Institution-independent Model Theory pp 189-221 | Cite as

# Interpolation

Chapter

## 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
where
which meets the intuition that

**PL**(propositional logic):$$
p_1 \wedge q \vDash p_2 \vee q
$$

*p*_{1},*p*_{2},*q*are propositional symbols (i.e., relation symbols of zero arity). The simplest justification for this deduction is by factoring it as$$
p_1 \wedge q \vDash q \vDash p_2 \vee q
$$

*p*_{1}is not involved in establishing the truth of*p*_{2}∨*q*. In general, the so-called ‘Craig interpolation’ (abbreviated CI) property can be formulated as follows: if ρ_{1}⊨ ρ_{2}for two sentences, then there exists a sentence ρ, called the*interpolant*of ρ_{1}and ρ_{2}, that uses logical symbols that appear both in ρ_{1}and ρ_{2}and such that ρ_{1}⊨ ρ ⊨ ρ_{2}. An equivalent expression of the above property assumes ρ_{1}⊨ ρ_{2}in the union signature Σ_{1}∪ Σ_{2}, and asks for ρ to be in the*intersection signature*Σ_{1}∩ Σ_{2}, 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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