Zeroth-order logic is first-order logic

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

 without quantifiers. A finitely axiomatizable

Axiomatic system

In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...

 zeroth-order logic is isomorphic to a propositional logic. Zeroth-order logic with axiom schema

Axiom schema

In mathematical logic, an axiom schema generalizes the notion of axiom.An axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which...

 is a more expressive system than propositional logic. An example is given by the system Primitive recursive arithmetic

Primitive recursive arithmetic

Primitive recursive arithmetic, or PRA, is a quantifier-free formalization of the natural numbers. It was first proposed by Skolem as a formalization of his finitist conception of the foundations of arithmetic, and it is widely agreed that all reasoning of PRA is finitist...

, or PRA.

Example

The well-known syllogism

Syllogism

A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...

• All men are mortal
• Socrates is a man
• Therefore, Socrates is mortal

cannot be formalized in propositional logic, because of the use of predicate

Predicate (grammar)

There are two competing notions of the predicate in theories of grammar. Traditional grammar tends to view a predicate as one of two main parts of a sentence, the other being the subject, which the predicate modifies. The other understanding of predicates is inspired from work in predicate calculus...

s like "is a man" and "is mortal". The obvious formalization in first-order logic uses universal quantification

Universal quantification

In predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing....

 to model the use of "All".

The following weak version of the syllogism can be formalized in propositional logic:

• If Socrates is a man, then Socrates is mortal
• Socrates is a man
• Therefore, Socrates is mortal

This can be done by introducing propositional constants SMN (for "Socrates is a man") and SML (for "Socrates is mortal"), and the two axioms

• SMN → SML, and
• SMN.

Together with the usual rule of modus ponens

Modus ponens

In classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...

 the conclusion, SML, follows.

In this weak version most of the essence of the original syllogism has been lost. In predicate logic one can instead introduce predicates Man (for "is a man'), Mortal (for "is mortal"), constants A (for "Aristotle"), S (for "Socrates"), Z (for "Zeus"), and so on, and use a multitude of axioms, one for each individual:

• Man(A) → Mortal(A)
• Man(S) → Mortal(S)
• Man(Z) → Mortal(Z)
• ...
• Man(S)
• ¬Mortal(Z)

Again, modus ponens allows to conclude Mortal(S). If the axioms for contraposition

Contraposition

In traditional logic, contraposition is a form of immediate inference in which from a given proposition another is inferred having for its subject the contradictory of the original predicate, and in some cases involving a change of quality . For its symbolic expression in modern logic see the rule...

 are added, also ¬Man(Z) becomes a theorem.

By using an axiom schema

Axiom schema

In mathematical logic, an axiom schema generalizes the notion of axiom.An axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which...

, the above can be collapsed into:

• Man(x) → Mortal(x)
• Man(S)
• ¬Mortal(Z)

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.

Relation to general first-order 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.