Sometimes it happens that a person doesn't want to accept your argument, because he claims not all the inferences are valid. There's a famous example of Lewis Carroll, namely What the Tortoise Said to Achilles, where nonacceptance of a basic inference rule leads to infinite regression.
Sometimes, it's the other way around, where a student doesn't want to acknowledge your proof. It might be more subtle, like with the non-constructive proof that an irrational raised to the power of another irrational can be rational, or it could be less subtle, e.g. when pupil doesn't concur that $$(\forall \varepsilon > 0\;. \ |x| < \varepsilon) \implies x = 0$$ for $x \in \mathbb{R}$ (I remember one guy that agreed on $x \neq 0 \implies |x| \not<\frac12|x| > 0$, but wouldn't accept the statement above).
To give a different kind of example, consider the following. For propositions $P(x)$ and $Q(x)$ we could infer that $\exists x. P(x) \land Q(x)$ from $\exists x. P(x)$ and $\exists x.\ Q(x)$. This is not true in general, but there are cases where it works, e.g. if $P$ and $Q$ are independent (this could be formalized as, for any considered model $\mathcal{M}$ and element $x \in \mathcal{M}$ we have $\mathcal{M},P(x) \not\models Q(x)$ and $\mathcal{M},P(x) \not\models \neg Q(x)$, etc.). While a scrutinizing approach would be to work out all the details, one might want the students to apply a more intuitive, higher-level perspective; in my opinion, such exercises are helpful to develop a deeper understanding. Yet, some don't even want to start participating.
This happens also in non-math fields, like psychology or biology where papers on non-widely-accepted theories (like group selection) are ridiculed, for example, because the audience doesn't handle well the concept of "assumption", or because the author insists on something bizarre without considering it might be false.
Question: How can we resolve conflicts that stem from disagreements on the fundamentals?
