what does " calculus" mean when used in logic? and more broadly in mathematics? ( not a question dealing specially with derivatives, integrals, etc.) Sentence logic is sometimes called " propositional calculus". 
I'd be interested in knowing what the word " calculus" means precisely/ technically here, and what are the ( historical) roots of this use of the word " calculus". 
As a pure hypothesis, I would say that any system of objects could be called a "calculus" provided : 
(1) there are some operations defined for these objects 
(2) these operations are purely extensional, that is, all extensionnaly equivalent expressions can be substituted for one another salva veritate 
Is there a link, as I suppose, between " calculus" and extensionality? 
Is it possible to give an example of a mathematical system that  cannot be called a " calculus"? 
 A: Personally I don't think it has a precise meaning; "concrete set of rules for manipulating syntactic expressions" is close to the best thing I can think of. Of course, it's both  vague and broad.

That said, we usually speak of (logical) calculi in the context of a fixed semantics for our language, in which case we're interested in calculi which are sound and complete with respect to that semantics. In this case, for various meanings of the word "concrete," we can indeed rule out the existence of calculi for various logics. Second-order logic (with the standard semantics) is a prime example, since basic questions about its entailment relation are set-theoretically contingent - e.g. there is a sentence $\varphi$ in second-order logic which is true in every structure, hence entailed by $\top$, iff the continuum hypothesis holds.
And then there are logics which are intermediate. For example, no non-compact logic has a finitary proof system, so in that sense doesn't have a calculus; however, some non-compact logics - the prime example being the infinitary logic $\mathcal{L}_{\omega_1,\omega}$, which indeed satisfies a technical analogue of compactness - do still have very well-behaved "infinitary" proof systems. Do these count as calculi?
A: "Calculus" in this sense is not really a technical term.  The OED says
"A system or method of calculation, ‘a certain way of performing mathematical investigations and resolutions’ (Hutton); a branch of mathematics involving or leading to calculations, as the differential, integral calculus, etc. The differential calculus is often spoken of as ‘the calculus’."
See Earliest Known Uses of Some of the Words of Mathematics
