Equivalence of Semantic Consequence and Logical Implication

Theorem
Let $$U = \left\{{\phi_1, \phi_2, \ldots, \phi_m, \ldots}\right\}$$ be a countable set of logical formulas.

Let $$\psi$$ be a logical formula.

Then $$U \models \psi$$ iff $$U \vdash \psi$$.

That is, logical consequence is equivalent to logical implication.

Necessary Condition
This is a statement of the Extended Soundness Theorem of Propositional Calculus:

Let $$\mathbf H$$ be a countable set of logical formulas.

Let $$\mathbf A$$ be a logical formula.

If $$\mathbf H \vdash \mathbf A$$, then $$\mathbf H \models \mathbf A$$.

Sufficient Condition
Let $$\mathbf H$$ be a countable set of logical formulas.

Let $$\mathbf H \models \mathbf A$$.

Then $$\mathbf H \cup \left\{{\neg \mathbf A}\right\}$$ has no models.

By the Compactness Theorem of Propositional Calculus‎, there is a finite subset $$\mathbf H_0 \subseteq \mathbf H$$ such that $$\mathbf H_0 \cup \left\{{\neg \mathbf A}\right\}$$ has no models.

Then $$\mathbf H_0 \models \mathbf A$$.

By the statement of the Extended Completeness Theorem of Propositional Calculus $$\mathbf H_0 \vdash \mathbf A$$.

Hence $$\mathbf H \vdash \mathbf A$$.

Comment
There are two things being proved here:


 * Suppose we have a sequent $$\phi_1, \phi_2, \ldots, \phi_m, \ldots \vdash \psi$$, the validity of which has been established, for example, by a tableau proof.

The result:
 * "if $$\phi_1, \phi_2, \ldots, \phi_m, \ldots \vdash \psi$$ then $$\left\{{\phi_1, \phi_2, \ldots, \phi_m, \ldots}\right\} \models \psi$$"

establishes that if all the propositions $$\phi_1, \phi_2, \ldots, \phi_m, \ldots$$ evaluate to true, then so does $$\psi$$.

This establishes that propositional logic is sound.


 * Suppose we have determined that $$\left\{{\phi_1, \phi_2, \ldots, \phi_m, \ldots}\right\} \models \psi$$.

The result:
 * "if $$\left\{{\phi_1, \phi_2, \ldots, \phi_m, \ldots}\right\} \models \psi$$ then $$\phi_1, \phi_2, \ldots, \phi_m, \ldots \vdash \psi$$"

establishes that if we can show that there is a model for a proposition, then we will be able to find a tableau proof for it.

This establishes that propositional logic is complete.