$$\alpha $$ α -Paramodulation method for a lattice-valued logic $$L_nF(X)$$

In this paper, $$\alpha $$ α -paramodulation and $$\alpha $$ α -GH paramodulation methods are proposed for handling logical formulae with equality in a lattice-valued logic $$L_nF(X)$$ L n F ( X ) , which has unique ability for representing and reasoning uncertain information from a logical point of view. As an extension of the work of He et al. (in: 2015 10th international conference on intelligent systems and knowledge engineering (ISKE), pp 18–20. IEEE, 2015; Uncertainty modelling in knowledge engineering and decision making: proceedings of the 12th international FLINS conference, pp 477–482. World Scientific, 2016), a new form of $$\alpha $$ α -equality axioms set is proposed. The equivalence between $$\alpha $$ α -equality axioms set and $$E_{\alpha }$$ E α -interpretation in $$L_nF(X)$$ L n F ( X ) with an appropriate level is also established, which may provide a key foundation for equality reasoning in lattice-valued logic. Based on its equivalence, $$E_{\alpha }$$ E α -unsatisfiability equivalent transformation is given. Furthermore, $$\alpha $$ α -paramodulation and its restricted method (i.e., $$\alpha $$ α -GH paramodulation) are given. The soundness and completeness of the proposed methods are also examined.

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