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2 ideas
10704 | We can formalize second-order formation rules, but not inference rules [Potter] |
Full Idea: In second-order logic only the formation rules are completely formalizable, not the inference rules. | |
From: Michael Potter (Set Theory and Its Philosophy [2004], 01.2) | |
A reaction: He cites Gödel's First Incompleteness theorem for this. |
10703 | Supposing axioms (rather than accepting them) give truths, but they are conditional [Potter] |
Full Idea: A 'supposition' axiomatic theory is as concerned with truth as a 'realist' one (with undefined terms), but the truths are conditional. Satisfying the axioms is satisfying the theorem. This is if-thenism, or implicationism, or eliminative structuralism. | |
From: Michael Potter (Set Theory and Its Philosophy [2004], 01.1) | |
A reaction: Aha! I had failed to make the connection between if-thenism and eliminative structuralism (of which I am rather fond). I think I am an if-thenist (not about all truth, but about provable truth). |