Demonstrable sentences are called theorems of the system and are normally intended to be some or all of the sentences which are true relative to some intended semantics for the sentences. Some, but not all of these systems are normally described as logic s, others may be called calculi. Logics - Those formal systems normally described as logic s fall into two main groups. Firstly there are simple systems such as propositional and modal logics, which, though limited in their expressiveness, are clearly concerned with reasoning rather than computation. Since large parts of mathematics involve such claims, a logic proper, rather than a mere calculus, is needed for the formalisation of mathematics.
Logics fall into three groups, free logics, foundation systems, and intermediate systems.
Foundation Systems - Foundation Systems embody sufficiently lavish ontological presuppositions that a mathematician using them is unlikely to find further ontological premises necessary. A foundation system therefore provides a framework within which issues of consistency may be resolved by formal means. The term calculus is used mainly for systems where the notation involved is essentially equivalent to a programming language, and the sentences demonstrated are typically equations asserting the claim that two expressions evaluate to the same value.