Equivalence of Semantic Consequence and Logical Implication

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Theorem

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

Let $\psi$ be a propositional formula.


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


That is, semantic consequence is equivalent to provable consequence.


Direct Proof

Necessary Condition

This is a statement of the Extended Soundness Theorem for Propositional Tableaus and Boolean Interpretations:

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

Let $\mathbf A$ be a propositional formula.


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


Sufficient Condition

This is the statement of the Extended Completeness Theorem for Propositional Tableaus and Boolean Interpretations.

$\blacksquare$


Comment

There are two things being proved here:

$(1): \quad$ 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.


$(2): \quad$ 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.


Sources