Full Idea
The main role of substitutional semantics is to reduce ontology. As an alternative to model-theoretic semantics for formal languages, the idea is to replace the 'satisfaction' relation of formulas (by objects) with the 'truth' of sentences (using terms).
Gist of Idea
We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4)
Book Reference
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.243
A Reaction
I find this very appealing, and Ruth Barcan Marcus is the person to look at. My intuition is that logic should have no ontology at all, as it is just about how inference works, not about how things are. Shapiro offers a compromise.
Related Ideas
Idea 13675 Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro]
Idea 13678 The most popular account of logical consequence is the semantic or model-theoretic one [Sider]
Idea 13679 Maybe logical consequence is more a matter of provability than of truth-preservation [Sider]
Idea 17988 Quantification can't all be substitutional; some reference is obviously to objects [Hofweber]