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All the ideas for 'Intro to Naming,Necessity and Natural Kinds', 'works' and 'A Tour through Mathematical Logic'

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31 ideas

2. Reason / D. Definition / 1. Definitions
5831 The new view is that "water" is a name, and has no definition [Schwartz,SP]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
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 / 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 / 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 / n. Axiom of Comprehension
13526 Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS]
4. Formal Logic / G. Formal Mereology / 1. Mereology
10397 Abelard's mereology involves privileged and natural divisions, and principal parts [Abelard, by King,P]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
13534 In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS]
13535 First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS]
5. Theory of Logic / F. Referring in Logic / 1. Naming / b. Names as descriptive
5829 We refer to Thales successfully by name, even if all descriptions of him are false [Schwartz,SP]
5830 The traditional theory of names says some of the descriptions must be correct [Schwartz,SP]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
13532 Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS]
13519 Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
13533 First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS]
13531 Model theory reveals the structures of mathematics [Wolf,RS]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
13537 An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
13539 The LST Theorem is a serious limitation of first-order logic [Wolf,RS]
5. Theory of Logic / K. Features of Logics / 4. Completeness
13538 If a theory is complete, only a more powerful language can strengthen it [Wolf,RS]
5. Theory of Logic / K. Features of Logics / 10. Monotonicity
13525 Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13530 An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
13518 Modern mathematics has unified all of its objects within set theory [Wolf,RS]
8. Modes of Existence / E. Nominalism / 1. Nominalism / b. Nominalism about universals
10396 If 'animal' is wholly present in Socrates and an ass, then 'animal' is rational and irrational [Abelard, by King,P]
10395 Abelard was an irrealist about virtually everything apart from concrete individuals [Abelard, by King,P]
8. Modes of Existence / E. Nominalism / 3. Predicate Nominalism
15384 Only words can be 'predicated of many'; the universality is just in its mode of signifying [Abelard, by Panaccio]
10. Modality / A. Necessity / 4. De re / De dicto modality
8481 The de dicto-de re modality distinction dates back to Abelard [Abelard, by Orenstein]
18. Thought / C. Content / 8. Intension
5826 The intension of "lemon" is the conjunction of properties associated with it [Schwartz,SP]
18. Thought / E. Abstraction / 8. Abstractionism Critique
15385 Abelard's problem is the purely singular aspects of things won't account for abstraction [Panaccio on Abelard]
19. Language / C. Assigning Meanings / 3. Predicates
15383 Nothing external can truly be predicated of an object [Abelard, by Panaccio]
26. Natural Theory / B. Natural Kinds / 7. Critique of Kinds
10398 Natural kinds are not special; they are just well-defined resemblance collections [Abelard, by King,P]