From the medium hard question, we have “Löb’s Theorem shows that, whenever we have ((â»C)->C), we can prove C.” From the cartoon, the hypothesis of Lob’s theorem is, ” If PA proves ‘PA proves X, then X’ then PA proves X”, so to apply the theorem to the situation outlined in the question we would be required to write instead of “((â»C)->C)” in the first sentence the following “PA proves ‘((â»C)->C)’ ”. Then by the deduction theorem, we’d have the following conclusion: ‘â»(‘(â»C)->C’)->â»C’ and so the trouble with material implication wouldn’t arise because the first ‘->’ is in a quoted sentence.
From the medium hard question, we have “Löb’s Theorem shows that, whenever we have ((â»C)->C), we can prove C.” From the cartoon, the hypothesis of Lob’s theorem is, ” If PA proves ‘PA proves X, then X’ then PA proves X”, so to apply the theorem to the situation outlined in the question we would be required to write instead of “((â»C)->C)” in the first sentence the following “PA proves ‘((â»C)->C)’ ”. Then by the deduction theorem, we’d have the following conclusion: ‘â»(‘(â»C)->C’)->â»C’ and so the trouble with material implication wouldn’t arise because the first ‘->’ is in a quoted sentence.