Sentence Satisfaction under one Assignment means Satisfaction under all Assignments

Last updated Nov 1, 2022

Let $\mathcal{L}$ be a Language and let $\phi$ be an $\mathcal{L}$-Sentence. Let $\mathcal{M}$ be an $\mathcal{L}$-structure Then The following are Equivalent:

1. $\mathcal{M} \models_{\sigma} \phi$ for some Assignment $\sigma$
2. $\mathcal{M} \models_{\sigma} \phi$ for all Assignments $\sigma$

TODO - (2) implies (1) easily. (1) implies (2) by the Coincidence Lemma. See Marker - An Invitation Mathematical Logic Book Corollary 1.30.