18189
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ZFC could contain a contradiction, and it can never prove its own consistency [MacLane]
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Full Idea:
We have at hand no proof that the axioms of ZFC for set theory will never yield a contradiction, while Gödel's second theorem tells us that such a consistency proof cannot be conducted within ZFC.
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From:
Saunders MacLane (Mathematics: Form and Function [1986], p.406), quoted by Penelope Maddy - Naturalism in Mathematics
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A reaction:
Maddy quotes this, while defending set theory as the foundation of mathematics, but it clearly isn't the most secure foundation that could be devised. She says the benefits of set theory do not need guaranteed consistency (p.30).
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17896
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We need to know the meaning of 'and', prior to its role in reasoning [Prior,AN, by Belnap]
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Full Idea:
For Prior, so the moral goes, we must first have a notion of what 'and' means, independently of the role it plays as premise and as conclusion.
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From:
report of Arthur N. Prior (The Runabout Inference Ticket [1960]) by Nuel D. Belnap - Tonk, Plonk and Plink p.132
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A reaction:
The meaning would be given by the truth tables (the truth-conditions), whereas the role would be given by the natural deduction introduction and elimination rules. This seems to be the basic debate about logical connectives.
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17898
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Prior's 'tonk' is inconsistent, since it allows the non-conservative inference A |- B [Belnap on Prior,AN]
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Full Idea:
Prior's definition of 'tonk' is inconsistent. It gives us an extension of our original characterisation of deducibility which is not conservative, since in the extension (but not the original) we have, for arbitrary A and B, A |- B.
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From:
comment on Arthur N. Prior (The Runabout Inference Ticket [1960]) by Nuel D. Belnap - Tonk, Plonk and Plink p.135
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A reaction:
Belnap's idea is that connectives don't just rest on their rules, but also on the going concern of normal deduction.
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7903
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The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna]
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Full Idea:
The six perfections are of giving, morality, patience, vigour, meditation, and wisdom.
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From:
Nagarjuna (Mahaprajnaparamitashastra [c.120], 88)
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A reaction:
What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate').
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