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Math and science::Theory of Computation::Modal theory

# $$\mathcal{L}$$-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 $$\Delta \vDash \Gamma$$.