I do think there is a link, I just can't quite catch it.
I have looked it up but I'm pretty new to logic so the explanations all seemed a little confusing. I think a system needs to be sound and complete in order to rely on truth tables to make proofs but I couldn't be less sure.
Truth tables are used in defining the notion of logical. consequence. A conclusion $C$ is a logical consequence of a set $\mathcal H$ of hypotheses if, whenever an assignment of truth values to the propositional variables (a row of a truth table) makes all the hypotheses in $\mathcal H$ true, it also makes $C$ true. Notice that this definition of logical consequence is entirely about truth values; there is no mention of axioms or rules of inference.
A conclusion $C$ is deducible, in a formal system $X$, from a set $\mathcal H$ of hypotheses if there is a deduction of $C$ from hypotheses in $\mathcal H$ and axioms of $X$ by means of the rules of inferencee of $X$. Notice that this definition of deducible is entirely about axioms and rules of inference; there is no mention of truth values.
The notions of "logical consequence" and "deducible in $X$", despite their entirely different provenance, turn out to be equivalent provided $X$ was intelligently designed.
In more detail, $X$ is said to be sound if "deducible in $X$" implies "logical consequence", and $X$ is said to be complete if "logical consequence" implies "deducible in $X$". So the combination of soundness and completeness is the equivalence mentioned in the previous paragraph.
