University of Sistan and BaluchestanIranian Journal of Fuzzy Systems1735-065416520191001Equality propositional logic and its extensions125137491210.22111/ijfs.2019.4912ENX. L.GaoSchool of Mathematics, Northwest University, Xi'an,710127, ChinaX. L.XinSchool of Mathematics, Northwest University, Xi'an,710127, ChinaJournal Article20180717We 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.https://ijfs.usb.ac.ir/article_4912_f333b256a3857446705654eb1cb0a78a.pdf