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All the ideas for 'Mahaprajnaparamitashastra', 'A Tour through Mathematical Logic' and 'What Does It Take to Refer?'

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41 ideas

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 ∀
13521 Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS]
13522 Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [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 / 6. Free Logic
10455 Free logic at least allows empty names, but struggles to express non-existence [Bach]
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]
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 / C. Ontology of Logic / 1. Ontology of Logic
10454 In first-order we can't just assert existence, and it is very hard to deny something's existence [Bach]
5. Theory of Logic / E. Structures of Logic / 3. Constants in Logic
10453 In logic constants play the role of proper names [Bach]
5. Theory of Logic / F. Referring in Logic / 1. Naming / b. Names as descriptive
10452 Proper names can be non-referential - even predicate as well as attributive uses [Bach]
5. Theory of Logic / F. Referring in Logic / 1. Naming / c. Names as referential
10456 Millian names struggle with existence, empty names, identities and attitude ascription [Bach]
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / a. Descriptions
10440 An object can be described without being referred to [Bach]
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / b. Definite descriptions
10444 Definite descriptions can be used to refer, but are not semantically referential [Bach]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
13519 Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
13531 Model theory reveals the structures of mathematics [Wolf,RS]
13532 Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS]
13533 First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [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]
19. Language / B. Reference / 1. Reference theories
10446 Fictional reference is different inside and outside the fiction [Bach]
10447 We can refer to fictional entities if they are abstract objects [Bach]
10443 You 'allude to', not 'refer to', an individual if you keep their identity vague [Bach]
19. Language / B. Reference / 4. Descriptive Reference / b. Reference by description
10439 What refers: indefinite or definite or demonstrative descriptions, names, indexicals, demonstratives? [Bach]
10441 If we can refer to things which change, we can't be obliged to single out their properties [Bach]
10442 We can think of an individual without have a uniquely characterizing description [Bach]
10445 It can't be real reference if it could refer to some other thing that satisfies the description [Bach]
10457 Since most expressions can be used non-referentially, none of them are inherently referential [Bach]
10463 Just alluding to or describing an object is not the same as referring to it [Bach]
19. Language / B. Reference / 5. Speaker's Reference
10459 Context does not create reference; it is just something speakers can exploit [Bach]
10460 'That duck' may not refer to the most obvious one in the group [Bach]
10461 What a pronoun like 'he' refers back to is usually a matter of speaker's intentions [Bach]
10462 Information comes from knowing who is speaking, not just from interpretation of the utterance [Bach]
19. Language / F. Communication / 5. Pragmatics / a. Contextual meaning
10458 People slide from contextual variability all the way to contextual determination [Bach]
23. Ethics / C. Virtue Theory / 3. Virtues / a. Virtues
7903 The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna]