I agree—modus ponens works, even though there are some minds who will reject it with internally coherent criteria. Even criteria as simple as “modus ponens works, except when it will lead to belief in the primness of 7 being added to your belief pool”—this definition defends itself because if it was wrong, you could prove 7 was prime, therefore it’s not wrong.
You could be put in a room with one off these 7-denialists, and no argument you made could convince them that they had the wrong form of modus ponens, and you had the right one.
But try seeing it from their perspective. To them, 7 not being prime is just how it is. To them, you’re the 7-denialist, and they’ve been put in a room with you, yet are unable to convince you that you have the wrong form of modus ponens, and they have the right one.
Suppose you try to show that a universe where 7 isn’t prime is internally inconsistent. What would the proof look like? Well, it would look like some axioms of arithmetic, which you and the 7-denialists share. Then you’d apply modus ponens to these axioms, until you reached the conclusion that 7 is prime, and thus any system with “7 is not prime” added to the basic axioms would be inconsistent.
What would the 7-denialist you’re in a room with say to that? I think it’s pretty clear—they’d say that you’re making a very elementary mistake, you’re just applying modus ponens wrong. In the step where you go from 7 not being factorable into 2, 3, 4, 5 or 6, to 7 being prime, you’ve committed a logical fallacy, and have not shown that 7 is prime from the basic axioms. Therefore you cannot rule out that 7 is not prime, and your version of modus ponens is therefore not true in every possible universe.
Just because you can use something to prove itself, they say, doesn’t mean it’s right in every possible universe. You should try to be a little more cosmopolitan and seriously consider that 7 isn’t prime.
I’m guessing you disagree with Elizier’s thoughts on Reductionism, then?
The 7-denialists are making a circular argument with your first defence of their posistion. Circular arguments aren’t self-evidently wrong, but they are self-evidently not evidence as there isn’t justification for believing any of them. The argument for conventional modus ponens is not a circular argument.
The second argument would be that the 7-denialists are making an additional assumption they haven’t proven, whilst the Foundationalist Skeptic starts with no assumptions. That there is an inconsistency in 7 being prime needs demonstrating, after all. If you redefine Prime to exclude 7 then it is strictly correct and we don’t have a disagreement, but we don’t need a different logic for that. (And the standard defintition of Prime is more mathematically useful)
Finally, the Foundationalist Skeptic would argue that they aren’t using something to prove itself- they are starting from no starting assumptions whatsoever. I have concluded, as I mentioned, that there is a problem with their posistion, but not the one you claim.
I agree—modus ponens works, even though there are some minds who will reject it with internally coherent criteria. Even criteria as simple as “modus ponens works, except when it will lead to belief in the primness of 7 being added to your belief pool”—this definition defends itself because if it was wrong, you could prove 7 was prime, therefore it’s not wrong.
You could be put in a room with one off these 7-denialists, and no argument you made could convince them that they had the wrong form of modus ponens, and you had the right one.
But try seeing it from their perspective. To them, 7 not being prime is just how it is. To them, you’re the 7-denialist, and they’ve been put in a room with you, yet are unable to convince you that you have the wrong form of modus ponens, and they have the right one.
Suppose you try to show that a universe where 7 isn’t prime is internally inconsistent. What would the proof look like? Well, it would look like some axioms of arithmetic, which you and the 7-denialists share. Then you’d apply modus ponens to these axioms, until you reached the conclusion that 7 is prime, and thus any system with “7 is not prime” added to the basic axioms would be inconsistent.
What would the 7-denialist you’re in a room with say to that? I think it’s pretty clear—they’d say that you’re making a very elementary mistake, you’re just applying modus ponens wrong. In the step where you go from 7 not being factorable into 2, 3, 4, 5 or 6, to 7 being prime, you’ve committed a logical fallacy, and have not shown that 7 is prime from the basic axioms. Therefore you cannot rule out that 7 is not prime, and your version of modus ponens is therefore not true in every possible universe.
Just because you can use something to prove itself, they say, doesn’t mean it’s right in every possible universe. You should try to be a little more cosmopolitan and seriously consider that 7 isn’t prime.
I’m guessing you disagree with Elizier’s thoughts on Reductionism, then?
The 7-denialists are making a circular argument with your first defence of their posistion. Circular arguments aren’t self-evidently wrong, but they are self-evidently not evidence as there isn’t justification for believing any of them. The argument for conventional modus ponens is not a circular argument.
The second argument would be that the 7-denialists are making an additional assumption they haven’t proven, whilst the Foundationalist Skeptic starts with no assumptions. That there is an inconsistency in 7 being prime needs demonstrating, after all. If you redefine Prime to exclude 7 then it is strictly correct and we don’t have a disagreement, but we don’t need a different logic for that. (And the standard defintition of Prime is more mathematically useful)
Finally, the Foundationalist Skeptic would argue that they aren’t using something to prove itself- they are starting from no starting assumptions whatsoever. I have concluded, as I mentioned, that there is a problem with their posistion, but not the one you claim.