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Calculi for Many-Valued Logics

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Abstract

We present a number of equivalent calculi for many-valued logics and prove soundness and strong completeness theorems. The calculi are obtained from the truth tables of the logic under consideration in a straightforward manner and there is a natural duality among these calculi. We also prove the cut elimination theorems for the sequent-like systems.

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Notes

  1. We use the term “\(\varvec{L}\)-sequent” to distinguish between sequents from [2] and “just” sequents in our calculi.

  2. In [8, 15, 16] such expressions are called “matrices.”

  3. The value \(v(\varphi )\) of a formula \(\varphi \) under a valuation v is defined by a standard recursion from the truth tables.

  4. Actually, the rule in [2, Definition 3.3] is a bit different, but follows from this one and (2.3) below.

  5. In [2], this rule is called “cut” and in [5], it is called “coordination.” We reserve the term “cut” for a different rule that looks like an ordinary cut.

  6. This semantics is common to all calculi under consideration and will be referred to as just \({\varvec{MVL}}\) semantics.

  7. Here the subscript “A” stands for the axiom description of \({\varvec{MVL}}\).

  8. In fact, the natural deduction for \(\varvec{L}\) also interprets in \(\varvec{MVL}_{\varvec{A}}\) (and all other calculi considered in this paper) in the following manner: a set of formulas \(X_n\) is derivable from \(X_1 \, | \, \cdots \, | \, X_{n-1}\) in the former if and only if the sequent \(\bigcup \nolimits _{k=1}^{n-1} X_k \times \{ k \} \rightarrow X_n \times \{ n \}\), where \(X_k \times \{ k \} = \{ (\varphi ,k), \varphi \in X_k \}\), is derivable in the latter.

  9. We say that a labelled formula \((\varphi ,k)\) is a subformula of a labelled formula \((\varphi ^\prime , k^\prime )\), if \(\varphi \) is a subformula of \(\varphi ^\prime \).

  10. In other words, v satisfies a sequent \(\Gamma \rightarrow \Delta \), if the metavalue of the classical metasequent \(\{ v(\varphi ) = v_k : (\varphi ,k) \in \Gamma \} \rightarrow \{ v(\varphi ) = v_k : (\varphi ,k) \in \Delta \}\) is “true.”

  11. In other words, \(\varvec{\Gamma }\) is complete, cf. [8, paragraph 3.63] and the definition of the “classical” negation completeness.

  12. That is, (4.1) are the rules of introduction for the poly-sequent calculus from [5].

  13. This theorem provides us with an alternative proof of (weak) decidability of \(\varvec{MVL}_{\varvec{A}}\), cf. the note following Theorem 3.11.

  14. Note that \(*(\varphi _1,\ldots ,\varphi _\ell )\) may be introduced into the succeedent, only.

  15. That is, Sect. 5.1 contains the distributively dual counterparts of the respective results from Sects. 3.1 and 4.1.

  16. Note that \(s\) depends both on \(*\) and k.

  17. Note the form of \(\Theta _q\): for each \(j = 1,\ldots ,\ell \) it contains exactly one labelled formula with the first component \(\varphi _j\).

  18. Recall that \((\varphi _{j_q} ,k_{j_q}) \in \varvec{\Gamma }\) and \((\varphi _{j_q} , k_{j_q,q}) \in \Theta _q\), see (5.2) and (5.3).

  19. Since \(k^\prime \not \in K\) and \(k^\prime \ne k^{\prime \prime }\), \(k^\prime \not \in K \cup \{ k^{\prime \prime } \}\). Thus, \(K \cup \{ k^{\prime \prime } \} \ne \{ 1,\ldots , n\}\).

  20. Note that \(k^\prime \notin K\).

  21. This is why we need (3.10) in \({\varvec{MVL}_{\varvec{R}\varvec{S}\varvec{D}}}\) and \({\varvec{MVL}}_{\varvec{R}\varvec{D}\varvec{D}\varvec{S}\varvec{D}}\). Note that (3.4) is not sufficient for such cut elimination. See also Remark 3.5 in the end of Sect. 3.1.

  22. That is, without rules (5.17) in \({\varvec{MVL}_{\varvec{R}\varvec{S}\varvec{D}}}\) and without rules (5.19) in \({\varvec{MVL}}_{\varvec{R}\varvec{D}\varvec{D}\varvec{S}\varvec{D}}\).

  23. Recall that we are in the case of (5.21) with \(K \cup \{ k^\prime \} = \{1,\ldots ,n \}\).

  24. Namely, if \(\Theta = \{ (\varphi _1,k_1),\dots ,(\varphi _\ell ,k_\ell ) \}\), then k is such that \(*(v_{k_1},\ldots ,v_{k_\ell }) = v_k\).

  25. Cf. the definition on the top of p. 66 in [10].

  26. Whereas (3.2) is an axiom of \(\varvec{MVL}_{\varvec{A}}\) and (5.8) is an axiom of \({\varvec{MVL}}_{\varvec{A}\varvec{D}\varvec{D}}\), they are not axioms of any of the logics \({\varvec{MVL}}\) defined in the beginning of this section.

  27. That is, \(\{ \Theta _1,\ldots ,\Theta _{s} \} = (*(\varphi _1,\ldots ,\varphi _\ell ),k)^{-1}\).

  28. Cf. rules (5.17) for introduction of \(*\) to the antecedent and the elimination rules in [2, Definition 4.1] and [5, Sect. 3.1].

  29. This is a generalization of general elimination rules to multi-valued logics, cf. [4].

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  1. Department of Computer Science, Technion – Israel Institute of Technology, Haifa, 3200003, Israel

    Michael Kaminski & Nissim Francez

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  1. Michael Kaminski
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  2. Nissim Francez
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Correspondence to Michael Kaminski.

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Kaminski, M., Francez, N. Calculi for Many-Valued Logics. Log. Univers. 15, 193–226 (2021). https://doi.org/10.1007/s11787-021-00274-5

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