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 1 we explain the main physical and philosophical motivations of RQML. Next, in Sections 2 and 3, respectively, we present RQML and FDE syntax and semantics and explain the relation between both logics. Section 4 introduces \(\varvec{\mathcal {Q}}\) calculus, an FDE-based tableaux calculus for RQML. In Section 5 we prove that \(\varvec{\mathcal {Q}}\) calculus is sound and complete with respect to RQML three-valued semantics. Finally, in Section 6 we consider some of the main advantages of \(\varvec{\mathcal {Q}}\) calculus and we apply it to Reichenbach’s analysis of causal anomalies.
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Notes
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.
Yet, since \(\lnot \) can be defined as \(\sim [\sim (\sim \sim A \vee A) \vee \sim \sim (\sim A \vee A)]\), the set \(\{\lnot , \sim , \vee \}\) is not independent. However, we include \(\lnot \) 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 \(\{\sim , \vee \}\) is functionally complete was given by Post [16]. Given that RQML and Post’s three-valued logic with \(\sim \) and \(\vee \) as primitive connectives (hereafter P3) are both functionally complete three-valued logics and take 1 as the only designated value, then \(\Sigma \vDash _{\text {RQML}} A\) iff \(\Sigma \vDash _{\text {{\textbf {P3}}}} A\) (where \(\Sigma \) can be empty). This entails that RQML and P3 are equivalent.
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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.
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Funding for open access publishing: Universidad de Sevilla/CBUA. The authors declare that no funding was received for the preparation of the present article.
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Appendices
Appendix A
Definition 4
A set S of truth-functional connectives is functionally complete in an n-valued logic L (for finite n) iff any truth-function f in L different from the ones in S can be defined in terms of S (Theorem 5 provides a functional completeness criterion).
Theorem 5
Footnote 1 If \(A = \{0, i, 1\}\), with order \(0< i <1\), the set containing the monadic functions
and the binary functions \(\wedge \) and \(\vee \) is functionally complete.
Proof
Any n-ary connective g defined over \(A= \{1, i, 0\}\) can be expressed by means of the following normal form:
(for \(a_{k} \in A\)), where \(L_1 = f_{100}\), \(L_i = f_{010}\) and \(L_0 = f_{001}\) (\(f_{jkm}\) –for \(j,k,m\in \{0,i,1\}\)– represents the unary connective h such that \(h(1)=j\), \(h(i)=k\) and \(h(0)=m\)), and \(g(a_1, . . . , a_n)\) for any \(a_{1}, ..., a_{n}\) is 0, i or 1, that can be obtained from \(f_{111}\), \(f_{iii}\) and \(f_{000}\) applied to any variable \(x_1,...,x_n\). \(\square \)
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 K3\(^4\) (equivalent to RQML) is given in Theorem 39, and the functional completeness of \(\{\sim , \vee \}\) is given in (Post [16]) –see footnote 21.
Corollary 5.2
The set \(\{\lnot , \sim , \vee \}\) of the primitive connectives in RQML is functionally complete.
Proof
The function \(\vee \) is a primitive connective in RQML (see Section 2), while \(\wedge \) is definable in terms of \(\lnot \) and \(\vee \); also, the six monary functions in Theorem 5 can be defined in terms of \(\lnot , \sim \) and \(\vee \):
-
\(f_{111} = A \vee (\sim A \ \vee \sim \sim A)\)
-
\(f_{iii} \ = \ \sim f_{111}\)
-
\(f_{000}=\lnot f_{111}\)
-
\(f_{100}= \ \lnot \sim \sim [\sim (\sim A \ \vee \sim \sim A) \ \vee \sim \sim (\sim \ A \ \vee \sim \sim A)]\)
-
\(f_{010} = \ \lnot \sim \lnot (A\ \vee \sim A)\)
-
\(f_{001}=\ \lnot (A \ \vee \lnot \sim A)\)
Since \(\vee \) is a primitive connective in RQML and, on the other hand, the truth-functions \(\wedge \), \(f_{000}, f_{iii}, f_{111}, f_{100}, f_{010}\) and \(f_{001}\) are definable in terms of our primitive connectives, the set \(\{\lnot , \sim , \vee \}\) is functionally complete in RQML.Footnote 2 Naturally, it follows that the ten connectives taken as primitives by Reichenbach also form a functionally complete set. \(\square \)
Appendix B
Theorem 6
(Deduction Theorem) If \(\Sigma \vdash _{\mathcal {Q}} A\), where \(\Sigma = \{B_{1}, ... \ B_{n}\}\), then
Proof
First, suppose \(\Sigma \vdash _{\mathcal {Q}} A\). This corresponds to a closed tableau of type
Now consider a tableau starting with \(\bigwedge _{i\le n} B_i \rightarrow A,-\). Given that \(\Sigma \vdash _{\mathcal {Q}} A\), the tableau
closes too, since it will contain \(B_{i}, +\) for every \(B_{i} \in \Sigma \) and \(A, -\) (that is, the same elements as the tableau for \(\Sigma \vdash _{\mathcal {Q}} A\)). Hence, if \(\Sigma \vdash _{\mathcal {Q}} A\), where \(\Sigma = \{B_{1}, ... \ B_{n}\}\), then
as required. \(\square \)
Theorem 7
There exists deductions \(\Sigma \vdash _{\mathcal {Q}} A\), where \(\Sigma = \{B_{1}, ... \ B_{n}\}\), such that
or
Proof
By means of \(\mathcal {Q}\) calculus one can check that
-
\(A \vee A \vdash _{\mathcal {Q}} A\), but \(\nvdash _{\mathcal {Q}}(A \vee A) \rightsquigarrow A\)
-
\(A \supset \ \sim A \vdash _{\mathcal {Q}} \sim A\), but \(\nvdash _{\mathcal {Q}} (A \supset \ \sim A) \supset \ \sim A\)
Hence, Deduction Theorem only applies to \(\rightarrow \), and neither to \(\rightsquigarrow \) nor to \(\supset \). \(\square \)
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Caballero, P., Valencia, P. A Sound and Complete Tableaux Calculus for Reichenbach’s Quantum Mechanics Logic. J Philos Logic 53, 223–245 (2024). https://doi.org/10.1007/s10992-023-09730-7
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DOI: https://doi.org/10.1007/s10992-023-09730-7