For (finitary) deductive systems, we formulate a signature‐independent abstraction of the weak excluded middle law (WEML), which strengthens the existing general notion of an inconsistency lemma (IL). Of special interest is the case where a quasivariety K$\mathsf {K}$ algebraizes a deductive system ⊢. We prove that, in this case, if ⊢ has a WEML (in the general sense) then every relatively subdirectly irreducible member of K$\mathsf {K}$ has a greatest proper K$\mathsf {K}$‐congruence; the converse holds if ⊢ has an inconsistency lemma. The result extends, in a suitable form, to all protoalgebraic logics. A super‐intuitionistic logic possesses a WEML iff it extends KC$\mathsf {KC}$. We characterize the IL and the WEML for normal modal logics and for relevance logics. A normal extension of sans-serifSsans-serif4$\mathsf {S4}$ has a global consequence relation with a WEML iff it extends sans-serifSsans-serif4.sans-serif2$\mathsf {S4.2}$, while every axiomatic extension of sans-serifRsans-serift$\mathsf {R^t}$ with an IL has a WEML.
We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of
$\omega $
implies that the modal logic
$\mathbf {S4.1.2}$
is complete with respect to the Čech–Stone compactification of the natural numbers, the space
$\beta \omega $
. In the same fashion we prove that the modal logic
$\mathbf {S4}$
is complete with respect to the space
$\omega ^*=\beta \omega \setminus \omega $
. This improves the results of G. Bezhanishvili and J. Harding in [4], where the authors prove these theorems under stronger assumptions (
$\mathfrak {a=c}$
). Our proof is also somewhat simpler.
We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of ω implies that the modal logic S4.1.2 is complete with respect to the Čech-Stone compactification of the natural numbers, the space βω. In the same fashion we prove that the modal logic S4 is complete with respect to the space ω * = βω \ω. This improves the results of G. Bezhanishvili and J. Harding in [4], where the authors prove these theorems under stronger assumptions (a = c). Our proof is also somewhat simpler.
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