Knowledge is to cut, divide, separate (etymologically).
Logic is to collect (etymologically).
Knowledge requires logic to arrive at truth.
I think the above demonstrates a rule that is previously ignored which would prevent the tool of logic from leading someone off course. The user uses a tool but if the rules of how to use it aren't known, then it can be misused. Logic will work on a soundly constructed origin.
But when using a=b, that violates the law of identity. It's not explicitly shown, but that's what the above states. 'ab' can only equal 'a^2' if a=b. Constructing a false setup and using logic on a false setup will produce a false result. The logic isn't the problem. In the end, with the false result, one can recognize its falsity through logic that collects and compares the data for consistency and non-contradiction.
The logical problem in my example of 1=2 would actually be the last step mathematically. I agree that it may violate the law of identity, but in math, if a = 1 and b also = 1 then a would be equal to b. The problem is to cancel out the a^2 - ab you would need to divide each side. Technically this would be a division by 0 which causes the problem. We cannot divide by 0 since we cannot define the quotient, so it's not legitimate to divide by 0. My post was mainly to prove what you stated, "Constructing a false setup and using logic on a false setup will produce a false result". I'm more of a programmer than a philosopher, but I personally feel like philosophy and programming share similar roots. Philosophy being the love of wisdom or the seeking of wisdom. I think that we find answers to questions we cannot fully answer by using logical deduction to achieve the best possible answer and widen our understanding of things. Thank you for explaining your point of view, I read your stuff often and it is quite interesting.