First-Order Logic: Terms and Normal Forms

Abstract

The formulas in first-order logic that we have defined are sufficient to express many interesting properties. Consider, for example, the formula:

$$ \forall x \forall y \forall z\, (p\,(x,y) \wedge p\,(y,z) \mathbin{\rightarrow} p\,(x,z)). $$

Under the interpretation:

$$ \{Z,\{ < \} ,\{ \} \} , $$

it expresses the true statement that the relation less-than is transitive in the domain of the integers. Suppose, now, that we want to express the following statement which is also true in the domain of integers:

$$ \mathit{for} \:\mathit{all}\; x, y, z: (x < y) \mathbin{\rightarrow} (x+z < y+z). $$

The difference between this statement and the previous one is that it uses the function +.