Lev Beklemishev (Russian Academy of Sciences Moscow) gives a talk at the MCMP Colloquium (12 November, 2015) titled "Positive Reflection Calculi". Abstract: We deal with the fragment of propositional modal logic consisting of implications of formulas built up from the variables and the constant `true' by conjunction and diamonds only. We call such fragments strictly positive. The interest towards strictly positive modal logics independently emerged around 2010 in two different disciplines: the work on description logic by Zakharyaschev, Kurucz, et al., and the work on proof-theoretic applications of provability logic by myself, Dashkov, et al. The advantages of considering such fragments are twofold. On the one hand, strictly positive fragments of modal logics are usually (and not surprisingly) much simpler than the original logics. Typically, strictly positive fragments of standard modal logics are polytime decidable. On the other hand, the strictly positive language, being weaker than the standard modal language, allows for many more meaningful interpretations. In this talk we review basic results on strictly positive logics, their syntax and semantics. Furthermore, we develop the framework of reflection calculus, that is, a logic in which the diamonds are interpreted as reflection schemata in arithmetic, possibly of unrestricted logical complexity. This framework allows for a natural treatment of extensions of arithmetic by Tarskian truth predicates and the corresponding reflection principles.