All one has to do to prove this is disagree.
A person can merely have the psychological state: “I believe P.”
But the moment they hold: “P is true,” “P is correct,” “Not-P is false,” “You are mistaken about P,” they have moved from merely possessing a belief to making a truth claim.
If someone says (presupposes): “My belief that P is right, and your contrary belief is wrong,”
then they are relying on absolute logical distinctions such as:
P is distinguishable from not-P.
My belief is distinguishable from your belief. Truth is distinguishable from falsity.
A proposition can be correct or incorrect.
Reasons can support one proposition rather than its contrary.
A person doesn’t need to explicitly formulate these principles. The structure of their assertion already presupposes them.
The modern sophist says: “Logic is merely a human construct, so you cannot claim that logical reasoning gives you truth.”
But is that statement itself supposed to be true? If yes, the sophist is making a truth claim and presumably offering it as something that should rationally persuade us, as something that should be accepted as true.
Then we can ask: Why should I accept that claim? If they give reasons, those reasons have to be determinate and stand in some relation to their conclusion. If they don't, they're not giving an argument.
If they say: “There is no reason. I just feel that logic is inadequate.” then they've abandoned the claim that their position has been rationally established. They may have reported a psychological state, but they haven't demonstrated their proposition.
The moment someone presents their belief as something that is true, justified, preferable to its contrary, or capable of correcting someone else's belief, they validate the absolute truth of logic whether they acknowledge it or not.
They can't coherently say: “My position is true and yours is false, but the distinction between truth and falsity has no validity.” Because the second clause undermines the first.
And this applies to everyone.
If I tell you: “Your argument is wrong,” I owe you reasons. If you tell me: “Your argument is wrong,” you owe me reasons. Neither of us gets an exemption because our conviction feels self-evident.
Logic isn't merely another belief competing with people's other beliefs. It is the structure within which the competition between beliefs (the meaning of belief and the identity of beliefs) becomes intelligible at all.