6 reasons we know logic is not merely a man-made convention. Pay special attention to reason no. 2.
1. A world where logic wasn’t real would be an incoherent world.
If someone wants to say that logic is a human convention, then he'll have to defend that claim.
He'll have to present a worldview with a metaphysic that isn't fundamentally logical and where the laws of logic aren't universal and objective (so in that imaginary world A can equal not-A, no excluded middle, a thing is not itself).
2. Laws of logic that are merely “conventions” are actually false.
Our skeptical friend would have to explain how and why, in a world that is not inherently logic—where the laws of logic aren’t real and didn’t exist without humans...
—humans suddenly came along and developed laws of logic, “conventions” which, in that kind of universe actually fly in the face of reality.
These “conventions” would not only inadequately describe reality, they would functionally lie about reality. After all, in this hypothetical world, logic isn’t actual.
And since human conventions say they are, they are false. (Of course, in a non-logical world, what’s false can be true in the same sense at the same time, so the “conventional” laws of logic could be lies and true, and, well you get it. It’s a great big incoherent mess.)
3. Laws of logic that are merely conventional aren’t really “laws” at all—they have no teeth. Why would anyone be obligated to follow a standard (these hypothetically man-made laws of logic) that go against reality?
Remember, in a non-logical world, laws of logic lie about the way things are. So in this hypothetical scenario, those who “violate logic” through contradictions are actually acting in agreement with the way really are.
In this world, being logical is going against reality, and following logic is following falsehoods. Violating logic is tantamount to refusing to follow lies. Is the skeptic really going to get mad at people who don’t follow lies?
If not, then he can’t get mad at people who don’t follow logic. After all, according to him, logic is merely a convention. That’s not how the world really is.
If our skeptic says logic is manmade and not objective, you could simply reply, "So it isn't manmade and objective." He couldn't argue. After all, the two of you are clearly operating by different conventions.
And he couldn’t appeal to his conventional laws of logic in any kind of objective way to prove you wrong. (Side note: in our transgressive culture, the avant-garde take pride in smashing conventions, and they’re often rewarded for it.
If logic were purely conventional, one of the “best things” someone could do, culturally speaking, might be to make a career out of being illogical and contradictory. And our skeptical friend couldn’t say boo about it).
4. By proposing the argument, the skeptic is showing he doesn’t really believe it. He wants to be understood logically. All the while the conversation is happening, he'll be expecting you to understand him logically, as if the laws of logic are real.
This undermines his claim that logic is merely conventional; it is rather the behavior of someone who believes logic is universal and objective, and applies to both you and him (and, one assumes, anyone else who overhears the conversation or anyone else he might converse with).
In this way he is showing that he doesn't really believe logic is purely conventional. Remember, if logic is just a convention, then it isn’t objective and real in the sense that it is universal and absolute, reflecting the way reality actually is.
In terms of apologetical discussion, the key is to show your opponent that he doesn't live like he actually believes what he claims to believe. He doesn't live like logic is purely conventional (and not objective and universal).
5. Human experience reveals that logic is discovered, not invented.
Human conventions are useful for organizing things humans can control (like the grammatical rules of language, certain societal laws, etc.). But are laws of logic really like that? No. Logic is something most people don't study, and yet they conform to it all the time.
They naturally and instinctively revile contradictions. They understand that a thing is what it is. In other words, people naturally think and act logically—not perfectly, but normatively—and expect others to do the same.
Our natural bent is not to treat logic as a manmade convention but as the way things are.
We instinctively think and act logically. It's only later, upon reflection, that they learn about logic and realize that this is the way they've been operating in the world all along.
Logic is discovered, not invented.
6. If logic is a human convention, then it would not exist where no humans existed.
There are no humans on Mars. If you went to Mars, could you invent your own logic? Could you stand on Mars and say "I am here and not here at the same time in the same sense" and be correct?
Obviously, not—and not only not in terms of our (supposedly) subjective earth-bound conventions. It is plain that our martian would be uttering a false statement, and that the statement is false because it’s illogical. But if logic is purely a human convention, it could be true.
Since that’s wrong, we can see that logical is not merely conventional.
CONCLUSION: TRYING TO MAKE LOGIC “CONVENTIONAL” DOESN’T SOLVE ANYTHING
In short, by attempting to make logic a merely human convention, the skeptic is showing that he doesn’t understand logic (which is not invented but discovered), but at the same time he is undercutting his ability to attack the Bible for supposedly containing contradictions.
After all, if logic were merely a manmade convention, then why accuse the Bible of having a different convention? And whose to say the skeptic’s is the “right” logic, and the Bible’s is “wrong?” They’d just be “different.”
The truth is that logic is real, and it is discovered by humans, not invented by us. If logic were not universal and objective, then logic would be universal and objective. And if trying to understand that seems like an exercise in futility, you’re right, it is.
Just like trying to deny the reality of logic, and the reality of the Triune God of Scripture.