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The Class of All Natural Implicative Expansions of Kleene’s Strong Logic Functionally Equivalent to Łkasiewicz’s 3-Valued Logic Ł3

Abstract

We consider the logics determined by the set of all natural implicative expansions of Kleene’s strong 3-valued matrix (with both only one and two designated values) and select the class of all logics functionally equivalent to Łukasiewicz’s 3-valued logic Ł3. The concept of a “natural implicative matrix” is based upon the notion of a “natural conditional” defined in Tomova (Rep Math Log 47:173–182, 2012).

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Acknowledgements

This work is supported by the Spanish Ministry of Economy, Industry and Competitiveness [FFI2017-82878-P]. We sincerely thank three referees of the Journal of Logic, Language and Information for their comments and suggestions on a previous draft of this paper.

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Correspondence to Gemma Robles.

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Appendices

A Appendix I: The 27 Tables in TI

The 54 tables in TI are divided in two groups: 27 tables in TIII and 27 tables in TIV. The 27 tables in TIII are the following (designated values are starred). We refer to the tables in this set by t\(n_{2}\) or simply by t\( n (1\le n\le 27)\).

$$\begin{aligned}&\hbox {t1.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t2.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t3.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array}\\&\hbox {t4.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 1 &{}\quad 2 \end{array} \quad \hbox {t5.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 1 &{}\quad 2 \end{array} \quad \hbox {t6.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 1 &{}\quad 2 \end{array}\\&\hbox {t7.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 2 &{}\quad 2 \end{array} \quad \hbox {t8.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 2 &{}\quad 2 \end{array} \quad \hbox {t9.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 2 &{}\quad 2 \end{array}\\&\hbox {t10.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t11.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t12.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array}\\&\hbox {t13.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 1 &{}\quad 2 \end{array} \quad \hbox {t14.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 1 &{}\quad 2 \end{array} \quad \hbox {t15.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 1 &{}\quad 2 \end{array}\\&\hbox {t16.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 2 &{}\quad 2 \end{array} \quad \hbox {t17.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 2 &{}\quad 2 \end{array} \quad \hbox {t18.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 2 &{}\quad 2 \end{array}\\&\hbox {t19.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t20.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t21.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array}\\&\hbox {t22.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 1 &{}\quad 2 \end{array} \quad \hbox {t23.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 1 &{}\quad 2 \end{array} \quad \hbox {t24.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 1 &{}\quad 2 \end{array}\\&\hbox {t25.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 2 &{}\quad 2 \end{array} \quad \hbox {t26.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 2 &{}\quad 2 \end{array} \quad \hbox {t27.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 2 &{}\quad 2 \end{array} \end{aligned}$$

Then, the 27 tables in TIV are obtained by replacing \(1\rightarrow 1=1\) by \( 1\rightarrow 1=2\) in each one of the tables recorded in the list above. We refer to the tables in this set by t\(n_{1} (1\le n\le 27)\). Thus, for example, t8\(_{1}\) is:

$$\begin{aligned} t8_{1}. \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ {*}1 &{}\quad 0 &{}\quad 1 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 2 &{}\quad 2 \end{array} \end{aligned}$$

B Appendix II: The 54 Tables in TII

The 27 tables in which \(b_{1}=0\) are the following (designated values are starred). We refer to the tables in this set by \(tn_{0} (1\le n\le 27)\).

$$\begin{aligned}&\hbox {t1.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ 1 &{}\quad 0 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t2.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ 1 &{}\quad 0 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t3.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ 1 &{}\quad 0 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array}\\&\hbox {t4.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ 1 &{}\quad 0 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t5.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ 1 &{}\quad 0 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t6.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ 1 &{}\quad 0 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array}\\&\hbox {t7.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ 1 &{}\quad 0 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t8.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ 1 &{}\quad 0 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t9.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ 1 &{}\quad 0 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array}\\&\hbox {t10.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ 1 &{}\quad 1 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t11.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ 1 &{}\quad 1 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t12.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ 1 &{}\quad 1 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array}\\&\hbox {t13.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ 1 &{}\quad 1 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t14.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ 1 &{}\quad 1 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t15.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ 1 &{}\quad 1 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array}\\&\hbox {t16.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ 1 &{}\quad 1 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t17.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ 1 &{}\quad 1 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t18.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ 1 &{}\quad 1 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array}\\&\hbox {t19.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t20.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t21.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 0 &{}\quad 2 \\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array}\\&\hbox {t22.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t23.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t24.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array}\\&\hbox {t25.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 0 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t26.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \quad \hbox {t27.} \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ {*}2 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{array} \end{aligned}$$

Then, the 27 tables in which \(b_{1}=1\) are obtained by replacing \( 2\rightarrow 1=1\) by \(2\rightarrow 1=0\) in each one of the tables recorded in the list above. We refer to the tables in this set by \(tn_{1} (1\le n\le 27)\). Thus, for example, t17\(_{1}\) is:

$$\begin{aligned} \hbox {t17}_{1}. \begin{array}{l|lll} \rightarrow &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ \hline 0 &{}\quad 2 &{}\quad 2 &{}\quad 2 \\ 1 &{}\quad 1 &{}\quad 2 &{}\quad 1 \\ {*}2 &{}\quad 0 &{}\quad 1 &{}\quad 2 \end{array} \end{aligned}$$

C Appendix III

It is proved that all implicative expansions of MK3 defined by Tomova are not relevant logics in the sense that they lack the “variable-sharing property” (vsp). However, 11 of the implicative expansions of MK3 we have introduced are relevant, as they have the vsp. Also, we examine the behavior of the 108 logics considered in the present paper w.r.t. two vsp-related properties, the “quasi variable-sharing property” (qvsp) and the “weak relevant property” (wrp).

The vsp, the qvsp and the wrp are defined as follows (cf. Definition 2.3).

Definition C.1

(Variable-sharing property—vsp) Let L be a logic defined upon the matrix M. L has the “variable-sharing property” (vsp) if in all M-valid wffs of the form \(A\rightarrow B\), A and B share at least a propositional variable.

Definition C.2

(Quasi variable-sharing property—qvsp) Let L be a logic defined upon the matrix M. L has the “quasi variable-sharing property” (qvsp) if in all M-valid wffs of the form \(A\rightarrow B\), A and B share at least a propositional variable or else both A and B are M-valid.

Definition C.3

(Weak relevant property—wrp) Let L be a logic defined upon the matrix M. L has the “weak relevant property” (wrp) if in all M-valid wffs of the form \(A\rightarrow B\), A and B share at least a propositional variable or else both \(\lnot A\) and B are M-valid.

To the best of our knowledge, the qvsp is a new property introduced here for the first time. As far as we know, the wrp is first defined in Anderson and Belnap (1975) (cf. p. 117), while the vsp is a well-known property (cf. Belnap (1960), Anderson and Belnap (1975) and references in the last item). Nevertheless, notice that the vsp and the wrp are customarily defined leaning upon the notion of L-theoremhood, not upon that of M-validity. Of course, both versions of the vsp and the wrp are equivalent in the presence of a soundness and completeness theorem, as it is the case with the logics mentioned in the first paragraph of this third appendix (cf. conclusion 7 below).

Next, we proceed to the examination of the 108 logics considered in the paper w.r.t. the properties just defined.

Proposition C.4

(TII-logics, the vsp, the qvsp and the wrp) Let L be a TII-logic-Then, L lacks the vsp, the qvsp and the wrp.

Proof

Immediate since the wff \(\lnot (p\rightarrow p)\rightarrow \lnot (q\rightarrow q)\) is, for any propositional variables p, q, M-valid in any implicative expansion of MK3, M, built upon any table in TII. \(\square \)

Proposition C.5

(TIII-logics, the vsp, the qvsp and the wrp) Let L be a TIII-logic. Then, L lacks the vsp, the qvsp and the wrp.

Proof

Immediate by using the wff \(\lnot (p\rightarrow p)\rightarrow \lnot (q\rightarrow q)\) similarly as in the precedent proposition. \(\square \)

Thus, concerning the three properties we are interested in in this appendix, we are left with the 27 tables in TIV. In what follows, some facts about TIV-logics w.r.t. these properties are proven. Let us refer by TIV\(_{i} (1\le i\le 27)\) to the TIV-logic built upon ti.

Proposition C.6

(TIV-logics and the vsp I) Let L be the logic TIV\(_{i}\) where \(i\in \{5,6,8,9,11,12,14,15,17,18,20,21,23,24,26,27\}\). Then, L lacks the vsp.

Proof

  1. (a)

    TIV\(_{i}\) (\(i\in \{5,6,8,9\}\)). By using the wff \((p\rightarrow p)\rightarrow (q\rightarrow q)\) (p, q are distinct propositional variables) verified by tables t5, t6, t8 and t9.

  2. (b)

    TIV\(_{i}\) (\(i\in \{11,12,14,15,17,18,20,21,23,24,26,27\}\)). By using the wff \(\lnot (p\rightarrow p)\rightarrow (q\rightarrow q)\) (p, q are distinct propositional variables) verified by tables t11, t12, t14, t15, t17, t18, t20, t21, t23, t24, t26 and t27. \(\square \)

Proposition C.7

(TIV-logics and the qvsp I) Let L be the logic TIV\(_{i}\) where \(i\in \{11,12,14,15,17,18,20,21,23,24,26,27\}\). Then, L lacks the qvsp.

Proof

By using the wff \((p\wedge \lnot p)\rightarrow (q\vee \lnot q)\) (p, q are distinct propositional variables) similarly as in the preceding proposition. \(\square \)

Proposition C.8

(TIV-logics and the wrp I) Let L be the logic TIV\(_{i}\) where \(i\in \{5,6,8,9,14,15,17,18,23,24,26,27\}\). Then, L lacks the wrp.

Proof

By using the wff \((p\rightarrow p)\rightarrow (q\rightarrow q)\) (p, q are distinct propositional variables) similarly as in Propositions C7 and C8. \(\square \)

On the other hand, we have the following propositions.

Proposition C.9

(TIV-logics and the vsp II) Let L be the logic TIV\(_{i}\) where \(i\in \{1,2,3,4,7,10,13,16,19,22,25\}\). Then, L has the vsp.

Proof

  1. (a)

    TIV\(_{i}\) (\(i\in \{1,4,7,10,13,16,19,22,25\}\)). Let M be the implicative expansion of MK3 determining the logic L. Suppose that there are wffs A and B such that \(A\rightarrow B\) is M-valid but A and B do not share propositional variables. Let I be an M-interpretation assigning 1 (resp., 0) to each propositional variable in A (resp., B). Then \(I(A)=1\) and \( I(B)\in \{0,2\}\), since \(\{1\}\) and \(\{0,2\}\) are closed under \(\rightarrow ,\wedge ,\vee \) and \(\lnot \). Consequently, \(I(A\rightarrow B)=0\), contradicting the M-validity of \(A\rightarrow B\).

  2. (b)

    TIV\(_{i}\) (\(i\in \{1,2,3\}\)). The proof is similar to that of case (a) by using now the fact that \(f_{\rightarrow }(0,1)=f_{\rightarrow }(2,1)=0\). \(\square \)

Proposition C.10

(TIV-logics and the qvsp II) Let L be the logic TIV\(_{i}\) where \(i\in \{5,6,8,9\}\). Then, L has the qvsp.

Proof

Let M be the implicative expansion of MK3 determining the logic L. Suppose that there are wffs A and B not having propositional variables in common and such that \(A\rightarrow B\) is M-valid but either A or B is not.

  1. (a)

    A is not M-valid: Then, there is an M-interpretation I such that \( I(A)=0\). Let \(I^{\prime }\) be exactly as I except that for each propositional variable p in B, \(I^{\prime }(p)=1\). Clearly, \(I^{\prime }(B)=1\) since \(\{1\}\) is closed under \(\rightarrow ,\wedge ,\vee \) and \( \lnot \), and \(I^{\prime }(A)=0\) since A and B do not share propositional variables. Then, \(I^{\prime }(A\rightarrow B)=0\), contradicting the M-validity of A.

  2. (b)

    B is not M-valid: Then, there is an M-interpretation I such that \( I(B)=0\). Let \(I^{\prime }\) be exactly as I except that for each propositional variable p in A, \(I^{\prime }(p)=1\). Similarly as in case (a), \(I^{\prime }(A)=1\), \(I^{\prime }(B)=0\), and then \(I^{\prime }(A\rightarrow B)=0\), contradicting the M-validity of \(A\rightarrow B\). \(\square \)

Proposition C.11

(TIV-logics and the wrp II) Let L be the logic TIV\(_{i}\) where \(i\in \{11,12,20,21\}\). Then, L has the wrp.

Proof

Let M be the implicative expansion of MK3 determining the logic L. Suppose that there are wffs A and B not having propositional variables in common and such that \(A\rightarrow B\) is M-valid but either \(\lnot A\) or B is not. The proof that, given the suppositions just stated, \(A\rightarrow B\) is actually not M-valid is similar to that of the preceding proposition leaning now on the fact that \(f_{\rightarrow }(1,0)=f_{\rightarrow }(2,1)=0\). \(\square \)

The results we have obtained are summarized below.

  1. 1.

    All TII-logics and TIII-logics lack the vsp, the qvsp and wrp (Propositions C4 and C5).

  2. 2.

    The only TI-logics having the vsp are TIV-logics. In particular, the following 11 TIV-logics: TIV\(_{i}\) where \(i\in \{1,2,3,4,7,10,13,16,19,22,25\}\) (Propositions C4, C5, C6 and C9).

  3. 3.

    The only TI-logics having the qvsp are TIV-logics. In particular, the following 4 TIV-logics, in addition to the 11 TIV-logics having the vsp: TIV\( _{i}\) where \(i\in \{5,6,8,9\}\) (Propositions C4, C5, C7 and C10).

  4. 4.

    The only TI-logics having the wrp are TIV-logics. In particular, the following 4 TIV-logics, in addition to the 11 TIV-logics having the vsp: TIV\( _{i}\) where \(i\in \{11,12,20,21\}\) (Propositions C5, C6, C8 and C11).

  5. 5.

    Tomova’s 30 tables are generally defined in tables TI\(^{\prime }\) and TII\(^{\prime }\). The tables with two designated values are also defined in TIII and TIV. Tomova’s 24 tables in TIII and TIV are the following: t11, t12, t14, t15, t17, t18, t20, t21, t23, t24, t26 and t27. All these TIII-logics and TIV-logics lack the vsp and the qsvp. Also, all lack the wrp, except TIV\(_{11}\), TIV\(_{12}\), TIV\(_{20}\) and TIV\(_{21}\) (cf. 1, 2, 3 and 4 above; notice that TIV\(_{21}\) is the quasi-relevant logic RM3—cf. Brady (1982), Anderson and Belnap (1975) and references therein).

  6. 6.

    The 78 new tables we have introduced together with Tomova’s 30 tables are generally defined in tables TI (30 new tables) and TII (48 new tables). The tables with two designated values are also defined in TIII and TIV. These 30 tables are the following: t1 through t9, t10, t13, t16, t19, t22 and t25. Of these, the new 15 TIV-logics are the only members in the set of 108 logics considered in this paper with either the vsp or the qvsp: TIV\(_{i}\) (\(i\in \{1,2,3,4,7,10,13,16,19,22,25\}\)) has the vsp; TIV\(_{i}\) (\( i\in \{5,6,8,9\}\)) has the qvsp.

  7. 7.

    As pointed out in the Concluding remarks (Sect. 8), Hilbert-style axiomatic systems corresponding to Tomova’s logics and the 15 new TIV-logics we have introduced are defined in a general and unified way in Robles and Méndez (2019a), Robles et al. (2019) and Robles and Méndez (in preparation).

The appendix is ended by remarking that none of the new 15 TIV-logics (with either the vsp or the qvsp) we have introduced is included in RM3, the 3-valued extension of the logic R-Mingle, commonly considered as the strongest logic in the relevant logic family (cf. Avron 1991, p. 276). (But notice that RM3—cf. table TIV\(_{21}\)—does not have the vsp.)

Proposition C.12

(The 15 new TIV-logics are not incl. in RM3) Let L be the logic TIV\(_{i}\) where \(i\in \{1,2,3,4,5,6,7,8,9,10,13,16,19,22,25\}\). Then, L is not included in RM3.

Proof

Let L be any of the TIV\(_{i}\)-logics except TIV\(_{3}\), TIV\(_{6}\), TIV\(_{9}\). Then, the rule “If \(A\wedge \lnot A\), then \(\lnot (A\rightarrow B)\)” holds in L. Next, “If \( B\wedge \lnot B\), then \(\lnot (A\rightarrow B)\)” holds in TIV\(_{3}\) and TIV\(_{6}\) and the thesis \((A\wedge B)\rightarrow (A\rightarrow B)\) holds in TIV\(_{9}\). However, both aforementioned rules and the thesis just quoted fail in RM3. \(\square \)

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Robles, G., Méndez, J.M. The Class of All Natural Implicative Expansions of Kleene’s Strong Logic Functionally Equivalent to Łkasiewicz’s 3-Valued Logic Ł3. J of Log Lang and Inf (2019). https://doi.org/10.1007/s10849-019-09306-2

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Keywords

  • Many-valued logic
  • 3-Valued logic
  • Kleene’s strong logic
  • Łukasiewicz’s 3-valued logic
  • Functional equivalence
  • Natural conditionals

Mathematics Subject Classification

  • 03B47