Ykhjuy

From what I can figure: A deductive system is a language $L$, a set of logical axioms $Delta_L$ that are formulas of the language $L$, and a set of ordered pairs of rules of inference $(Gamma, phi)$.

$text{Deductive System } = (L, Delta_L, {(Gamma, phi)})$

Is this the correct definition or am I misunderstanding something?

There is not really a single agreed-on definition of "deductive system".

It's more of a stepping-stone system that each author will set up with the details they need for being able to define the actual systems they want to teach about. Other authors who want to speak about other actual systems may need to choose different details for their definition.

This is one of the few places in mathematics where the generalization is much less important than the concrete instances.

Everybody agrees quite well on what the entailment relations of, say, intuitionistic propositional logic or classical first-order logic are. But the precise systems that realize these entailment relations vary from author to author, so -- if they bother to define a general concept of "deductive system" at all -- their helper concepts are slightly different too.

What you're suggesting is close enough to the general kind of abstractions people tend to call "deductive system", that it ought to enable you to follow their train of thought when they use the term. Just as long as you don't take the details too seriously.

To expand on Henning's Answer:

There are many 'deductive systems'.

Your definition will be able to 'fit' a good number of deductive systems, but there are also some that do not allow such a nice generalization:

Some systems define subproofs (see, for example, Fitch systems) and an inference is based on whole subproofs, rather than just statements

And systems of truth trees (sometimes called semantic tableaux) work quite differently yet.

Of course, some may argue that those that don't fit your definition are not 'deductive systems'. But others construe them more broadly. I really don't think there is any universally agreed upon nice and clean definition. The same seems to be true for 'systems of natural deduction': some have a more narrow definition for those than others (I am sure to get comments on this :) )

Point is: there is a whole taxonomy of these kinds of systems. The best thing to do is just to learn how they all work (indeed, you can really gain some deeper insights into logic if you look at different systems, rather than sticking to just one).

While it is true what Hennings Makolm and Bram28 have said in their answers it is also true that there is a general definition of what a deductive system should be given by Lambek.

Lambek idea is that logical systems are given by sequent, that can be thought as expressions of the form $A to B$ and that can be composed via some operations, the rules of inference.

So according to Lambek a deductive system is nothing but some sort of graph, whose directed edges are sequents of the calculus, with operations defined over them, that is a graph-algebra.

Of course this is one possible definition, but one that capture most of the known logical systems.