Robinson's joint consistency theorem is an important theorem of

mathematical logic Mathematical logic is the study of 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 formal ...

. 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 sy ...

and

Beth definability In mathematical logic, Beth definability is a result that connects implicit definability of a property to its explicit definability. Specifically Beth definability states that the two senses of definability are equivalent. First-order logic has th ...

. 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

T_1

and

T_2

be first-order

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

T_1

and

T_2

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 consisten ...

and the intersection

T_1 \cap T_2

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 ...

(in the common language of

T_1

and

T_2

), then the union

T_1 \cup T_2

is consistent. A theory

T

is called

if it decides every formula, meaning that for every sentence

\varphi,

the theory contains the sentence or its negation but not both (that is, either

T \vdash \varphi

T \vdash \neg \varphi

). Since the completeness assumption is quite hard to fulfill, there is a variant of the theorem: Let

T_1

and

T_2

be first-order theories. If

T_1

and

T_2

are consistent and if there is no formula

\varphi

in the common language of

T_1

and

T_2

such that

T_1 \vdash \varphi

and

T_2 \vdash \neg \varphi,

then the union

T_1\cup T_2

is consistent.

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 {{mathlogic-stub

See also

References