Combining Philosophers

Ideas for David Bostock, E.J. Lowe and Robert S. Wolf

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27 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]
13520 A 'tautology' must include connectives [Wolf,RS]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
13524 Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS]
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 / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀
13522 Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS]
13521 Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃
13523 Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS]
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]
8309 A set is a 'number of things', not a 'collection', because nothing actually collects the members [Lowe]
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]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
8322 I don't believe in the empty set, because (lacking members) it lacks identity-conditions [Lowe]
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 / e. Axiom of the Empty Set IV
13529 Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS]
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]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension
13526 Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS]
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]