Combining Philosophers

Ideas for H.Putnam/P.Oppenheim, David Bostock and Shaughan Lavine

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33 ideas

4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
13439 Venn Diagrams map three predicates into eight compartments, then look for the conclusion [Bostock]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
13421 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope [Bostock]
13422 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope [Bostock]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
13355 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [Bostock]
13350 'Assumptions' says that a formula entails itself (φ|=φ) [Bostock]
13351 'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock]
13356 The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [Bostock]
13352 'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock]
13353 'Negation' says that Γ,¬φ|= iff Γ|=φ [Bostock]
13354 'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ [Bostock]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
13610 A logic with ¬ and → needs three axiom-schemas and one rule as foundation [Bostock]
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
18122 Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock]
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
13846 A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
18114 There is no single agreed structure for set theory [Bostock]
15945 Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
15914 An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
18107 A 'proper class' cannot be a member of anything [Bostock]
15921 Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
15937 Those who reject infinite collections also want to reject the Axiom of Choice [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
18115 We could add axioms to make sets either as small or as large as possible [Bostock]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
15936 The Power Set is just the collection of functions from one collection to another [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
15899 Replacement was immediately accepted, despite having very few implications [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
15930 Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
18139 The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock]
15920 Pure collections of things obey Choice, but collections defined by a rule may not [Lavine]
15898 The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
15919 The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
15900 The iterative conception of set wasn't suggested until 1947 [Lavine]
15931 The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine]
15932 The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
18105 Replacement enforces a 'limitation of size' test for the existence of sets [Bostock]
15933 Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine]
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
15913 A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine]