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چکیده
Our article ‘‘A map of dependencies among three-valued logics’’ (Information Sciences, vol. 250, 161–177, 2013) contains some typing mistakes that may affect a proper understanding of the paper. They are corrected here. Table 1 (Table 8 in [1]) Relations among implications. N. How it can be obtained Name 1 Residuation applied to p⁄1q = :((:p? 12:q)? 12:q) 2 (p? 3q)^(:q? 3:p), (p? 4q)^(:q? 4:p) Sobociński 3 :p_(p? 2q) 4 q_(p? 1q), q_(p? 2q) Jaśkowski 5 :p_(p? 1q), (p? 3q)_(:q? 3:p), q_(p? 3q) (strong) Kleene (p? 4q)_(:q? 4:p), :p_(p? 4q) 6 In terms of ?11 (see Remark 4 of [1]) Sette Residuation applied to p⁄6q = :((:p? 8:q)? 8:q) 7 :p_(p? 6q) 8 (p? 6q)_(:q? 6:p), (p? 7q)_(:q? 7:p) p? 7(p? 7q), (p? 9q)_(:q? 9:p) (p? 12q)_(:q? 12:p) 9 p? 11(p? 11q), q_(p? 12q) Nelson 10 q_(p? 14q) Gödel 11 (p? 7q)^(:q? 7:p), (p? 9q)^(:q? 9:p) Łukasiewicz (p? 10q)_(:q? 10:p), :p_(p? 10q), (p? 13q)_(:q? 13:p), p_(p? 13q) 12 p? 12q = p? 13(p? 13q) Bochvar external 13 :p_(p? 14q) 14 (p? 6q)^(:q? 6:p), (p? 10q)^(:q? 10:p) Gaines–Rescher (p? 12q)^(:q? 12:p), (p? 13q)^(:q? 13:p) ⇑ Corresponding author at: DISCo, Università di Milano-Bicocca, Viale Sarca 336/14, I-20126 Milano, Italy. E-mail addresses: [email protected] (D. Ciucci), [email protected] (D. Dubois). Definition 3 of our paper [1], Section 2.1.3, p. 166, should read as follows: Definition 3. [2] A binary operator ⁄ on a finite scale {F < x1 < < T} is named t-operator if it is associative, commutative, such that F F = F,T T = T and it satisfies 1-smoothness: xi xjÿ1 6 xi xj and if xi xj = xk then {xiÿ1 xj,xi xjÿ1} # {xk,xkÿ1}. Moreover in Table 8, due to the lack of horizontal lines separating the items in the published version, the left-hand column containing the numbering of implications has been erroneously shifted down and items in this column no longer correspond to the items in the two other columns. This is also due to the presence of a blank item in Table 8, line 1, column 2 of original manuscript, that looked misleadingly erroneous in the proofs. Below we provide a completed Table 8 where blank lines are filled with information present in Remark 4, p. 169 (for Sette implication 6) and also in Section 3.3 (for implications 1 and 6) of our paper [1] (see Table 1).
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