Combining Texts

All the ideas for 'De Legibus Naturae', 'Philosophies of Mathematics' and 'Prior Analytics'

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62 ideas

2. Reason / A. Nature of Reason / 7. Status of Reason
6222 If a decision is in accord with right reason, everyone can agree with it [Cumberland]
2. Reason / D. Definition / 7. Contextual Definition
9955 Contextual definitions replace a complete sentence containing the expression [George/Velleman]
2. Reason / D. Definition / 8. Impredicative Definition
10031 Impredicative definitions quantify over the thing being defined [George/Velleman]
4. Formal Logic / A. Syllogistic Logic / 1. Aristotelian Logic
22271 Aristotle was the first to use schematic letters in logic [Aristotle, by Potter]
11060 Aristotelian syllogisms are three-part, subject-predicate, existentially committed, with laws of thought [Aristotle, by Hanna]
18909 Aristotelian sentences are made up by one of four 'formative' connectors [Aristotle, by Engelbretsen]
8080 Aristotelian identified 256 possible syllogisms, saying that 19 are valid [Aristotle, by Devlin]
13912 Aristotle replaced Plato's noun-verb form with unions of pairs of terms by one of four 'copulae' [Aristotle, by Engelbretsen/Sayward]
8071 Aristotle listed nineteen valid syllogisms (though a few of them were wrong) [Aristotle, by Devlin]
4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
13819 Aristotle's said some Fs are G or some Fs are not G, forgetting that there might be no Fs [Bostock on Aristotle]
4. Formal Logic / D. Modal Logic ML / 4. Alethic Modal Logic
9403 There are three different deductions for actual terms, necessary terms and possible terms [Aristotle]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
10098 The 'power set' of A is all the subsets of A [George/Velleman]
10099 The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman]
10101 Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [George/Velleman]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes
10103 Grouping by property is common in mathematics, usually using equivalence [George/Velleman]
10104 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
10096 Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
10097 Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
10100 Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
17900 The Axiom of Reducibility made impredicative definitions possible [George/Velleman]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
10109 ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman]
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
10108 As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman]
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
11148 Deduction is when we suppose one thing, and another necessarily follows [Aristotle]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
10111 Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman]
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
18896 Aristotle places terms at opposite ends, joined by a quantified copula [Aristotle, by Sommers]
5. Theory of Logic / E. Structures of Logic / 7. Predicates in Logic
3300 Aristotle's logic is based on the subject/predicate distinction, which leads him to substances and properties [Aristotle, by Benardete,JA]
5. Theory of Logic / G. Quantification / 1. Quantification
11149 Affirming/denying sentences are universal, particular, or indeterminate [Aristotle]
5. Theory of Logic / G. Quantification / 3. Objectual Quantification
8079 Aristotelian logic has two quantifiers of the subject ('all' and 'some') [Aristotle, by Devlin]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
10129 A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
10105 Differences between isomorphic structures seem unimportant [George/Velleman]
5. Theory of Logic / K. Features of Logics / 2. Consistency
10119 Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman]
10126 A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman]
5. Theory of Logic / K. Features of Logics / 3. Soundness
10120 Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman]
5. Theory of Logic / K. Features of Logics / 4. Completeness
10127 A 'complete' theory contains either any sentence or its negation [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
10106 Rational numbers give answers to division problems with integers [George/Velleman]
10102 The integers are answers to subtraction problems involving natural numbers [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10107 Real numbers provide answers to square root problems [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
9946 Logicists say mathematics is applicable because it is totally general [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
10125 The classical mathematician believes the real numbers form an actual set [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
17899 Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
10128 The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
17902 A successor is the union of a set with its singleton [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
10133 Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
10130 Set theory can prove the Peano Postulates [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
10089 Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
10131 If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
10092 In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman]
10094 The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman]
10095 Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman]
17901 Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 8. Finitism
10114 Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman]
10134 Much infinite mathematics can still be justified finitely [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
10123 The intuitionists are the idealists of mathematics [George/Velleman]
10124 Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman]
10. Modality / A. Necessity / 4. De re / De dicto modality
14641 A deduction is necessary if the major (but not the minor) premise is also necessary [Aristotle]
15. Nature of Minds / C. Capacities of Minds / 5. Generalisation by mind
18911 Linguistic terms form a hierarchy, with higher terms predicable of increasing numbers of things [Aristotle, by Engelbretsen]
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
10110 Corresponding to every concept there is a class (some of them sets) [George/Velleman]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / d. Biological ethics
6217 Natural law is supplied to the human mind by reality and human nature [Cumberland]
22. Metaethics / B. Value / 1. Nature of Value / f. Ultimate value
6221 If there are different ultimate goods, there will be conflicting good actions, which is impossible [Cumberland]
23. Ethics / E. Utilitarianism / 1. Utilitarianism
6218 The happiness of individuals is linked to the happiness of everyone (which is individuals taken together) [Cumberland]
6220 The happiness of all contains the happiness of each, and promotes it [Cumberland]
25. Social Practice / D. Justice / 2. The Law / c. Natural law
6216 Natural law is immutable truth giving moral truths and duties independent of society [Cumberland]