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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.## References

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

Let $T\_1$ and $T\_2$ be first-order theories. If $T\_1$ and $T\_2$ are consistent and the intersection $T\_1cap\; T\_2$ is complete (in the common language of $T\_1$ and $T\_2$), then the union $T\_1cup\; T\_2$ is consistent. Note that a theory is complete if it decides every formula, i.e. either $T\; vdash\; varphi$ or $T\; vdash\; negvarphi$.

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 $T\_2\; vdash\; negvarphi$, then the union $T\_1cup\; T\_2$ is consistent.

- Boolos, George S.; Burgess, John P.; Jeffrey, Richard C. Computability and Logic. Cambridge University Press.

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Last updated on Thursday April 17, 2008 at 16:41:44 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday April 17, 2008 at 16:41:44 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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