$\mathcal{L}$-formula implication

Model

Let $\mathcal{L}$ be a first-order language. Let $\mathfrak{U}$ be an $\mathcal{L}$-structure for $\mathcal{L}$ and let $\varphi$ be an $\mathcal{L}$-formula of $\mathcal{L}$.

If $\mathfrak{U}\models \varphi [s]$ for every [what?], then we say that $\mathfrak{U}$ is a model of $\varphi$ (or $\mathfrak{U}$ models $\varphi$). We write $\mathfrak{U}\models \varphi$.

Extendion to sets of $\mathcal{L}$-formula: if $\mathfrak{U}$ is a model for every $\mathcal{L}$-formula in a set of $\mathcal{L}$-formulas, $\mathrm{\Phi }$, then we say $\mathfrak{U}$ is a model of $\mathrm{\Phi }$, and we write $\mathfrak{U}\models \mathrm{\Phi }$.

Logical implication, under a structure

Let $\mathrm{\Delta }$ and $\mathrm{\Gamma }$ be two sets of $\mathcal{L}$-formulas from the same language $\mathcal{L}$. Let $\mathfrak{U}$ be an $\mathcal{L}$-structure for $\mathcal{L}$.

If $\mathfrak{U}\models \mathrm{\Delta }$ implies $\mathfrak{U}\models \mathrm{\Gamma }$, we say that $\mathrm{\Delta }$ logically implies $\mathrm{\Gamma }$ under $\mathfrak{U}$.

In other words, if whenever all the [what?] are satisfied by $\mathfrak{U}$ so too are the [what?].

Logical implication

Let $\mathrm{\Delta }$ and $\mathrm{\Gamma }$ be two sets of $\mathcal{L}$-formulas from the same language $\mathcal{L}$.

If [what?] for any [what?] of $\mathcal{L}$, then we say that $\mathrm{\Delta }$ logically implies $\mathrm{\Gamma }$. We write this as $\mathrm{\Delta }\models \mathrm{\Gamma }$.