Combining Texts

All the ideas for 'Introduction to Mathematical Logic', 'How Things Might Have Been' and 'Identity and Reference'

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38 ideas

4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
17749 Post proved the consistency of propositional logic in 1921 [Walicki]
17765 Propositional language can only relate statements as the same or as different [Walicki]
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
17764 Boolean connectives are interpreted as functions on the set {1,0} [Walicki]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
17752 The empty set is useful for defining sets by properties, when the members are not yet known [Walicki]
17753 The empty set avoids having to take special precautions in case members vanish [Walicki]
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
17759 Ordinals play the central role in set theory, providing the model of well-ordering [Walicki]
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
17741 To determine the patterns in logic, one must identify its 'building blocks' [Walicki]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
17747 A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
17748 The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17763 Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki]
17761 A compact axiomatisation makes it possible to understand a field as a whole [Walicki]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
17757 Members of ordinals are ordinals, and also subsets of ordinals [Walicki]
17758 Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki]
17755 Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki]
17756 The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki]
17760 Two infinite ordinals can represent a single infinite cardinal [Walicki]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
17762 In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
17754 Inductive proof depends on the choice of the ordering [Walicki]
9. Objects / A. Existence of Objects / 5. Individuation / a. Individuation
11897 A principle of individuation may pinpoint identity and distinctness, now and over time [Mackie,P]
11898 Individuation may include counterfactual possibilities, as well as identity and persistence [Mackie,P]
9. Objects / A. Existence of Objects / 5. Individuation / d. Individuation by haecceity
11883 A haecceity is the essential, simple, unanalysable property of being-this-thing [Mackie,P]
9. Objects / D. Essence of Objects / 1. Essences of Objects
11889 Essentialism must avoid both reduplication of essences, and multiple occupancy by essences [Mackie,P]
9. Objects / D. Essence of Objects / 3. Individual Essences
11877 An individual essence is the properties the object could not exist without [Mackie,P]
11882 No other object can possibly have the same individual essence as some object [Mackie,P]
11886 There are problems both with individual essences and without them [Mackie,P]
9. Objects / D. Essence of Objects / 5. Essence as Kind
11909 Unlike Hesperus=Phosophorus, water=H2O needs further premisses before it is necessary [Mackie,P]
11899 Why are any sortals essential, and why are only some of them essential? [Mackie,P]
9. Objects / D. Essence of Objects / 8. Essence as Explanatory
11906 The Kripke and Putnam view of kinds makes them explanatorily basic, but has modal implications [Mackie,P]
9. Objects / E. Objects over Time / 12. Origin as Essential
11894 Origin is not a necessity, it is just 'tenacious'; we keep it fixed in counterfactual discussions [Mackie,P]
10. Modality / A. Necessity / 2. Nature of Necessity
17742 Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki]
10. Modality / E. Possible worlds / 3. Transworld Objects / a. Transworld identity
11887 Transworld identity without individual essences leads to 'bare identities' [Mackie,P]
10. Modality / E. Possible worlds / 3. Transworld Objects / c. Counterparts
11890 De re modality without bare identities or individual essence needs counterparts [Mackie,P]
11892 Things may only be counterparts under some particular relation [Mackie,P]
11893 Possibilities for Caesar must be based on some phase of the real Caesar [Mackie,P]
10. Modality / E. Possible worlds / 3. Transworld Objects / d. Haecceitism
11884 The theory of 'haecceitism' does not need commitment to individual haecceities [Mackie,P]
14. Science / D. Explanation / 2. Types of Explanation / k. Explanations by essence
11905 Locke's kind essences are explanatory, without being necessary to the kind [Mackie,P]
18. Thought / B. Mechanics of Thought / 5. Mental Files
16362 An identity statement aims at getting the hearer to merge two mental files [Lockwood]
26. Natural Theory / B. Natural Kinds / 6. Necessity of Kinds
11907 Maybe the identity of kinds is necessary, but instances being of that kind is not [Mackie,P]