Combining Texts

All the ideas for 'Individuals without Sortals', 'A Tour through Mathematical Logic' and 'Wittgenstein's 'Philosophical Investigations''

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35 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 / 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 / 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 / A. Nature of Mathematics / 4. Using Numbers / d. Counting via concepts
17518 Counting 'coin in this box' may have coin as the unit, with 'in this box' merely as the scope [Ayers]
17516 If counting needs a sortal, what of things which fall under two sortals? [Ayers]
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]
7. Existence / B. Change in Existence / 4. Events / a. Nature of events
17520 Events do not have natural boundaries, and we have to set them [Ayers]
9. Objects / A. Existence of Objects / 5. Individuation / a. Individuation
17519 To express borderline cases of objects, you need the concept of an 'object' [Ayers]
9. Objects / A. Existence of Objects / 5. Individuation / e. Individuation by kind
17510 Speakers need the very general category of a thing, if they are to think about it [Ayers]
17522 We use sortals to classify physical objects by the nature and origin of their unity [Ayers]
17515 Seeing caterpillar and moth as the same needs continuity, not identity of sortal concepts [Ayers]
17511 Recognising continuity is separate from sortals, and must precede their use [Ayers]
9. Objects / B. Unity of Objects / 1. Unifying an Object / a. Intrinsic unification
17517 Could the same matter have more than one form or principle of unity? [Ayers]
9. Objects / B. Unity of Objects / 3. Unity Problems / c. Statue and clay
17513 If there are two objects, then 'that marble, man-shaped object' is ambiguous [Ayers]
9. Objects / D. Essence of Objects / 5. Essence as Kind
17523 Sortals basically apply to individuals [Ayers]
9. Objects / E. Objects over Time / 5. Temporal Parts
17521 You can't have the concept of a 'stage' if you lack the concept of an object [Ayers]
17514 Temporal 'parts' cannot be separated or rearranged [Ayers]
9. Objects / F. Identity among Objects / 1. Concept of Identity
17509 Some say a 'covering concept' completes identity; others place the concept in the reference [Ayers]
9. Objects / F. Identity among Objects / 3. Relative Identity
17512 If diachronic identities need covering concepts, why not synchronic identities too? [Ayers]
15. Nature of Minds / A. Nature of Mind / 4. Other Minds / d. Other minds by analogy
14644 If my conception of pain derives from me, it is a contradiction to speak of another's pain [Malcolm]