What is really the conceptual difference between a calculus and an algebra. 
Eg. Is SKI combinator calculus really a calculus?


A friend claims that free variables are fundamental for a calculus, and as such that SKI is not a calculus, but an algebra.



Difference between a 'calculus' and an 'algebra'
In logic, the terminology seems to have been influenced by two factors. The very early development of various deductive systems was done by people who were more philosophers than mathematicians and who seem to have used "calculus" to refer to anything that looked mathematical. Also, that development took place before "algebra" had acquired all of its current meanings. 


My impression is that the use of "calculus" in logic is restricted to the meaning of "formal deductive system" --- and usually rather old systems. As for the SKI system of combinators, I would call it a calculus if you're talking about rules of inference. But if you mean the system of all combinators, with the operation of application, generated by S and K (I is redundant), then this is an algebra.


