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 ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ be first-order theories. If ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ are consistent and the intersection ${\displaystyle T_{1}\cap T_{2}}$ is complete (in the common language of ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$), then the union ${\displaystyle T_{1}\cup T_{2}}$ is consistent. A theory ${\displaystyle T}$ is called complete if it decides every formula, meaning that for every sentence ${\displaystyle \varphi ,}$ the theory contains the sentence or its negation but not both (that is, either ${\displaystyle T\vdash \varphi }$ or ${\displaystyle T\vdash \neg \varphi }$).

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

Let ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ be first-order theories. If ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ are consistent and if there is no formula ${\displaystyle \varphi }$ in the common language of ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ such that ${\displaystyle T_{1}\vdash \varphi }$ and ${\displaystyle T_{2}\vdash \neg \varphi ,}$ then the union ${\displaystyle T_{1}\cup T_{2}}$ is consistent.

See also

• Łoś–Vaught test

References

• 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.