Does it reveal ignorance to speak of an axiom as being true?

Logic Personified: To speak of an axiom as being "true" is to reveal ignorance.

Boojatta: Do you agree that if there exists even one instance of a rule of inference that involves an assumption that is true and a conclusion that is false, then that rule of inference is faulty?

Logic Personified: Yes, of course. You can rely on me.

Boojatta: Is it possible for an instance of a rule of inference to involve an assumption that is an axiom of standard mathematics?"

Logic Personified: Yes, of course. Otherwise, the axioms of standard mathematics would be cut off from everything else. You would be unable to use the axioms to prove any theorems.

Boojatta: Okay, let's put that together now, shall we? Suppose that there exists at least one instance of a rule of inference that involves a true assumption that is an axiom of standard mathematics and that involves a false conclusion. Is such a rule of inference faulty?

Logic Personified: Yes, so what?

Boojatta: "Yes" isn't good enough. Please say it as your own utterance. I, Logic, do solemnly believe...

Logic Personified: Very well, but as a Quaker I refuse to take the oath. Suppose that there exists at least one instance of a rule of inference that involves a true assumption that is an axiom and that involves a false conclusion. Then that particular rule of inference is faulty.

Boojatta: Are you just saying that without conviction, but merely to be agreeable?

Logic Personified: No, I meant it.

Boojatta: You spoke of an axiom as being true. You have revealed your ignorance.

Socrates was mortal.
Therefore all men are Socrates.

then I would see how you obtain your conclusion following the word "therefore." However, that wasn't stated as one of your assumptions.

Also, it's obvious that the conclusion "all men are Socrates" is false, so it's unlikely that anyone would put forward the argument as anything but a joke. On the other hand, the claim "To speak of an axiom as being true is to reveal ignorance" has been made seriously.

You have LP state: Suppose that there exists at least one instance of a rule of inference that involves a true assumption that is an axiom and that involves a false conclusion. Then that particular rule of inference is faulty.

But, let: All A is B be an axiom.

Then using standard deductive logic, we can form the syllogism:

All A is B
All B is C
Therefore, All A is C.

Now, assume All B is C is a false propostition, and further that some of the B that is not C belongs to A.

Then we have met the criteria: Suppose that there exists at least one instance of a rule of inference that involves a true assumption that is an axiom and that involves a false conclusion; but, there is no reason to conclude that the rule of inference is faulty.

However, I would characterize it as failure to articulate an idea carefully enough, rather than as lack of knowledge of the idea. You are correct, of course. It should be rewritten to say...



... unless of course there is also some other problem lurking in there that needs to be addressed to make the formulation both clear and correct.