We extend the constructive dependent type theory of the Logical Framework $\mathsf{LF}$ with monadic, dependent type constructors indexed with predicates over judgements, called Locks. These monads capture various possible proof attitudes in establishing the judgment of the object logic encoded by an $\mathsf{LF}$ type. Standard examples are factoring-out the verification of a constraint or delegating it to an external oracle, or supplying some non-apodictic epistemic evidence, or simply discarding the proof witness of a precondition deeming it irrelevant. This new framework, called Lax Logical Framework, $\mathsf{LLF}_{\cal P}$, is a conservative extension of $\mathsf{LF}$, and hence it is the appropriate metalanguage for dealing formally with side-conditions in rules or external evidence in logical systems. $\mathsf{LLF}_{\cal P}$ arises once the monadic nature of the lock type-constructor, ${\cal L}^{\cal P}_{M,\sigma}[\cdot]$, introduced by the authors in a series of papers, together with Marina Lenisa, is fully exploited. The nature of the lock monads permits to utilize the very Lock destructor, ${\cal U}^{\cal P}_{M,\sigma}[\cdot]$, in place of Moggi's monadic $let_T$, thus simplifying the equational theory. The rules for ${\cal U}^{\cal P}_{M,\sigma}[\cdot]$ permit also the removal of the monad once the constraint is satisfied. We derive the meta-theory of $\mathsf{LLF}_{\cal P}$ by a novel indirect method based on the encoding of $\mathsf{LLF}_{\cal P}$ in $\mathsf{LF}$. We discuss encodings in $\mathsf{LLF}_{\cal P}$ of call-by-value $\lambda$-calculi, Hoare's Logic, and Fitch-Prawitz Naive Set Theory. Comment: Accepted for publication in LMCS

@article{HonsellLMS17,
  title = {$\mathsf{LLF}_{\cal P}$: a logical framework for modeling external evidence, side conditions, and proof irrelevance using monads},
  author = {Furio Honsell and Luigi Liquori and Petar Maksimovic and Ivan Scagnetto},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 13, Issue 3},
  year = {2017},
  doi = {10.23638/lmcs-13(3:2)2017}
}