High Quality Content by WIKIPEDIA articles! Zeroth-order logic is first-order logic without quantifiers. A finitely axiomatizable zeroth-order logic is isomorphic to a propositional logic. Zeroth-order logic with axiom schema is a more expressive system than propositional logic. An example is given by the system Primitive recursive arithmetic, or PRA. The first line uses the variable x, which can be instantiated by any constant for an individual, such as S. The axioms are then the substitution instances of the schema. An equivalent approach is to declare the schema to be a plain axiom and to make variable substitution a special inference rule of the logic. At first glance it might appear that by using axiom schemata as in the example any first-order logic can be made zeroth-order. However, in general only universal quantifiers at the outermost level can be eliminated this way.

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