Why are validity and satisfiability important for a logical calculus?

In metalogic, validity and satisfiability are fundamental semantic concepts used to evaluate the adequacy and behavior of a logical calculus. Their importance lies in how they bridge the gap between the syntactic structure of a system and its semantic meaning.12

• Validity is defined as a sentence being true in all possible interpretations or possible worlds. It is essential because a primary goal of a logical calculus is completeness—ensuring that every valid sentence is provable as a theorem within the system.321
• Satisfiability refers to a sentence having at least one interpretation or model where it is true. It is critical for establishing consistency; if a calculus is complete, then any consistent sentence (one whose negation is not a theorem) is necessarily satisfiable.12

Together, these concepts allow logicians to prove that a system is free from contradiction and that its syntactic rules—such as axioms and rules of inference—correctly capture logical truths. By aligning semantic validity with syntactic derivability, these concepts provide the standard for determining whether a formal system is an accurate representation of logical reasoning.312