On canonicity and strong completeness conditions in intermediate propositional logics (Q1970920)

Necessary conditions for canonicity, strong completeness, \(\omega\)-canonicity and strong \(\omega\)-completeness of intermediate propositional logics are proved. These necessary conditions are formulated in algebraic-categorical terms (namely, using inverse limits of co-chain diagrams in the category of posets and order-preserving maps, the dual category of distributive lattices etc.). By applying these conditions, the following results are proved: (i) Gabbay-de Jongh logics \(D_n\) (of finite branching \(\leq n+1\), for \(n \geq 1\)) are not strongly \(\omega\)-complete; (ii) all intermediate logics axiomatized by one-variable formulas are not strongly complete, except the four logics axiomatized by the following formulas (these four logics are strongly complete and even canonical): \[ p \vee \neg p , \neg p \vee \neg \neg p , \neg \neg p \vee ( \neg \neg p \rightarrow p), ( \neg \neg p \rightarrow p) \vee (( \neg \neg p \rightarrow p) \rightarrow p \vee \neg p). \]