We've considered various algorithms (tableaux, resolution, etc.) for verifying that a first-order formula is logically valid, if indeed it is. But these will not in general tell us when a formula is not valid. We'll see in Chapter 7 that there is no systematic procedure for doing so. However, there are procedures that work for certain special classes of formulas, or for validity in certain special (classes of) models, and we discuss some of the more important ones in this chapter. Often these naturally generalize common decision problems in mathematics and universal algebra such as equation-solving or the ‘word problem’.

The decision problem

There are three natural and closely connected problems for first-order logic for which we might want an algorithmic solution. By negating the formula, we can according to taste present them in terms of validity or unsatisfiability.

1. Confirm that a logically valid (or unsatisfiable) formula is indeed valid (resp. unsatisfiable), and never confirm an invalid (satisfiable) one.

2. Confirm that a logically invalid (or satisfiable) formula is indeed invalid (resp. satisfiable), and never confirm a valid (unsatisfiable) one.

3. Test whether a formula is valid or invalid (or whether it is satisfiable or unsatisfiable).

Evidently (3) encompasses both (1) and (2). Conversely, solutions to both (1) and (2) could be used together to solve (3): just run the verification procedures for validity and invalidity (or satisfiability and unsatisfiability) in parallel. Now, we have presented explicit solutions to (1), such as tableaux or resolution. But these do not solve (3). Given a satisfiable formula, these algorithms, while at least not incorrectly claiming they are unsatisfiable, will not always terminate.