Is the system of propositional logic itself induced, beyond its manipulated assumptions? It seems to be the case that we use propositional logic on the basis of two facts - firstly that it seems to work when applied correctly and with sufficient knowledge, which is an a postiori justification seemingly from induction, and secondly that the specific rules of inference can be induced easily. Conditionals are induced from the fact that certain events seem to connect to one other predictably, the "and" function is induced from the observation that certain events seem to connect to one another necessarily to create a valid conditional, the "or" function is induced from the observation that certain conditionals can be valid in multiple distinct or combined cases, and so on. It seems then that our use of propositional logic is itself validated only by induction. Another way to phrase this question is through a thought experiment. If a human lived in a world in which events connected randomly, where the connection of events did not seem to follow deductive rules, would they be able to imagine up deductive reasoning at all? If so, how would they arrive at this thought? If not, doesn't this demonstrate that propositional logic is justified only by induction? Keep in mind that claiming that "our world does not function this way" does not answer why a human in such a world would not be able to arrive at the idea of deduction, if deduction is not based on induction. Likewise, claiming that "deduction is useful" seems to be an argument from induction of the results of using deduction.
Potentially, you could be asking one of two different questions. One is how did our ability to do logic get started? i.e. a question about origins. The other is how do we consider logic to be grounded or justified? i.e. a question about the epistemology of logic.
In the case of the former question, we really just don't know. We know from studies of animals that some species have a basic ability to reason, and some are able to count. Mothers with a litter of six young usually know when one is missing. But how this developed in humans we don't have much idea: it happened in prehistoric times and we don't have records, so we can only guess. I think we can safely say though, that by the time the stoic philosophers in the third century BCE were devising the logic of propositions (i.e. 'and', 'or', 'if') we were a long way past the simple idea that we are just generalising from experience.
A comparison with counting is perhaps appropriate: parents teach their children to count by showing them real things such as stones or trees or ducks and getting them to count them. But once we progress beyond the nursery stage, we don't need to count ducks to know how arithmetic works, nor do we believe that arithmetic is correct just because it works on ducks, no matter how many times we practice counting them. Similarly, the words 'and', 'or' and 'if' are part of the English language, and we learn how to use them when we learn our first language. The examples that we were presented with when we learned how to use them are no longer relevant to our understanding.
In the case of the second question, there is much disagreement about the epistemology of logic. Some positions are that it is an innate and privileged form of a priori knowledge, that it is grounded in the grammar of the language we use, that it is derived from the inferential relationships that we consider semantically compelling, that it is grounded in a theory of meaning, that it is justified by its close relationship with the concept of computation, that it is justified by virtue of being the product of natural selection, that it is justified in an indirect way by the contibution it makes to our scientific knowledge.
However, the idea that we simply proceed by induction from observations is implausible. For one thing, propositional logic is concerned with the connection between the truths of propositions, not between events. There is nothing intrinsically causal about the connections. For another, once a logical connection is correctly grasped, further examples become irrelevant, except for illustrative purposes. By contrast, with inductive support, the more data you have the better, because the data is needed to support the proposed connection.
As to your thought experiment, I would be inclined to say that a world that is so chaotic that no connections of any kind are observable would be a world in which I, or any living thing, could not exist, so the question is moot. The most that you might say is that since we use logic to organise information, if we have no information there is nothing to organise, so logic would be a pointless exercise.