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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]].

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

Let <math>T_1</math> and <math>T_2</math> be [[first-order logic|first-order]] theories. If <math>T_1</math> and <math>T_2</math> are [[consistent]] and the intersection <math>T_1\cap T_2</math> is [[complete theory | complete]] (in the common language of <math>T_1</math> and <math>T_2</math>), then the union <math>T_1\cup T_2</math> is consistent. Note that a theory is complete if it decides every formula, i.e. either <math>T \vdash \varphi</math> or <math>T \vdash \neg\varphi</math>.

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

Let <math>T_1</math> and <math>T_2</math> be [[first-order logic|first-order]] theories. If <math>T_1</math> and <math>T_2</math> are consistent and if there is no formula <math>\varphi</math> in the common language of <math>T_1</math> and <math>T_2</math> such that <math>T_1 \vdash \varphi</math> and <math>T_2 \vdash \neg\varphi</math>, then the union <math>T_1\cup T_2</math> is consistent.

==References==
*{{cite book|last = Boolos|first = George S.|coauthors = Burgess, John P.; Jeffrey, Richard C.|title = Computability and Logic|publisher = Cambridge University Press|date = 2002|pages = 264|isbn = 0-521-00758-5|url = http://books.google.com/books?id=Yy14JSjPyY8C}}

[[Category:Mathematical logic]]
[[Category:Theorems in the foundations of mathematics]]

{{logic-stub}}
{{mathlogic-stub}}

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