Sherlock Holmes used _both_ inductive and deductive logic, indeed, as a few commenters have noted, more formally, abductive logic. The process went somewhat as follows in sequence: observation --> inference --> deduction. The inference, rather Sherlock himself was the inference engine, as he raised several theories (he used the word "theory" on more than one occasion), where the theory would be for explaining EVERY fact that he observed. Here is the crux: note that in Sherlock Holmes stories, his observations are so meticulous that he does not omit a single fact. Then he generalises or infers from these facts one or more than one theories which he then tests with hypotheses. This is where deduction comes into the picture, i.e., from generalisable truths to particular instances that he would then test with gathering more facts.

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