"Ordinary reasoning" makes little use of any formal logic. It's based on heuristics of what usually worked in the past. It goes like this: "In condition A, thinking in manner X was helpful in the past. We are now dealing with a condition similar to A. So, we will try thinking in manner X." No part of this involves formal deductions. Both A and X are huge patterns of neural activations, not neat little propositions.
Your brain is a massive, continuous-weighted, randomized neural network, highly tolerant to faults and damage. Formal deduction involves a small number of axioms, works with discrete propositions, deals only with logical necessity, and will completely break if given a single false proposition. They are deeply contrasting paradigms.
There is little we can say about the real world that follows as a matter of pure deduction. We cannot deductively predict what will result from any physical situation; our judgments about this are heuristic and inductive. "It worked that way in the past, so I suppose it will probably do it again." Deduction is used to make predictions about idealized models of physical situations. These models sometimes may be applicable to real situations, but in doing so there is always going to be the possibility of error, where the real world doesn't match the model.
GOFAI (Good Old-Fashioned AI) often involves modeling the world using propositions about it, from which the computer makes logical deductions. If deductive logic really worked for everyday situations, this approach would have been very successful. But what we've seen, instead, is that GOFAI works on simple toy scenarios or in factories where it will only encounter carefully controlled situations, and has great difficulty with the real world. The real world is full of edge cases that make the GOFAI's "logical deductions" incorrect.
There have been attempts to make logic more flexible, with defeasible reasoning. But this does not work that well. It's still a human writing the logical rules, which drastically limits the size of the model. And a lot of the world just doesn't map that neatly onto logical propositions. It's much more effective, in practice, for AI researchers to mimic some of what the brain does: tweak the weights of a huge neural network until its behavior is approximately what you wanted.
Not to say these neural networks are really brain-like; they aren't. There's a lot about human thinking they fail to capture. But they are more brain-like than GOFAI, and they do tend to be vastly more capable than GOFAI programs in the same domains. ML translation > GOFAI translation; ML text to speech > GOFAI text to speech; ML image generation > GOFAI image generation; ML chess playing > GOFAI chess playing; ML common-sense reasoning > GOFAI common-sense reasoning; and so on. The areas where GOFAI still has a hold tend to be relatively simple domains where it's important for a human to be able to interpret why the machine made its judgment.
