We introduce a general framework for solving the problem of a computer collecting and combining information from various sources. Unlike previous approaches to this problem, in our framework the sources are allowed to provide information about complex formulae too. This is enabled by the use of a new tool—non-deterministic logical matrices. We also consider several alternative plausible assumptions concerning the framework which lead to various logics. We provide strongly sound and complete proof systems for all the basic logics induced in this way.