There are approaches to logic that would give an affirmative answer to your question. Keynes and Carnap, in particular, developed the concept of logical probability, under which the premises of an argument support the conclusion, or provide a partial entailment of it, expressed as the conditional probability of the conclusion given the premises. Deductive entailment is then the limiting case in which the support is total, and the conditional probability goes to unity.
Carnap's theory is purely syntactical in nature and runs into the standard objection, expressed by Nelson Goodman among others, that the degree of support of some premises for a conclusion depends on the choice of predicates with which the argument is expressed. Today there is little support for Carnap's approach, though there is still strong support for the closely related Bayesian approach to confirmation theory.
As to inductive 'logic' being fundamental, this could be argued, though in practice it is contentious. There has been a lot of discussion around the issue of the epistemology of logic, i.e. how do we know that logical truths are true, or that logical arguments are valid? A number of different positions on this issue have been defended, which broadly fall into 'internalist' and 'externalist' positions. One form of internalism is that we judge logical truths to be true because we cannot coherently conceive them being otherwise. One common form of externalism is that logical truths are those that are completely reliable and give structure to the way we organize and systematize our knowledge.
Sometimes you will see people criticize inductive justifications for inductive reasoning on the grounds that this is circular, but exactly the same objection could be made about deduction. One cannot justify deductive reasoning without using deduction in the process. One possible position you might adopt is to argue that deductive reasoning is justified inductively, because we observe that it works reliably, though again, this is only one of many positions that have been defended.
There are some references that you might find useful in my answer to this question:
References for the justification of the use of Logic