# Theory (mathematical logic)

In

**first-order theory**is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms.

## General theories (as expressed in formal language)

When defining theories for foundational purposes, additional care must be taken, as normal set-theoretic language may not be appropriate.

The construction of a theory begins by specifying a definite non-empty *conceptual class* , the elements of which are called *statements*. These initial statements are often called the *primitive elements* or *elementary* statements of the theory—to distinguish them from other statements that may be derived from them.

A theory is a conceptual class consisting of certain of these elementary statements. The elementary statements that belong to are called the *elementary theorems* of and are said to be *true*. In this way, a theory can be seen as a way of designating a subset of that only contain statements that are true.

This general way of designating a theory stipulates that the truth of any of its elementary statements is not known without reference to . Thus the same elementary statement may be true with respect to one theory but false with respect to another. This is reminiscent of the case in ordinary language where statements such as "He is an honest person" cannot be judged true or false without interpreting who "he" is, and, for that matter, what an "honest person" is under this theory.^{[1]}

### Subtheories and extensions

A theory * is a ***subtheory** of a theory * if ** is a subset of **. If ** is a subset of ** then ** is called an ***extension** or a **supertheory** of *
*

### Deductive theories

A theory is said to be a *deductive theory* if is an

^{[1]}

### Consistency and completeness

A **syntactically consistent theory** is a theory from which not every sentence in the underlying language can be proven (with respect to some

A **satisfiable theory** is a theory that has a

A **consistent theory** is sometimes defined to be a syntactically consistent theory, and sometimes defined to be a satisfiable theory. For

A

*such that for every sentence φ in its language, either φ is provable from*

*or*

*{φ} is inconsistent. For theories closed under logical consequence, this means that for every sentence φ, either φ or its negation is contained in the theory.*

^{}[3] An

**incomplete theory**is a consistent theory that is not complete.

(see also **ω-consistent theory** for a stronger notion of consistency.)

### Interpretation of a theory

An *interpretation of a theory* is the relationship between a theory and some subject matter when there is a

*full interpretation*, otherwise it is called a

*partial interpretation*.

^{[4]}

### Theories associated with a structure

Each structure has several associated theories. The **complete theory** of a structure *A* is the set of all first-order sentences over the signature of *A* that are satisfied by *A*. It is denoted by Th(*A*). More generally, the **theory** of *K*, a class of σ-structures, is the set of all first-order σ-sentences that are satisfied by all structures in *K*, and is denoted by Th(*K*). Clearly Th(*A*) = Th({*A*}). These notions can also be defined with respect to other logics.

For each σ-structure *A*, there are several associated theories in a larger signature σ' that extends σ by adding one new constant symbol for each element of the domain of *A*. (If the new constant symbols are identified with the elements of *A* that they represent, σ' can be taken to be σ A.) The cardinality of σ' is thus the larger of the cardinality of σ and the cardinality of *A*.^{[further explanation needed]}

The **diagram** of *A* consists of all atomic or negated atomic σ'-sentences that are satisfied by *A* and is denoted by diag_{A}. The **positive diagram** of *A* is the set of all atomic σ'-sentences that *A* satisfies. It is denoted by diag^{+}_{A}. The **elementary diagram** of *A* is the set eldiag_{A} of *all* first-order σ'-sentences that are satisfied by *A* or, equivalently, the complete (first-order) theory of the natural

*A*to the signature σ'.

## First-order theories

A first-order theory is a set of sentences in a first-order formal language .

### Derivation in a first-order theory

There are many formal derivation ("proof") systems for first-order logic. These include

### Syntactic consequence in a first-order theory

A formula *A* is a **syntactic consequence** of a first-order theory if there is a derivation of *A* using only formulas in as non-logical axioms. Such a formula *A* is also called a theorem of . The notation "" indicates *A* is a theorem of .

### Interpretation of a first-order theory

An **interpretation** of a first-order theory provides a semantics for the formulas of the theory. An interpretation is said to satisfy a formula if the formula is true according to the interpretation. A **model** of a first-order theory is an interpretation in which every formula of is satisfied.

### First-order theories with identity

A first-order theory is a first-order theory with identity if includes the identity relation symbol "=" and the reflexivity and substitution axiom schemes for this symbol.

### Topics related to first-order theories

- Compactness theorem
- Consistent set
- Deduction theorem
- Enumeration theorem
- Lindenbaum's lemma
- Löwenheim–Skolem theorem

## Examples

One way to specify a theory is to define a set of

A second way to specify a theory is to begin with a

**R**, +, ×, 0, 1, =) was shown by Tarski to be

## See also

- Axiomatic system
- Interpretability
- List of first-order theories
- Mathematical theory

## References

- ^
^{a}^{b}Haskell Curry,*Foundations of Mathematical Logic*, 2010. **^**Weiss, William; D'Mello, Cherie (2015). "Fundamentals of Model Theory" (PDF).*University of Toronto — Department of Mathematics*.**^**"Completeness (in logic) - Encyclopedia of Mathematics".*www.encyclopediaofmath.org*. Retrieved 2019-11-01.**^**Haskell Curry (1963).*Foundations of Mathematical Logic*. Mcgraw Hill. Here: p.48

## Further reading

- ISBN 0-521-58713-1.