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## Preview

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