Hypothetico-deductive method, abductive reasoning, induction, deduction, reduction to absurdity, dialectic.... There's reasoning in mathematics, too.
What is the fundamental reasoning that creates all these others?
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The brain created all those reasoning methods. When you ask for the "most basic" reasoning, you're asking how the brain works innately, according to our genes.
Somehow, the brain builds models of the world and itself, updates and corrects or discards the models as new information comes in, and uses these models to achieve objectives like "get food."
How the brain does this is a mystery to all. If we knew, we would already have an AI able to think like a human, and we don't.
If logical pluralism is true, there might not be a fundamental kind of reasoning. The consequence relation might differ from case to case so that not only do reasons vary from case to case, but how those reasons are reasons varies, too. In mathematics, for example, it might make sense to apply normal deductive logic to relatively infinite sets but paraconsistent logic to absolute infinity.
Generalizing just a tad bit more (or going a little sideways), it could be that foundationalism was the true pattern of some reasoning, while coherentism fit better with another kind of reasoning, and then infinitism worked better for yet another rational sphere. (For example, if reflective equilibrium is coherentistic in structure, and if it is the most fitting structure for ethical reflection, then...)
Reasoning is a fairly general and vague term. At ultimate neural science level, our reasoning may be just eliminated to an illusion, since at this synaptic/neural level our understanding and reasoning (at least in the right hemisphere of our brain) may not be based on any symbolic propositional language via any formal logic system instead it may be closer to some recurrent feedforward and backpropagation input/output complex nonlinear dynamical system.
In philosophy, reasoning more or less implies some superiority over animals for most rationalists. For empiricists like David Hume, reasoning may not be fundamental at all as he famously proposed "Reason is, and ought only to be the slave of the passions". For me, it seems abductive reasoning is more basic than others in your taxonomy above.
Personally I view the most basic and distinct reasoning as self-critique. Advanced animals can no doubt see the sky, enjoy some styles of music or even abductively reason about where to find foods based on its past experience, but seems none will doubt itself seriously, as we never see a healthy dog rejects eating simply because it's meditating or self-criticizing about something it just perceived or conceived of moment ago. Self-critique is the sign of the beginning of any kind of dialectical thinking, and clearly René Descartes valued this unique human aspect so much that he considered this is the true criterion of human existence, also this is why he's considered a rationalist and the founder of modern western philosophy. We all see people say a lot only to support their own view at the end of the day, but we seldom see one says a lot in order to rationally and critically conclude he or she may be wrong...