Robinson's joint consistency theorem is an important theorem of
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
. It is related to
Craig interpolation In mathematical logic, Craig's interpolation theorem is a result about the relationship between different logical theories. Roughly stated, the theorem says that if a formula φ implies a formula ψ, and the two have at least one atomic variable s ...
and
Beth definability.
The classical formulation of Robinson's joint
consistency
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
theorem is as follows:
Let
and
be
first-order
In mathematics and other formal sciences, first-order or first order most often means either:
* "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
theories
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...
. If
and
are
consistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
and the intersection
is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
(in the common language of
and
), then the union
is consistent. A theory
is called
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
if it decides every formula, meaning that for every
sentence the theory contains the sentence or its negation but not both (that is, either
or
).
Since the completeness assumption is quite hard to fulfill, there is a variant of the theorem:
Let
and
be
first-order
In mathematics and other formal sciences, first-order or first order most often means either:
* "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
theories. If
and
are consistent and if there is no formula
in the common language of
and
such that
and
then the union
is consistent.
See also
*
References
*
*
Robinson, Abraham, 'A result on consistency and its application to the theory of definition', Proc. Royal Academy of Sciences, Amsterdam, series A, vol 59, pp 47-58.
Mathematical logic
Theorems in the foundations of mathematics
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