Well, I am doing mathematical logic instead of philosophical logic

You seem to have a notion there's a signifigant difference, maybe confusing them with formal and informal logic. No one's doing Kantian, Hegelian or even Aristotelian logic here. Logic hasn't been philosophical since maybe Frege.

so you have to excuse my unfamiliarity with terms used in philosophy.

Model theory is considerd pure maths by most practisioners.

Perhaps you could redirect your energy to instead providing what you would consider a satisfactory answer to the OP's question?

Why should I? People seem to think the answer they recognize is good enough. It's a pedagogical criticism.

Usesbigwords already gave them a short answer. Like showing the square root of two is irrational, by simply saying, it's a contradiction. It might provokes some sort of epiphany, by luck, but that just seems haphazard and poor style. Without confirmation from the OP, it's doubtful they fully appreciate it. They may possibly even not understand it, except as a rote response to their question. Without their confirmation, there's no way to know. It's debatable whether simply just giving out answers like that is ever of much benefit anyway.

Incidentally if you're studying mathematical logic only deductively you're missing another important approach. Churchs book 'Introduction to mathematical Logic' though a bit old, might be useful. Jeffrey is ok. Mathematic Logic by Kleene is a really good book for a rigorous introduction to model theory, though the later sections are graduate level. There's others like Crossley's but it's short and not a gentle introduction. Suppes Introduction to logic is good, and in case you have some weird prejudice against philosophy departments, he was a mathematician. His style is remarkably clear, precise and it's a gentle introduction, but very comprehensive and progresses quite far. There's other good books if you wanted to persue axiomatic set theory.