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Logic has very little to do with critical thinking. The amount of logic needed is simply what it means for one thing to imply another, and not necessarily the other way around.
What critical thinking is really about is being able to identify what assumptions you and others are making, as well of course as being able to realize that you and another person are not necessarily attaching the same meanings to the words you're using.
These assumptions are often called "hidden assumptions" because it is often not immediately apparent what they are.
An example of critical thinking would be trying to identify the hidden assumptions in a statement like "Compulsory schooling is a good thing." The point here is not to argue, but to actually try to understand what kind of model-of-the -world someone has to have in order to be able to make a statement like that.
(And a possibly hidden assumption here is that the person is actually trying to make a meaningful statement based upon their own beliefs and not simply parroting back something they've heard with no meaningful engagement on their part - an assumption that is not always warranted!)
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