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## Decidable problems

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’.

5.1 The decision problem

There are three natural and closely connected problems for first-order ...

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