There seem to be two questions in one here. The first is, what justification can we give for using deductive reasoning generally? The second is, what justification is there for one particular system over another? One cannot really address the second without attempting to answer the first.
The first question is seeking to understand what is the epistemology of logic. How do we know logical truths are true, or that valid arguments are valid? There have been many different responses to this question. Broadly, they divide into internalist and externalist categories, and each in turn has several variants.
Some have defended the idea that our intuition or insight as rational beings provides a kind of privileged a priori knowledge of logic that is incontrovertible. One might challenge this on many grounds. The history of philosophers claiming that various things are intuitively and certainly true is not a happy one: often the claimed truths turn out to be false or even absurd. Our intuitions aren't really that reliable. Also, if logic is incontrovertible, why is there so much disagreement between competent logicians about which logic is the 'correct' one, or even concerning what logic is fundamentally about?
Some hold that logic is justified inferentially by the relations of logical consequence that we consider correct. This is often defended by appealing to the concepts of logical harmony and logical stability. This runs into a circularity objection, pointed out by Lewis Carroll.
Some contend that logic is justified purely syntactically by the proofs it is capable of generating. Or that proofs correspond to computations, and so logic is justified indirectly by our understanding of computation.
Another approach is to try to base logic on a concept of analyticity. The idea is that some sentences have no empirical content and are therefore true 'come what may', or true in virtue of the meaning of the terms they contain. This position was popular with the logical positivists in the 1920s and 1930s and still has some defenders today, though it took a fair beating from Quine, Putnam and others, and does not appear to be especially popular with philosophers.
Another possible attempt at justification would be to appeal to natural selection. If we were not good at logical reasoning, we would be selected against and be less like to survive and propagate. This runs into the objection that we cannot be sure that logic always provides a strongly positive selection bias. Also, we know that humans are quite spectacularly bad at reasoning with probabilities and uncertainties, among other things, so it seems dubious to place too many expectations on natural selection.
Another approach is called anti-exceptionalism and maintains that logic is similar to a scientific theory. It has no special properties of a priority or analyticity and is potentially revisable in the light of empirical discoveries. On this account, logic is justified in the same way scientific theories are: we subject it to criticism and attempt to solve problems with it. If we succeed we keep trying more problems, and if we fail we look for something better. The logic or logics we are left with are the ones that work best because they have survived critical testing.
Which of the various systems of logic is 'correct'? On the rational intuition account, presumably we must just consult our intuitions on the matter. For myself, I don't see how this helps. Are the defenders of the various different logics just wrong in their intuitions? Is it intuitively obvious that the principle of explosion should hold, or that it should not? Is it intuitively obvious that universal statements have existential import, or that they do not?
On an analytic account, we would have to come up with a theory of meaning for natural language and argue that one particular logic does the best job of correctly accounting for, or at least conforming to, that theory. Michael Dummett took this approach in arguing for intuitionistic logic. Quine and the later Wittgenstein argued that this approach does not work.
Anti-exceptionalism potentially allows for logical pluralism, so the question of which system is 'correct' need not arise.
There are a number of useful references in my answer to this question.
