Consider writing a piece of music. You will have to admit that involves thought. You can argue there is a kind of logic, an aesthetic one maybe. But it is nothing like formal logic.
Natural language is full of https://en.m.wikipedia.org/wiki/Vagueness and formal logic is abstracted out of it, rather than preceding it, similarly to mathematics. Private language is as meaningless as private money, language and money derive their currency and value from communities of practice.
Even physics has to face this. Abstractions relate to real things, and can be evaluated as being more or less correct. Laws, including logic, then are created based on the given abstractions - but the laws are only as real as the abstractions are good, the laws don't truly have their own existence. This view is developed in 'How the Laws of Physics Lie by Nancy Cartwright' http://www.oxfordscholarship.com/mobile/view/10.1093/0198247044.001.0001/acprof-9780198247043
It must be noticed that abstraction always simplifies, it looks for core dynamics, it seeks to make things more computable. But reality remains ontically irreducible, and whatever it does that is what it is and how it behaves, however that violates our expectations or 'laws'.
"Meaning gets going because we move around and act on a world of other objects and agents; pragmatic engagements in the world, which logically precede language. It is these practical engagements, rather than the shared logical form of the Tractatus, that enable meaning. We do not mirror reality. We are enmeshed in it." https://www.the-tls.co.uk/articles/public/ludwig-wittgenstein-honesty-ground
It comes back to Popper also, who pointed out that science cannot be purely inductive, deriving results only from observations. Hypothesis generation is an essentially aesthetic practice, but only one held to the standard of being falsifiable but not being falsified. Mathematics also goes like this, with creativity, insight, and aesthetic judgements required, as well as logic.
Edit to add: You might also consider meditative states of mind, where a person shifts entirely to observation of mental formations, rather than engagement or participation with them. This state can be linked with 'flow', also an active way of being present in what you are doing, rather than analysing and critiquing it.
This topic put in mind also, of how philosophers have used logic to knock itself down, to draw out it's inadequacies and inconsistencies that show the limits of logical thought: https://absoluteirony.wordpress.com/2014/09/17/nagarjuna-nietzsche-rorty-and-their-strange-looping-trick/amp/For Wittgenstein the 'superset' is language games.