Continuing (Fuchino et al. in Arch Math Log, 2020. https://doi.org/10.1007/s00153-020-00730-x), we study the Strong Downward Löwenheim–Skolem Theorems (SDLSs) of the stationary logic and their variations. In Fuchino et al. (2020) it has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters \(\textsf {SDLS}({\mathcal {L}}^{\aleph _0}_{stat},{<}\,\aleph _2)\) down to \({<}\,\aleph _2\) is equivalent to the conjunction of CH and Cox’s Diagonal Reflection Principle for internally clubness. We show that the SDLS for the stationary logic without weak second-order parameters \(\textsf {SDLS}^-({\mathcal {L}}^{\aleph _0}_{stat},{<}\,2^{\aleph _0})\) down to \({<}\,2^{\aleph _0}\) implies that the size of the continuum is \(\aleph _2\). In contrast, an internal interpretation of the stationary logic can satisfy the SDLS down to \({<}\,2^{\aleph _0}\) under the continuum being of size \(>\aleph _2\). This SDLS is shown to be equivalent to an internal version of the Diagonal Reflection Principle down to an internally stationary set of size \({<}\,2^{\aleph _0}\). We also consider a

继续（Fuchino等人，《 Arch Math Log》，2020年。https://doi.org/10.1007/s00153-020-00730-x），我们研究了平稳逻辑的强向下Löwenheim-Skolem定理（SDLSs）及其变化。在Fuchino等。（2020）已证明具有弱二阶参数\（\ textsf {SDLS}（{\ mathcal {L}} ^ {\ aleph _0} _ {stat}，{<}的普通平稳逻辑的SDLS \，\ aleph _2）\）降到\（{<} \，\ aleph _2 \）等效于CH和Cox的内部对角线反射对角线反射原理的结合。我们证明了没有弱二阶参数\（\ textsf {SDLS} ^-（{\ mathcal {L}} ^ {\ aleph _0} _ {stat}，{<} \，2 ^ {\ aleph _0}）\）降到\（{<} \，2 ^ {\ aleph _0} \）表示连续体的大小为\（\ aleph _2 \）。相反，在大小为（（> \ aleph _2 \）的连续体下，固定逻辑的内部解释可以满足SDLS降至\（{<} \，2 ^ {\ aleph _0} \）。该SDLS等效于对角反射原理的内部版本，直至内部固定大小为\（{<} \，2 ^ {\ aleph _0} \）为止。我们还考虑了