%0 Journal Article
%T Equality propositional logic and its extensions
%J Iranian Journal of Fuzzy Systems
%I University of Sistan and Baluchestan
%Z 1735-0654
%A Gao, X. L.
%A Xin, X. L.
%D 2019
%\ 10/01/2019
%V 16
%N 5
%P 125-137
%! Equality propositional logic and its extensions
%K Equality algebra
%K Equality propositional logic
%K completeness
%K Delta equality propositional logic
%R 10.22111/ijfs.2019.4912
%X We introduce a new formal logic, called equality propositional logic. It has two basic connectives, $boldsymbol{wedge}$ (conjunction) and $equiv$ (equivalence). Moreover, the $Rightarrow$ (implication) connective can be derived as $ARightarrow B:=(Aboldsymbol{wedge}B)equiv A$. We formulate the equality propositional logic and demonstrate that the resulting logic has reasonable properties such as Modus Ponens(MP) rule, Hypothetical Syllogism(HS) rule and completeness, etc. Especially, we provide two ways to prove the completeness of this logic system. We also introduce two extensions of equality propositional logic. The first one is involutive equality propositional logic, which is equality propositional logic with double negation. The second one adds prelinearity which is rich enough to enjoy the strong completeness property. Finally, we introduce additional connective $Delta$(delta) in equality propositional logic and demonstrate that the resulting logic holds soundness and completeness.
%U http://ijfs.usb.ac.ir/article_4912_f333b256a3857446705654eb1cb0a78a.pdf