Subintuitionistic logics and their modal companions: a nested approach

Figures & data

Figure 1. The calculus $\mathbf{NSI}$.

Figure 2. The cut rule.

Figure 3. Admissible structural rules.

Table

$\mathbf{GSIK}$
Initial Sequents
Structural rules
Logical rules

Table

Initial Sequents
Logical Rules
Modal Rules
The calculus $\mathbf{NSS}\mathbf{4}$

Table

NSI	NSIK, NSIT, NSIK4	NSIB
$P^{\ast }=\mathrm{◻P}$	$P^{\ast }=P$	$P^{\ast }=\mathrm{◻P}\wedge P$
$(A\wedge B{)}^{\ast }=A^{\ast }\wedge B^{\ast }$	$(A\wedge B{)}^{\ast }=A^{\ast }\wedge B^{\ast }$	$(A\wedge B{)}^{\ast }=A^{\ast }\wedge B^{\ast }$
$(A\vee B{)}^{\ast }=A^{\ast }\vee B^{\ast }$	$(A\vee B{)}^{\ast }=A^{\ast }\vee B^{\ast }$	$(A\vee B{)}^{\ast }=A^{\ast }\vee B^{\ast }$
$(A\to B{)}^{\ast }=◻(A^{\ast }\to B^{\ast })$	$(A\to B{)}^{\ast }=◻(A^{\ast }\to B^{\ast })$	$(A\to B{)}^{\ast }=◻(A^{\ast }\to B^{\ast })$

Figure 4. The calculus $\mathbf{NSIC}$.