Your thinking is most certainly rational. The contraposition to Godel's incompleteness theorem tells us that any framework with sufficient explanatory power will necessarily contain contradictions. Since we attempt to reason about everything, our framework is necessarily large enough to be full of contradictions. Since we've got to deal with contradictions, they are not something to be avoided but rather
acknowledged. If you're not acknowledging the contradictions and the "opposite side" for the implications you visit, then you will miss when that "other side" starts making more sense than the chain you're following. Not doing this means ending up at a nonsensical position while ignoring its contradictory obvious truth, a result we call cognitive dissonance.
This dual-thinking is related to the computer security mindset - you can't naively write code thinking your assertions will simply hold as you intend, but rather you need to be continually examining what every assertion "gives away" to a hostile counterparty.
There are alternative systems of logic that attempt to formalize reasoning in the presence of contradictions, to keep a single contradiction from being able to prove everything. For example, intuitionistic logic and paraconsistent logic. These feel much more in line with reasoning in an open world where a lack of a negative doesn't necessarily imply truth. The focus on a singular "logic" that asserts that everything has some single rational "answer" is a source of much of our modern strife.