When do you call something "a calculus" vs. "a logic"?

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"?

Solution 1:

The short answer is, after a few more months of letting this sit, is that in reality there is no difference between algebra, logic, and calculus. They are all just saying "a formal collection of mathematical rules". But each of these words have a history, and so when authors use them, they are mentally invoking that history of the word. Because these words were used in the development of different ideas, the logic/calculus/algebra typically prioritize different ideas.

It is like saying "what is the difference between book and tome"? They both are the exact same thing, you are just highlighting different aspects of it by invoking mental imagery. In the algebra/logic/calculus case, the mental imagery is the history of it's use.

Solution 2:

I would elaborate on the answer above with respect to the etymology of logic. Logic is closely associated with philosophy since the days of Aristotle. Many philosophers are well-versed in logic, but not as something that you represent symbolically. This is what is known as informal logic and involves such things as logical fallacies and assessing the validity of an argument.

Then, at the very foundational level, we have the work of logicians like Frege, Russell, and Gödel, who saw logic as the foundation from which to derive arithmetic and thence, algebra and calculus. In this view, the difference between the terms it not only historical, but hierarchical as well.

Solution 3:

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.