Basic propositional logic and the weak excluded middle.

We study basic propositional logic |$\operatorname{\textbf{(BPC)}}$| augmented with the law of the weak excluded middle |$\operatorname{\textbf{(WEM)}}$|⁠, i.e. |$\operatorname{\textbf{BPW}} = \operatorname{\textbf{BPC}} + \operatorname{\textbf{WEM}}$|⁠. We show that the variety of the algebraic models of |$\operatorname{\textbf{BPW}}$| is canonical, and its Kripke completeness is proved via cononicity. Moreover, it is also proved that |$\operatorname{\textbf{BPW}}$| has the finite model property and is decidable. It is shown that |$\operatorname{\textbf{BPC}}$| and |$\operatorname{\textbf{BPW}}$| have the same behaviour on the |$\bot $| -free formulas and that |$\operatorname{\textbf{CPC}}$| and |$\operatorname{\textbf{BPW}}$| have the same behaviour on the negated formulas. [ABSTRACT FROM AUTHOR]