TITLE: When do you call something "a calculus" vs. "a logic"?
QUESTION [14 upvotes]: Similar to What do Algebra and Calculus mean?, what is the difference between a logic and a calculus?
I am learning about the different kinds of logics, and often when I look them up in a different resource, some people call it a logic, others call it a calculus (propositional calculus and propositional logic). Or some calculus is defined as a logical system, like the situation calculus:

The situation calculus is a logic formalism designed for representing and reasoning about dynamical domains.

When do you call something a calculus vs. a logic?
It seems that the definitions of "a logic" and "a calculus" are often circular. A logic is a calculus, and a calculus is a logic. Or a calculus is rules for calculating, while a logic is rules for inference. But in this sense, they're both systems of rules, so maybe they are both just generally "formal systems", and when focusing on inference it's a "logic", and when focusing on calculation it's a "calculus"?

REPLY [2 votes]: In proof theory there is a difference between logic and calculi
There might be one semantic consequence relation $\vDash$, but many different syntactic consequences relations $\vdash_1$, $\vdash_2$, $\vdash_3$, ... . Thats the usual modus operandi of mathematical logic in the formal sciences, especially proof theory.
The semantic consequence relation $\vDash$ might be viewed extensional and is a litmus test for the syntactic consequences relations. The $\vdash_1$, $\vdash_2$, $\vdash_3$, ... are then studied intensionally, whereby various properties can be studied individually and comparatively. 
For example Gerhard Gentzen in his landmark 1934 paper "Investigations in Logical Deduction" showed a cut elimination and subsequently did relate a natural deduction style calculus with a sequent calculus.