Robinson's joint consistency theorem
Encyclopedia
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
and 
be first-order
theories. If
and 
are consistent and the intersection 
is complete
(in the common language of
and 
), then the union 
is consistent. Note that a theory is complete if it decides every formula, i.e. either 
or 
.
Since the completeness assumption is quite hard to fulfill, there is a variant of the theorem:
Let
and 
be first-order
theories. If
and 
are consistent and if there is no formula 
in the common language of 
and 
such that 
and 
, then the union 
is consistent.
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
. It is related to Craig interpolation
Craig interpolation
In mathematical logic, Craig's interpolation theorem is a result about the relationship between different logical theories. Roughly stated, the theorem says that if a formula φ implies a formula ψ then there is a third formula ρ, called an interpolant, such that every nonlogical symbol...
and Beth definability
Beth definability
In mathematical logic, Beth definability states that if for any two models A, B of a first-order theory T in the language L' ⊇ L with A|L = B|L it is satisfied A ⊨ φ[a] if and only if B ⊨ φ[a] for φ a formula in L' and for all tuples a of A, then φ is equivalent modulo T to a formula ψ...
.
The classical formulation of Robinson's joint consistency theorem is as follows:
Let
and be first-orderFirst-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
theories. If
and are consistent and the intersection is completeComplete theory
In mathematical logic, a theory is complete if it is a maximal consistent set of sentences, i.e., if it is consistent, and none of its proper extensions is consistent...
(in the common language of
and ), then the union is consistent. Note that a theory is complete if it decides every formula, i.e. either or .Since the completeness assumption is quite hard to fulfill, there is a variant of the theorem:
Let
and be first-orderFirst-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
theories. If
and are consistent and if there is no formula in the common language of and such that and , then the union is consistent.