How can an inference form be tested for validity?

In formal logic, an inference form is considered valid if it is constructed such that every instance of it—produced by uniform replacement of its variables—guarantees that true premises will result in a true conclusion. Testing for this validity depends on the logical system being used:1

• Truth Tables: For propositional logic, truth tables provide an effective decision procedure. By mapping every possible combination of truth-values for the component propositions, one can determine if the conclusion is true in every case where the premises are true.2
• Model-Theoretic Approach: In more complex systems like the lower predicate calculus (LPC), validity is defined as truth in every possible model. Because LPC is not a decidable system, there is no general, pre-stated procedure to determine validity for any arbitrary formula, though the validity of many specific formulas can still be demonstrated.3
• Tableau Methods: These methods involve attempting to construct a countermodel—a scenario where the premises are true but the conclusion is false. If this construction is frustrated in all possible directions, the inference is proven valid.4

Formal logic focuses on these structures rather than specific subject matter to distinguish valid inference forms from invalid ones.1