. Proof Any n-ary connective gdefined over A={1,i,0}can be expressed by means of the following normal form: g(x1, ..., xn)= (a1,...,an)∈An [g(a1, ..., an)∧ j∈A Laj(xj)] 19 It is possible to know precisely when a tableaux is completed (see footnote 12). 20 We are grateful to Prof. José Pedro Úbeda Rives for his personal communication of Theorem 5–see also Úbeda [27] for more information on this topic. Corollary 5.2 below easily follows from Theorem 5. 123 241A Sound and Complete Tableaux Calculus for Reichenbach’s... (for ak∈A), where L1=f100,Li=f010 and L0=f001 (fjkm –for j,k,m∈ {0,i,1}– represents the unary connective hsuch that h(1)=j,h(i)=kand h(0)= m), and g(a1, ..., an)for any a1, ..., anis 0, ior 1, that can be obtained from f111,fiii and f000 applied to any variable x1, ..., xn. Remark 5.1 Another general criterion of functional completeness for many-valued logics is provided by Słupecki in (Słupecki [25]). Also, see Theorem 3.4 in (Omori & Sano [13]) for further developments on Słupecki’s criterion for functional completeness. Moreover, in (Omori & Wansing [15]), functional completeness of K34 (equivalent to RQML) is given in Theorem 39, and the functional completeness of {∼,∨} is given in (Post [16]) –see footnote 21. Corollary 5.2 The set {¬,∼,∨} of the primitive connectives in RQML is functionally complete. Proof The function ∨is a primitive connective in RQML (see Section 2), while ∧is definable in terms of ¬and ∨; also, the six monary functions in Theorem 5can be defined in terms of ¬,∼and ∨: f111 =A∨(∼A∨∼∼A) fiii =∼ f111 f000 =¬f111 f100 =¬∼∼[∼(∼A∨∼∼A)∨∼∼(∼A∨∼∼A)] f010 =¬∼¬(A∨∼A) f001 =¬(A∨¬∼ A) Since ∨is a primitive connective in RQML and, on the other hand, the truth-functions ∧,f000,fiii,f111,f100,f010 and f001 are definable in terms of our primitive connectives, the set {¬,∼,∨} is functionally complete in RQML.21 Naturally, it follows that the ten connectives taken as primitives by Reichenbach also form a functionally complete set.  Appendix B Theorem 6 (Deduction Theorem) If QA, where ={B1, ... Bn}, then Q i≤n Bi→A Proof First, suppose QA. This corresponds to a closed tableau of type 21 Yet, since ¬can be defined as ∼[∼(∼∼ A∨A)∨∼∼(∼A∨A)],theset{¬,∼,∨} is not independent. However, we include ¬as a primitive connective, given as it is commonly taken as an element of the formalism used in the calculus for FDE and FDE*. A proof that {∼,∨} is functionally complete was givenbyPost[16]. Given that RQML and Post’s three-valued logic with ∼and ∨as primitive connectives (hereafter P3) are both functionally complete three-valued logics and take 1 as the only designated value, then RQML Aiff P3 A(where can be empty). This entails that RQML and P3 are equivalent. 123 242 P. Caballero, P. Valencia B1 . . . Bi A,− . . .  Now consider a tableau starting with i≤nBi→A,−. Given that QA,the tableau i≤nBi→A,− i≤nBi,+ A,− . . .  closes too, since it will contain Bi,+for every Bi∈and A,−(that is, the same elements as the tableau for QA). Hence, if QA, where ={B1, ... Bn}, then Q i≤n Bi→A as required.  Theorem 7 There exists deductions QA, where ={B1, ... Bn}, such that Q i≤n BiA or Q i≤n Bi⊃A Proof By means of Qcalculus one can check that A∨AQA,butQ(A∨A)A A⊃∼AQ∼A,butQ(A⊃∼ A)⊃∼A Hence, Deduction Theorem only applies to →, and neither to nor to ⊃. 123 243A Sound and Complete Tableaux Calculus for Reichenbach’s... Acknowledgements The authors would like to thank Prof. José Pedro Úbeda Rives (Universidad de Valencia) for his numerous corrections and kind advice during the preparation and revision of this article. Thanks are also due to two anonymous reviewers of the Journal of Philosophical Logic for helpful comments and corrections on the first version of this article. Author Contributions Both Pablo Caballero and Pablo Valencia are principal authors of this article and have contributed equally to it. Funding Funding for open access publishing: Universidad de Sevilla/CBUA. The authors declare that no funding was received for the preparation of the present article. Declarations Ethical Approval The manuscript has not been submitted to more than one journal. Competing Interests The authors have no competing interests to declare. Open Access ThisarticleislicensedunderaCreativeCommonsAttribution4.0InternationalLicense,which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. 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