Also, I should add that I could just deny logic. For instance, I could just ask why I should even accept logic, or why I should accept anything at all. You could give me the most rigorous mathematical proof there ever was, and I could just go "But why should I accept that?" And the mathematician could go into all the deep mathematics, logic, etc., but you can always still say, "But why should I accept that?"
The thing with reason/logic, etc., is that it forces itself upon you. When you utter "But why should I believe that?", you are already betraying what you already believe: namely, that you are a competent language speaker, that you are talking to another person. You may not say, but you show that you affirm the principles of logic (i.e., when you ask "Why should I accept that?" you are implicitly also saying, "Why should I not not accept that?" and are differentiating your question from "What should I not accept that?" You've, in speaking, accepted certain things.
This is also the kind of realisation that underlies Descartes' famous cogito argument: I think therefore I am. You can doubt as much as you want, but you cannot ever doubt that you are a thinking thing. To doubt, you need to think.
And even on "doubting" (you should refer to Wittgenstein's On Certainty for this), there are certain things you must accept in order for you to even begin doubting.
There are different strategies for dealing with this. Here are some.
You can take the performative strategy and point out that in even speaking, someone is showing what they actually believe (e.g., that the world exists, other minds exist, etc.). Or else, why would they be talking to you or contemplating these things? So, right away, you know you're just dealing with a bullshitter, and you should just move on. In brief, this is the "Point out that the person is just being an insincere prick" strategy.
You can call into question what it even means to "believe" something, or to "know" something, or what it even means for something to be "true". This is what epistemologists normally do. So, when you even ask the question "How do I know that this is even objectively true? Why should I believe this? How could I know this?" You are using the terms "know", "believe", "true". Depending on what those mean, the philosopher could defeat you. Because (s)he can just shoot back at you: "What would it take for you to believe or know or think that these things are true?" And then you'd be in the position of articulating what you even mean when you started doubting these things, and chances are the philosopher has a better analysis of what knowing/believe/true even means. So, you're asking a question you aren't even sure means.
The philosopher can accept what you say, but just show that it doesn't really do the work you think it does. For example:
Wittgenstein does not try to refute skeptical doubts about the existence of an external world so much as he tries to sidestep them, showing that the doubts themselves do not do the work they are meant to do. By suggesting that certain fundamental propositions are logical in nature, Wittgenstein gives them a structural role in language: they define how language, and hence thought, works. “Here is a hand” is an ostensive definition, meaning that it defines the word by showing an example. That statement explains how the word hand is to be used rather than making an empirical claim about the presence of a hand. If we begin to doubt these sorts of propositions, then the whole structure of language, and hence thought, comes apart. If two people disagree over whether one of them has a hand, it is unclear whether they can agree on anything that might act as a common ground on which they can debate the matter.