In some ways "Aristotelian thinking" is an archaic misnomer that no longer fits our current model-theoretic paradigm. We need more to learn ways of reasoning, the appropriat arena for each way, how they fit together, and learn to recognize the majority of fallacies that people commonly use. But the most important activety is knowledge of, how to apply, and awareness in use, of the model-theoretic perspective that derives from Tarski and is applied to science by Popper, and which Korzybski's structural differential illustrates.
See this pdf article: http://time-binding.org/gsb/gsb47-mayper.pdf

Sane reasoners do not worry about the misused, fuzzy, and ill-defined label "Aristotelian". The abstraction is too high to be useful, pragmatic, or even to provide any kind of useful judgement. It seems to be most used by indivuals taking an "anti-Aristotelian" stance - and there is some documentation (outside of general semantics) in the time-binding record as to how this has been used.

We must learn the relationship between truth-preserving intensional logic as it has developed in mathematics and logic theory, and probabalistic reasoning, used in the context of observational research that seeks to model what is going on - in the hard sciences of physics and cosmology, in the not-so hard applied pragmatic disciplines such as engineering, in the softer crafts such as building, farming, illustration, etc,, and in the very soft disciplines such as psychology, anthropology, etc.

Nearly all scientist, and professionals in most other areas, are aware today that our ideas about what is going on is not the same as what's going on. This "map is not the territory" applied to virtually all knowledge is part and parcel of the modern model-theoretic paradigm. Polsters know it. The major news organizations know it.

We have gone from Walter Cronkite's "That's the way it is" of yesteryear to "Thats the news (we picked out)".

Model theoretic paradigm.

We abstract from what is going on.
We construct models to help predict.
We use the models to make predictions.
If the prediction fails, it's back to the drawing board with the model.
We revise the model and try again.
This process is understood to be never-ending, but we do stop worrying about prediction failures after a while. If I drop my coffee cup, I try to grab it because I do not hope that this time it will float up. I take my model for granted when it has achieved enough corroboration. BUT I know it might still fail sometime.

The general semantics cry "The word is not the thing" now falls mostly on deaf ears, because it is now part of the normal general model-theoretic paradigm. Tarski gave us one of the first formal mathematical models of such a model theory.

Karl Popper integrated it and explained it in science terms. It is still current philosophy of science, but we are now more interested in modeling the creation process in which new models are devised. The paradigm there is evolutionary epistemology.

I would say that most politicians are very aware that they can choose many different words to "hook" there prospective constituents; they can make something sound to have the opposite value. They can talk all around a question touching on point a listener wants to hear without actually making any commitment to or for something. Two valued they aren't. Politicians care about their ability to exercise power in such a way as to maintain supporters sufficient to remain in power.

However, in all fairness, there are "politicians" who champion a cause, and they stick with it, such as Al Gore on the environment. His cause takes presidence over obtaining and maintaining power.

Virtually every group has "boundary conditions" that distinguish its members from non-members. In the case of general semantics one is striving to understand and apply "non-Aristotelian" reasoning. But many novices have insufficient mathematics and logic experience and training to understand what is and what is not valid reasoning. Without that experiential knowledge, the use of the terms Aristotelian and non-Aristotelian become practically empty labels, but they carry, for the novices a "sting" and a "reward" simply because they are associated by two-valued orientation with the membership boundary conditions.

Valid logic "proves" propositions in the truth preserving reasoning methods, not to be confused with probabalistic descriptive and inferential methods which are not strictly truth preserving; non-Aristotelian reasoning knows when to use which and how to integrate these.

But if you challenge the so-called "laws of thought" you are attacking a straw-man that wasn't even Aristotle's.

We don't need to "recognize" "Aristotelian" reasoning, because the label is ill-defined.
What we do need to do is learn about and learn to avoid, and practice avoiding using fallacious reasoning. We need to learn valid two-valued logic. We need to learn valid multi-valued logic. We need to learn when to "correctly" use each.

The "hunt for" "aristotelian" logic as used by others becomes the thought police directed at others. Take responsibility for your own reasoning by learning how to recognize one-self using a fallacy to reason, and practice, practice, practice.