<p>We consider six variations of the linearity axioms along with well-known logical axioms. Then we completely solve both of the hierarchy of the intermediate propositional logics and the hierarchy of semi-classical arithmetic with respect to those logical axioms. In particular, the schema of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Sigma _{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Σ</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>-substitutions of a variation of the linearity axiom forms a new class in the hierarchy of semi-classical arithmetic. Our investigations reveal that the difference between the acceptable rules for the entailment relation for intermediate propositional logics and those for arithmetic sometimes yields a crucial difference in the hierarchy.</p>