Definitions

chopart's joint

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 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.

References

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