Combining Philosophers
Ideas for Rescher,N/Oppenheim,P, Mark Fisher and William D. Hart
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18 ideas
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
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13500
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Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent [Hart,WD]
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4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃
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13502
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∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...' [Hart,WD]
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4. Formal Logic / F. Set Theory ST / 1. Set Theory
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13456
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Set theory articulates the concept of order (through relations) [Hart,WD]
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13497
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Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe [Hart,WD]
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4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / a. Symbols of ST
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13443
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∈ relates across layers, while ⊆ relates within layers [Hart,WD]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
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13442
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Without the empty set we could not form a∩b without checking that a and b meet [Hart,WD]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
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13493
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In the modern view, foundation is the heart of the way to do set theory [Hart,WD]
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13495
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Foundation Axiom: an nonempty set has a member disjoint from it [Hart,WD]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
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13461
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We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD]
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13462
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With the Axiom of Choice every set can be well-ordered [Hart,WD]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L
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13516
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If we accept that V=L, it seems to settle all the open questions of set theory [Hart,WD]
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
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13441
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Naïve set theory has trouble with comprehension, the claim that every predicate has an extension [Hart,WD]
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
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13494
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The iterative conception may not be necessary, and may have fixed points or infinitely descending chains [Hart,WD]
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4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
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13457
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A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD]
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13458
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A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD]
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13460
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'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD]
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13490
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Von Neumann defines α<β as α∈β [Hart,WD]
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4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
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13481
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Maybe sets should be rethought in terms of the even more basic categories [Hart,WD]
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