Logic, polarity thinking, and factoring proportional dialectics...
Logic and argument are valid and important cognitive tools. But, like any tools, they have their appropriate uses and appropriate limitations.
The uses of logic....
Dissect
One appropriate use for logic is cognitive dissection. When we need to take apart a cognitive construct, logic is usually the best tool to use. Note: There are a variety of logics, (screwdrivers come in different sizes and shapes) for various types of constructs.
Construct
Another appropriate use for logic is cognitive construction. We can construct arguments using logical steps (again, various types of logic are more or less suited to the construction of various types of cognitive systems).
Resurrect
Occasionally, logics can be useful for heuristic reconstructions. For instance, logic can support an imaginative reconstruction of a past or future scenario.
The limitations of logic...
Appropriate limitations to logic have to do with logics' as yet essentially dual structure. Our logics are still similar to digital systems. Logics do their work by progressing through a series of yeas and/or nays. If x is yes then... but if x is no then.... is a basic logical construction pattern. And, as you can see from Vygotsky's schema, not a conceptual system at all.
Logics have spawned two other complex forms of digitally based cognitive constructs: dialectics and argument (debate). Dialectics and debate are based on a yea-or-nay analytic. Less obviously, they are also both examples of circular reasoning.
Because of the fundamentally digital nature of our logics, it is difficult to use any logic for compromise, negotiation, collaboration and consensus building, not to reaction logics famous ability to ruin any love making dynamic.
Logics have a great deal in common with mathematics. We owe a great deal to the affinity of those systems: Almost all of our present-day industries and technologies were developed from the synergy of math and logic.