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
be first-order theories.
are consistent and the intersection
is complete (in the common language of
), then the union
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
Since the completeness assumption is quite hard to fulfill, there is a variant of the theorem: Let
be first-order theories.
are consistent and if there is no formula
in the common language of
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