# 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