15927
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Definition just needs negation, known variables, conjunction, disjunction, substitution and quantification [Weyl, by Lavine]
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Full Idea:
For mathematics, Weyl arrived (by 1917) at a satisfactory list of definition principles: negation, identification of variables, conjunction, disjunction, substitution of constants, and existential quantification over the domain.
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From:
report of Hermann Weyl (works [1917]) by Shaughan Lavine - Understanding the Infinite V.3
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A reaction:
Lavine summarises this as 'first-order logic with parameters'.
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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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11214
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We learn 'not' along with affirmation, by learning to either affirm or deny a sentence [Rumfitt]
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Full Idea:
The standard view is that affirming not-A is more complex than affirming the atomic sentence A itself, with the latter determining its sense. But we could learn 'not' directly, by learning at once how to either affirm A or reject A.
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From:
Ian Rumfitt ("Yes" and "No" [2000], IV)
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A reaction:
[compressed] This seems fairly anti-Fregean in spirit, because it looks at the psychology of how we learn 'not' as a way of clarifying what we mean by it, rather than just looking at its logical behaviour (and thus giving it a secondary role).
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