"Since all cats are mammals, and all mammals feed their babies with milk, I think cats are also mammals"
Oh boy. This is no good. No good at all. It's neither an example for conscious thinking, nor a sample for logical thinking. I bet you meant to deduce that cats feed their babies with milk, this what you'd call logic is a formal logic.
However, your question has been answered several times by different authors. They all agreed, that under certain conditions and with the proper tasks, programming will make you think more logical in general and especially better in formal logic thinking.
The ongoing debate about what logic actually is, and where one may find samples for logic of all kind of (sub|un)consciouss thinking, is part from a huge other debate. Many subjects of that issue, consciousness, subconsciousness, free will, arts, logics and so on are actually subjects which dont have a proper definition. Despite the fact that wiki may explain all terms i have mentioned here, this doesnt automatically mean, that the debate in general (not ours here) is over yet. Neither modern psychology, classic psychology nor philosophy of mind and affected schools thereof (like philosophy of cognition) have statements on any of these topics.
You described the mentioned pieces from Lennon and Yoko as "their music aswell as their art" - in this particular case, i'd say it was rather art than music.