Induction and deduction are relatively artificial subsets of reasoning. Human reasoning is very complex and defeasible. A third form of reasoning recognized is that which is abductive generally considered equivalent to inference to best explanation. Is it "legal" to reduce various aspects of an argument into components? Yes, certaintly, with caveats.
The act of reasoning of the human mind is studied by informal logic. Informal logic differs remarkably from formal logic because it requires a more thorough analysis of natural language which is semantically more complex than formal languages. That being said, however, there are aspects of argumentation that lend themselves well to deductive and inductive analysis.
Let's take a real-world example. Can a person be in two places at once? Absolutely not according to physics as we understand it. Therefore in a criminal proceeding, evidence of a suspect's location can play into analyzing testimony of a percipient witness. Let's say such a witness said in two discontinuous portions of the deposition:
I witnessed the suspect pick up the knife in the kitchen and stabbed the man to death.
I was outside having a conversation with the police at the time of the death.
Here it is clear, a judge can use deductive logic to see that there is a contradiction.
P1. The witness claimed to be in the kitchen at the time of death.
P2. One cannot be in the kitchen and outside at the same time.
C. Therefore, the witness's testimony contradicts itself.
Is reducing some portion of a natural language exchange to deduction "legal" or sensible? Absolutely. The same can be said of reasoning regarding inferences of induction. The real mistake comes, however, when one tries to force other forms of reasoning into the wrong patterns. These patterns of argumentation, formal or informal, may even be persuasive; thus, the fallacy.
Stephen Toulmin and his Uses of Argument is a precursor to the modern movement of examining informal logic philosophically, and is known for his Toulmin model of argumentation. David Zarefsky of Northwestern University is one advocate of this model. Lastly, the case-based reasoning which inspired Toulmin is a place where deductive and inductive logic often don't hold.
So, is it possible to break down aspects of reasoning into deductive and inductive forms? Absolutely, but one always has to take care in doing so because human reasoning is notoriously complicated.