Doubly partially conservative sentences

The purpose of the present paper is to analyze several variants of Solovay’s theorem on the existence of doubly partially conservative sentences. First, we investigate $\mathrm{\Theta }$ sentences that are doubly $(\mathrm{\Gamma },\mathrm{\Lambda })$-conservative over $T$ for several triples $(\mathrm{\Theta },\mathrm{\Gamma },\mathrm{\Lambda })$. Among other things, we prove that the existence of a ${\mathrm{\Delta }}_{n+1}(\mathsf{P}\mathsf{A})$ sentence that is doubly $({\mathrm{\Sigma }}_n,{\mathrm{\Sigma }}_n)$-conservative over $T$ is equivalent to the ${\mathrm{\Sigma }}_{n+1}$-inconsistency of $T$ over $\mathsf{P}\mathsf{A}$. Secondly, we study $\mathrm{\Theta }$ sentences that are hereditarily doubly $(\mathrm{\Gamma },\mathrm{\Lambda })$-conservative over $T$ for several triples $(\mathrm{\Theta },\mathrm{\Gamma },\mathrm{\Lambda })$.