Graded many-valued resolution with aggregation.

*(English)*Zbl 1054.03021The aim of the paper is to define a resolution truth function \(f_\lor(x,y)\), \(x,y\in[0,1]\), for using it in a many-valued resolution procedure. This function \(f_\lor(x,y)\) provides a truth value \(z\) of a resolvent \([(C\lor B), z]\) of a pair of signed many-valued clauses \([(C\lor A), x]\) and \([(B\lor\neg A), y]\), i.e. clauses which are true with respect to some truth values \(x\) and \(y\) from the unit interval \([0,1]\).

The goal of the paper is achieved by algebraic analysis of such many-valued operators as t-(semi)conorms. The authors introduce a notion of aggregation deficit which is based on the construction of conjunction from residuated implication proposed by D. Dubois and H. Prade [Stochastica 8, 267–279 (1984; Zbl 0581.03016)].

There exist highly related papers that are not cited here:

(*) “Automated deduction for many-valued logics”, by M. Baaz, F. Fermüller and G. Salzer [in: A. Robinson et al. (eds.), Handbook of automated reasoning. In 2 vols. Amsterdam: North-Holland/Elsevier. 1355–1402 (2001; Zbl 0992.03015)];

(**) “\(\alpha\)-resolution principle based on first-order lattice-valued logic \(\text{LF}(X)\)”, by Y. Xu, D. Ruan, E. Kerre and J. Liu [Inf. Sci. 132, 221–239 (2001; Zbl 0997.03006)].

For instance, in (**) the authors develop a resolution calculus for first-order logic having a lattice-ordered set of truth values. It would be very interesting to establish wether this earlier paper by Xu et al. contains a more general result than the present paper.

The authors obtain a sufficient result which could be applied in many-valued theorem proving. However, they use algebraic notation which is not common among logicians, see e.g. (*). Of course it is not a defect of the paper, but it shows that the authors work in a very particular branch of many-valued logic without knowing important results concerning many-valued logic. Also it testifies the lack of commonly adopted symbolic notation and terminology among many-valued logic researchers in general. That constitutes the main methodological problem at the present stage of development of many-valued logic.

The goal of the paper is achieved by algebraic analysis of such many-valued operators as t-(semi)conorms. The authors introduce a notion of aggregation deficit which is based on the construction of conjunction from residuated implication proposed by D. Dubois and H. Prade [Stochastica 8, 267–279 (1984; Zbl 0581.03016)].

There exist highly related papers that are not cited here:

(*) “Automated deduction for many-valued logics”, by M. Baaz, F. Fermüller and G. Salzer [in: A. Robinson et al. (eds.), Handbook of automated reasoning. In 2 vols. Amsterdam: North-Holland/Elsevier. 1355–1402 (2001; Zbl 0992.03015)];

(**) “\(\alpha\)-resolution principle based on first-order lattice-valued logic \(\text{LF}(X)\)”, by Y. Xu, D. Ruan, E. Kerre and J. Liu [Inf. Sci. 132, 221–239 (2001; Zbl 0997.03006)].

For instance, in (**) the authors develop a resolution calculus for first-order logic having a lattice-ordered set of truth values. It would be very interesting to establish wether this earlier paper by Xu et al. contains a more general result than the present paper.

The authors obtain a sufficient result which could be applied in many-valued theorem proving. However, they use algebraic notation which is not common among logicians, see e.g. (*). Of course it is not a defect of the paper, but it shows that the authors work in a very particular branch of many-valued logic without knowing important results concerning many-valued logic. Also it testifies the lack of commonly adopted symbolic notation and terminology among many-valued logic researchers in general. That constitutes the main methodological problem at the present stage of development of many-valued logic.

Reviewer: Vladimir Komendantsky (Cork)

##### MSC:

03B50 | Many-valued logic |

03B35 | Mechanization of proofs and logical operations |

03B52 | Fuzzy logic; logic of vagueness |

68T15 | Theorem proving (deduction, resolution, etc.) (MSC2010) |

##### Keywords:

aggregation operators; triangular norms; additive generators; residual complicators; many-valued resolution; truth-functional fuzzy logic
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\textit{D. Smutná-Hliněná} and \textit{P. Vojtáš}, Fuzzy Sets Syst. 143, No. 1, 157--168 (2004; Zbl 1054.03021)

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