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All the ideas for 'Structures and Structuralism in Phil of Maths', 'Persistence, Change and Explanation' and 'A Completeness Theorem in Modal Logic'

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30 ideas

3. Truth / F. Semantic Truth / 2. Semantic Truth
10170 While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price]
4. Formal Logic / D. Modal Logic ML / 1. Modal Logic
10163 Propositional modal logic has been proved to be complete [Kripke, by Feferman/Feferman]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / a. Systems of modal logic
10760 With possible worlds, S4 and S5 are sound and complete, but S1-S3 are not even sound [Kripke, by Rossberg]
4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
16189 The variable domain approach to quantified modal logic invalidates the Barcan Formula [Kripke, by Simchen]
15132 The Barcan formulas fail in models with varying domains [Kripke, by Williamson]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
10166 ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price]
5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
10175 Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price]
5. Theory of Logic / L. Paradox / 2. Aporiai
13931 By using aporiai as his start, Aristotle can defer to the wise, as well as to the many [Haslanger]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10165 'Analysis' is the theory of the real numbers [Reck/Price]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
10174 Mereological arithmetic needs infinite objects, and function definitions [Reck/Price]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
10164 Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
10172 Set-theory gives a unified and an explicit basis for mathematics [Reck/Price]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
10167 Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
10169 Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price]
10179 There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price]
10181 Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price]
10182 There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
10168 Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price]
10178 Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
10176 Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price]
10177 Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
10171 The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price]
7. Existence / D. Theories of Reality / 1. Ontologies
13925 Ontology disputes rest on more basic explanation disputes [Haslanger]
8. Modes of Existence / E. Nominalism / 6. Mereological Nominalism
10173 A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price]
9. Objects / E. Objects over Time / 3. Three-Dimensionalism
13924 The persistence of objects seems to be needed if the past is to explain the present [Haslanger]
13930 Persistence makes change and its products intelligible [Haslanger]
9. Objects / E. Objects over Time / 5. Temporal Parts
13927 We must explain change amongst 'momentary entities', or else the world is inexplicable [Haslanger]
13928 If the things which exist prior to now are totally distinct, they need not have existed [Haslanger]
14. Science / D. Explanation / 2. Types of Explanation / g. Causal explanations
13929 Natural explanations give the causal interconnections [Haslanger]
14. Science / D. Explanation / 2. Types of Explanation / j. Explanations by reduction
13926 Best explanations, especially natural ones, need grounding, notably by persistent objects [Haslanger]