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**Robinson's joint consistency theorem** is an important theorem of mathematical logic. It is related to Craig interpolation and Beth definability.

The classical formulation of Robinson's joint consistency theorem is as follows:

Let and be first-order theories. If and are consistent and the intersection is complete (in the common language of and ), then the union is consistent. A theory is called complete 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 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.

- Boolos, George S.; Burgess, John P.; Jeffrey, Richard C. (2002).
*Computability and Logic*. Cambridge University Press. p. 264. ISBN 0-521-00758-5. - 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.