What does "counter-reasoning" exactly mean? I have got trouble finding its meaning in the web or in a dictionary.
According the "Harvard College Writing Center"
When you write an academic essay, you make an argument: you propose a thesis and offer some reasoning, using evidence, that suggests why the thesis is true. When you counter-argue, you consider a possible argument against your thesis or some aspect of your reasoning... it can be a persuasive and (in both senses of the word) disarming tactic. It allows you to anticipate doubts and pre-empt objections that a skeptical reader might have
According to Wikipedia:
... a counter argument is an objection to an objection
The same source gives this useful schematic (Graham's Hierarchy of Disagreement):
EDIT: I’ve just realised that I haven’t used the exact phrase counter-reasoning. The explanation involving these exact words is difficult to come by on the internet. To support the claim that the above does answer the question:
Counter (as an adjective) MW defines as:
marked by or tending toward or in an opposite direction or effect
There are not many examples of usage (about 825 on Google Books, and some false positive), but this example conveys the same meaning for counter-reasoning as the term counter-argument (explained above). It is a note in a translation of Plato’s Protagoras talking about Bonitz using counter-reasoning by Socrates as a proof of a certain statement being refuted. It’s the most direct explanation I managed to dig up.