Robinson's joint consistency theorem

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. Note that a theory is complete if it decides every formula, i.e. 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.

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.