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Interpolation

Part of the Studies in Universal Logic book series (SUL)

Abstract

Interpolation is one of the most important topics of logic and model theory. Below is a very simple example. Consider the following semantic deduction in PL (propositional logic):
$$ p_1 \wedge q \vDash p_2 \vee q $$
where p1, p2, q are propositional symbols (i.e., relation symbols of zero arity). The simplest justification for this deduction is by factoring it as
$$ p_1 \wedge q \vDash q \vDash p_2 \vee q $$
which meets the intuition that p1 is not involved in establishing the truth of p2q. In general, the so-called ‘Craig interpolation’ (abbreviated CI) property can be formulated as follows: if ρ1 ⊨ ρ2 for two sentences, then there exists a sentence ρ, called the interpolant of ρ1 and ρ2, that uses logical symbols that appear both in ρ1 and ρ2 and such that ρ1 ⊨ ρ ⊨ ρ2. An equivalent expression of the above property assumes ρ1 ⊨ ρ2 in the union signature Σ1 ∪ Σ2, and asks for ρ to be in the intersection signature Σ1 ∩ Σ2, where Σ i is the signature of ρ i .

Keywords

Closure Operator Semantic Operator Satisfaction Condition Interpolation Property Relation Symbol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2008

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