\( \mathcal{L} \)-formula implication
We wish to introduce an idea of implication between sets of \( \mathcal{L} \)-formula.
First, we must generalize the idea of truth/satisfaction that was introduced for \( \mathcal{L} \)-formula.
The idea of truth/satisfaction is recapped on the back side.
Model
Let \( \mathcal{L} \) be a first-order language. Let \( \mathfrak{U} \) be an \( \mathcal{L} \)-structure for \( \mathcal{L} \) and let \( \phi \) be an \( \mathcal{L} \)-formula of \( \mathcal{L} \).
If \( \mathfrak{U} \vDash \phi[s] \) for every [what?], then we say that \( \mathfrak{U} \) is a model of \( \phi \) (or \( \mathfrak{U} \) models \( \phi \)). We write \( \mathfrak{U} \vDash \phi \).
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, \( \Phi \), then we say \( \mathfrak{U} \) is a model of \( \Phi \), and we write \( \mathfrak{U} \vDash \Phi \).
If we compare two sets of \( \mathcal{L} \)-formula we arrive at a type of implication.
Logical implication, under a structure
Let \( \Delta \) and \( \Gamma \) 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 \( \mathfrak{U} \vDash \Delta \) implies \( \mathfrak{U} \vDash \Gamma \), we say that \( \Delta \) logically implies \( \Gamma \) under \( \mathfrak{U} \).
In other words, if whenever all the [what?] are satisfied by \( \mathfrak{U} \) so too are the [what?].
Finally, we remove the dependency on a specific structure.
Logical implication
Let \( \Delta \) and \( \Gamma \) 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 \( \Delta \) logically implies \( \Gamma \). We write this as \( \Delta \vDash \Gamma \).